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THE
SCHOLAR'S  ASSISTANT;
                       OR,


      A PLAIN, COMPREHENSIVE, AND PRACTICAL



SYSTEM   OF  ARITHMETIC:


                 TO WHICH IS PREFIXED


           AN  INTRODUCTION,


                     CONTAINING


  A PRACTICAL ILLUSTRATION OF THE PRIMARY RULES, AND OF THE
         TABLES OF MONEY, WEIGHTS, AND MEASURES.
<yo
                  DESIGNED


FOR THE USE OF SCHOOLS IN THE UNITED STATES.
BY THE
           REV. J.  G.  COOPER, A. M.
                      •«•
  EDITOR OF THE WORKS OF VIRGIL J AUTHOR OF A PLAIN, PRACTICAL
      LATIN GRAMMAR; ALSO, OF AN ABRIDGMENT OF MURRAY'S
           ENGLISH GRAMMAR, WITH IMPROVEMENTS.






                PUtlairelpfjta:

TOWAR, J. & D. M. HOGAN—PITTSBURG, HOGAN & CO.

               C. SHERMAN & CO., PRINTERS.


                       1830.
t 
(LIT..
  123^
Eastern District of Pennsylvania, to wit.
  Be it remembered, That on the twenty-first day of June, in the fifty-
fourth  year of the Independence of the United States  of  America,
A. D. 1830, Joab G. Cooper, A. M. of the said district, hath deposited
in this  office the title  of a book, the right whereof he claims as author,
in the words following, to wit:
'•The  Scholar's  Assistant; or, a  Plain,  Comprehensive, and  Practical
   System of Arithmetic: to which is prefixed, an Introduction, contain-
   ing a Practical Illustration of the Primary Rules, and of the Tables of
   Money, Weights, and Measures.  Designed for the Use  of Schools in
   the United States.   By the Rev. J.  G. Cooper, A. M. Editor  of the
   Works of Virgil, Author of a Plain, Practical Latin Grammar; also, of an
   Abridgment of Murray's English Grammar, with Improvements."
In conformity to the  Act of the Congress of the United States, entitled
" An act for the encouragement  of learning, by securing the copies of
maps,  charts, and books to the  authors  and proprietors of such copies
during the times therein  mentioned."   And also  to the act entitled
"An act supplementary to an act entitled 'An  act for the encourage-
ment of learning, by securing the copies of maps, charts, and books to
the authors and proprietors of such copies during the times  therein
mentioned,' and extending the benefits thereof to the arts of designing,
engraving, and etching historical  and other prints."
                                           D. CALDWELL,
                       Clerk of the Eastern District of Pennsylvania. 
CONTENTS.
  OF THE INTRODUCTION.
Addition—Table,  -
  Questions,    ...
  Examples,      ...
Subtraction—Table,  -
  Questions,      ...
  Examples,
Multiplication—Table,  -
 * Questions,    ...
  Exercises,      -
Division—Table,
  Questions,      ...
  Exercises,
Federal Currency—Table,
  Questions,
  Exercises,
 Pounds, Shillings and Pence—
     Table,
  Questions,      -    -
t-. Exercises,    -
 Troy Weight—Table,  -
   ftuestions    -
   Exercises,
 Avoirdupois Weight—Table,
   Questions,    -
   Exercises,
 Apothecaries' Weight—Table,
   Questions,    -
   Exercises,       -
 Long Measure—Table,
   Questions,      -
   Exercises,    .    -    -
 Cloth Measure—Table,
   Questions,
   Exercises,       -
 Land Measure—Table,
   Questions,
   Exercises,    -     -    -
 Liquid Measure—Table,
   Questions,
   Exercises,
  Dry Measure—Table,
   Questions,      -
   Exercises,    -
  Time—Table,
   Questions,
   Exercises,
Motion—Table,
  Questions,
  Exercises,
Page
   72
-   72
   73
OF THE ARITHMETIC.
   Numeration,       -     -     -76
     Questions,                  77
     Exercises,       -     -     -  77
   Addition,       ...    79
     Exercises,       -     -     -  80
   Subtraction,                   81
     Evereises,       -     -     -  81
   Multiplication,       -     -    82
     Exercises,      -     -     -  85
   Division,       ...    87
     Exercises,      -    -    -  90
47 Compound Addition,      -    91
47 Federal Currency,      -     -  92
48   Exercises,                  93
49 Subtraction,  &c.   -    -     -  93
50   Exercises,                  94
50 Multiplication, &c.     -     -  95
52    Exercises,                  96
52  Division, &c.     ...  97
53    Exercises,                 98
55  Pounds, Shillings, Pence,  &c.  99
55  Troy Weight,    -    -    -  104
56    Exercises,    -    -    -   105
57  Avoirdupois Weight,   -    -  106
57    Exercises,  -   -    -    -  108
58  Apothecaries' Weight,    -   108
60    Exercises,  -    -    -    -  110
61  Long Measure, -    -    -   110
61    Exercises,  -   -    -    -  112
62  Cloth Measure, -    -    -   113
62    Exercises,  -   -    -    -  114
63  Land Measure, -    -    -   115
64    Exercises, -   -    -    -  116
64  Liquid Measure, -    -    -   117
65    Exercises,  ....  118
66  Dry Measure,  ...   119
 67    Exercises, -    -    -    -  120
 67  Circle Measure,      -    -   121
 68   Exercises, -   -    -    -  122
 69 Time,                   -   122
 70   Exercises, -    -    -    - 123 
IV
CONTENTS.
Vulgar Fractions,
  Exercises, .    -    -
  Reduction,
  Addition,   -
  Subtraction,  -     -jj  -
  Multiplication,  -    ••
  Division,
  Questions, -    -    -
Decimal Fractions,
  Addition,   -
  Subtraction,  -
  Multiplication,  -
  Division, -    -     -    -
  Exercises, -
  Reduction,
  Questions,  -
Reduction of Integers,
  Questions,  -    -    -
Table of Foreign Currency,
  Do.  Domestic,  Do.
Page
  124
-  125
  126
-  135
  137
-  139
  139
-  141
  142
-  143
  145
-  146
  147
-  148
  149
- 153
  154
-  160
  161
  162
  Do. of Foreign and Domes-
    tic Coin,      ... 162
Reduction of Foreign Curren-
  cies,    -    -     -     -   163
Do. of Domestic      do.   - 169
Single Proportion,  or  Rule of
    Three,     -     -        183
  Questions,      ... 183
Compound Proportion, or Dou-
    ble Rule of Three     -   194
  Questions,      ... 198
Practice,
Tare and Trett,
Simple Interest,
Bank Interest,
Compound Interest,
  Questions,
Commission
Insurance,
Rebate, or Discount,
Barter,
Equation of Payments;
Loss and Gain,
Fellowship,
Alligation,
Single Position,
Double Position,
Permutation,
Combination,
Duodecimals,
Involution,
Evolution,
Page
  199
-  203
  206
-  211
  215
-  223
  223
-  225
  227
-  229
  231
-  233
  236
-  240
  244
-  246
  250
-  250
  251
-  254
  255
Extraction of the Square Root, 256
  Application,    ... 259
Extraction of the Cube Root,  261
  Application,  ...   264
Arithmetical Progression,    - 265
Geometrical Progression,     270
Annuities at Comp. Interest,   274
Annuities in Reversion,    -   284
Perpetuities,      ... 286
Perpetuities in Reversion,     287 
TO TEACHERS, AND INSTRUCTERS OF YOUTH.
   Gentlemen,—The favourable reception which many of
you have given to my edition of the works of Virgil,
and to my improved Latin and English Grammars,
leads me to hope,  that the Scholar's Assistant will
also meet your approbation.
   My experience in the business of instruction led me to
a knowledge of the wants of the pupil, in the department
of arithmetic, and made me  sensible of the defects of the
books in common use.  And,  as far as my acquaintance
extends, I believe many of you are persuaded, that import-
ant improvements might be made upon them.  Whether
I have fully supplied those defects,  and made the desired
improvements, must remain with you, and the public, to
judge.
   Arithmetic is of every  day  use in the business of life.
Hence arises its importance to the farmer, the mechanic, the
merchant, the gentleman, the scholar; and, I may add, to
every member of the community. Its operations are founded
upon mathematical principles;  and those principles should
be constantly  kept in view, fully explained,  elucidated,
and rendered intelligible to the pupil, in every stage of his
progress.  To effect this has been my chief aim.  Through
the whole work, I kept the pupil in my view, and gave
such explanations  of the subject before him,  and such
instructions, as I supposed he might require.
  The Scholar's Assistant may justly claim  the
following improvements:  1. It is a full and comprehensive
system of practical arithmetic.  2. The arrangement of its
several parts is calculated  to facilitate the progress of the
learner. 3. The principles of arithmetic are fully explained 
(vi)
  and elucidated.  4. The examples are  not mere obsolete
  and useless questions; but are interesting, instructive, and
  of a practical nature; a large portion of which are entirely
  new and original.   5. It explains  itself;  and, therefore,
  will save much labour on the part of the teacher, and will
  render the study pleasant to the pupil.
    For the correctness of these positions, 1 refer you to the
  work itself.  You will find Vulgar and Decimal Fractions,
  the  Reduction of Foreign and  Domestic Currencies, the
  Rules of Proportion, both Simple  and Compound,  the
  Extraction of the Square and  Cube Roots, Arithmetical
  and  Geometrical Progression,  Simple and  Compound
  Interest, Annuities,  &,c, more  comprehensive, and their
  principles better explained, and  rendered more intelligible,
  than they are in other works of the  kind.  In  addition to
  these, the  introduction contains more than eight hundred
  familiar, easy, and practical  exercises, designed to eluci-
  date the primary rules, to explain the  several  tables, and
  to show their use and application.
    Other works, that  I have seen, are deficient  in arrange-
  ment, and in suitable introductory  matter, calculated to
  engage the attention  of (he child, to  interest his mind, and
  to develope his reasoning faculties.   He is hurried through
  the primary rules, before his mind is  sufficiently disciplined
  in the property of numbers.
    The plan of the present work entirely remedies these
  defects.   It is somewhat novel,  and might seem to require
  an apology;  but, by adopting it, I  had an opportunity of
  explaining more fully the elementary principles, of dwelling
  upon them longer, and of giving that particular and minute
  instruction, which the child stands  in need of, in  his first
  essays in  numbers.   The questions upon the tables are
  designed  chiefly as a mental exercise; and, in this view,
  they will be  found very useful.   The instructer may add
  others, as he shall deem expedient.
    The division of the work necessarily  led to some repeti-
  tion.  But this can be no objection.   It is the very method
.  of useful instruction.   Children  must have " line upon line,
  precept  upon  precept, here  a little, and there  a little,'1 in
■j  order to induce thought and reflection, and to develope the
.  powers of his mind, 
( vii )
  In other books,  a large proportion, and, in some, even
the greater part, of the examples, are calculated in pounds,
shillings, pence, and farthings; a currency  obsolete and
useless, and even worse  than useless.   This is a material
defect.  As well might the pupil be  taught to perform the
ordinary calculations of life, in the  copecs and  rubles of
Russia, the candareens and tales of China, or in any other
foreign currency.  But such a course of instruction certainly
could find no advocates.   Here all the questions and exam-
ples are calculated in the easy and simple currency of the
United States.  Pounds, shillings,  and  pence  are intro-
duced only to show their nature and properties, and deter-
mine their values in the currency of our country.
   The  practical  utility of the Scholar's Assistant
I  have fully  tested.  The  manuscript was used by my
pupils for some months.  Their progress, in every respect,
fulfilled my expectations ; and I can say, with confidence,
that the saving of time, on their part, was full fifty  per cent.;
and the saving of labour, on the part of instruction, was in
a greater proportion.
   That a work of this magnitude, considerably larger than
the common books, and indeed larger than I contemplated,
should be free from errors, I cannot  suppose.  The circum-
stance of its being prepared during the daily occupations
of a school;  a great part of the examples  being original,
 and requiring calculation;  and the  whole printed from
 manuscript, precludes  the  expectation.  For   the errors
 of the  author, as  well as for those of the  press, I solicit
 your kind indulgence.   If the present work shall  meet your
 approbation, I have it in contemplation to furnish you with
 an " Introduction  to Algebra," upon a similar plan.
   That the Scholar's Assistant may  be  found to
 answer the purposes for which it was designed, namely, to
 facilitate the progress of the  pupil, and lighten the  labour
 of the instructer,  is the humble hope of, Gentlemen,
               Your obedient friend and servant,
                                     J. G. COOPER.
Philadelphia, June 20, 1830. 
EXPLANATION  OF  CHARACTERS.
        Signification.
equal, as, 100 cents=l dollar.
Addition, as, 4-f 6=10.
Subtraction, as, 10—4=6.
Multiplication, as,4x6=24.
Divison, as, 10-5-2 = 5.
Proportion as, 2 : 4 :: 6:12.
the Square Root as,  ^/9=3.
the Cube Root, as, -^27=3.
a  vinculum.   It shows  that the  numbers,  over
   which it is  drawn, are to be  considered as a
   simple quantity. 
INTRODUCTION.
  All the operations of Arithmetic are performed by the help
of the following characters or figures : 1, 2, 3, 4, 5, 6, 7> 8, 9, 0.
The first figure, when it is alone, represents a unit, or one of any
thing; as one  apple—one man—one  dollar.  The  figure  2
represents two units, or ones,  of any  thing; as two apples-
two dollars—two hats.   The figure 3 represents three units, or
ones.  The figure 4 represents four units, or ones.  The figure 5
represents five units, or ones.   The figure 6 represents six units,
or ones.  The figure 7 represents seven units,  or ones.  The
figure 8 represents eight units, or ones; and the figure 9 represents
nine units, or ones.  These are the simple values of the figures.
The cypher or nought has in itself no value.  In connection with
the preceding figures, it causes them  to assume different values,
by removing them to higher places.   Thus, in the combination
10, the nought or cypher removes the  one or unit to the ten's
place,  thereby  causing it  to assume  ten times the  value which
it would have, if it stood in the  unit's place, itself having, at the
same time, no value.   When any of the figures stand together,
they have a local value, which varies according to their situa-
tion.   The  first  figure on the right hand always represents
so many units or ones.   It is, therefore, said to stand in unit's
place.   The  next, on  the left hand, is said  to stand in ten's
place,  because a figure standing there represents ten  times as
much  as it would do, if it stood in unit's place.   Thus, in the
combination 21,  the figure 2 stands for two tens or twenty;
but, if it stood in  the place of the 1, it would represent two units
only.
  The simple  value of any  figure  is  increased ten  times by
removing it one place from the units;  one hundred times, by
removing it two places, a thousand times, by removing it three
places, from the units;  and so on, to any number of places : thus,
in the  combination 4321, the figure 4 stands for four thousand ;
the figure 3 stands  for three  hundred;  the figure 2 stands for 
10                  INTRODUCTION.

two tens or twenty ; and the figure 1 stands for a simple unit.
The whole combination represents four thousand three hundred
and twenty-one ;  and so  for any other combination of figures.
This is  called Numeration ; which means the expressing of
numbers by figures, and the giving to each its proper value.
  Question 1. How are the operations of arithmetic performed  ?
  2. What do you mean by a unit ?
  3. How many units does the figure 2 represent ?—how many
does the figure 3 represent?—how many does the figure 4 repre-
sent ?—how many does  the figure  5  represent ?—how many
does the figure 6 represent ?—how many does the figure 7 repre-
sent ?—how many does  the figure  8  represent?—how many
does the figure 9 represent ?
  4. When are the figures said to have their simple value ?
  5. When are they said to have a local value ?
  6. Has the cypher or nought any value in itself?
  7. What then is its use ?
  8. When  several figures  are combined,  in what proportion
do they  increase in value from the unit's place?
  9. Does the figure in the unit's place always retain its simple
value ?
  10. How would you set down, in figures, four thousand three
hundred and twenty-one ?
  11. Will you explain the  reason  of this, and assign to each
figure its local \alue?
  12. What is the expressing of numbers by figures called ?
                     ADDITION.
  Addition is the operation of finding the sum of two or more
numbers of the same kind or denomination.
ADDITION  TABLE.
2 and	1	are	3	3	and	1	are	4	4	and	1	are	5
2 "	2	tt	4	3	u	2	a	5	4	tt	2	it	6
2 "	3	it	5	3	tt	3	it	6	4	tt	3	n	7
2 «	4	it	6	3	it	4	tt	7	4	a	4	u	8
2 "	5	tt	7	3	it	5	tt	8	4	tt	5	tt	Q
2 "	6	u	8	3	tt	6	tt	9	4	a	6	ti	10
2 "	7	a	9	3	it	7	it	10	4	tt	7	u	11
2 «	8	it	10	3	tt	8	it	11	4	u	8	u	12
2 "	9	ti	11	3	it	9	tt	12	4	tt	9	a	n
2 "	10	it	12	3	tt	10	tt	13	4	tt	10	u	14
 
ADDITION.
11
5 and  1  are
5  "   2  "
 3
 4
 5
 6
 7
 8
 9
10
8 and 1 are
a
 6
 7
 8
 9
10
11
12
13
14
15

 9
10
11
12
13
14
15
6 and  1  are  7
6  "  2  "   8
      3  "   9
         u
         a
         a
       8  «  16
8  "   9  "  17
8  " 10  «  18
4  "  10
 5 "
 6 «
 7 "
 8 "
 9 "
10 "
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
7 and 1 are  8
7  "  2  "   9
   "  3
   ti  4
   "  5
   "  6
   "  7
 9
10
ire	10	lOanc
it	11	10 «
a	12	10 "
tt	13	10 "
it	14	10 "
tt	15	10 "
tt	16	10 "
ti	17	10 "
u	18	10 "
a	19	10 "
"  10
«  11
"  12
"  13
a  14
"  15
"  16
it  ,y

 are  11
  "  12
  "  13
  a  14
  "  15
  "  16
17
18
19
10  " 20
 1.
 2.
 3.
 4.
 5.
 6.
 7.
 8.
 9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
What
What
What
What
What
What
What
What
What
What
What
What
What
What
What
What
What
What
What
What
What
 on the Table to be answered by the learner.
s the sum of 2 and 4? Ans. 6.  Why is it 6?
s the sum of 2 and 7 ? Ans. 9.  Why is it 9 ?
is the sum of 3 and 5 ? Ans. 8.  Why is it 8 ?
is the sum of 4 and 6 ? Ans. 10.  Why is it 10 ?
is the sum of 5 and 7? Ans. 12.  Why is it 12 ?
is the sum of 6 and 8 ? Ans. 14.  Why is it 14 ?
s the sum of 7 and 9 ? Ans. 16.  Why is it 16 ?
is the sum of 8 and 8? Ans. 16.  Why is it 16?
is the sum of 9 and 5 ? Ans. 14.  Why is it 14 ?
s the sum of 10 and 4? Ans. 14.  Why Is it 14?
is the sum of 9 and 9 ? Ans. 18.  Why is it 18 ?
s the sum of 8 and 5 ? Ans. 13.  Why is it 13 ?
is the sum of 7 and 10? Ans.  17.  Why is it 17?
is the sum of 2,3 and 4 ? Ans. 9.  Why is it 9 ?
s the sum of 3, 5 and 4? Ans. 12.  Why is it 12?
s the sum of 4, 6 and 7? Ans. 17-  Why is it 17?
is the sum of 5, 6 and 4? Ans. 15.  Why is it 15?
is the sum of 6, 8 and 5 ? Ans. 19-  Why is it 19 ?
is the sum of7, 4 and 7? Ans. 18.  Why is it 18?
s the sum of 9, 3 and 6? Ans. 18.  Why is it 18?
is the sum of 10,4 and 6? Ans. 20. Why is it 20? 
 12                  INTRODUCTION.

   22.  What is the sum of 9, 5 and 3 ? Ans. 17-  Why is it 17 ?
   23.  What is the sum of 5, 6 and 7? Ans. 18.  Why is it 18?
   24.  What is the sum of 3, 4, 5 and 6?  Ans. 18-   Why is
 it  18?
   25.  What is the  sum of 4, 5, 6 and 4 ?  Ans. 19.   W'hy is
 it  19?
   26.  What is the sum of 3, 5, 2 and 1 ? Ans.  11.   Why is
 it  11?
   27.  What is the sum of 8, 4, 3 and 2 ? Ans.  17.   Why is
 it  17?
   28.  What is the sum of 9, 2, 3 and 4?  Ans. 18.   Why is
 it  18?
   29.  What is the sum of 2, 6, 10 and 2 ?  Ans. 20.   Why is
 it 20?
   30. What is the sum of 4, 7, 5 and 3 ?  Ans.  19.  Why is
 it 19?

   After this manner, the teacher may require the learner to add
together any other numbers that may be in the table, demanding
 at  the same time a reason for the answer.  This will be found a
 very useful exercise.

                  PRACTICAL EXAMPLES.
   31. Jane has 4 apples, and her two sisters have 3 apples each j
how  many have all three ? Ans. 10.  Why have they 10 ?
   32. Susan bought a slate for  10 cents, quills and a pencil
for 4 cents, and paper for 6 cents; how much did she spend
in  all?  Ans. 20 cents.  Can you prove that?
   33. William spends 2 cents a day for cakes, and John spends
 5 for nuts, and 3 for apples; how much  do they both spend in
a day? Ans. 10 cents.  Can you prove that ?
   34. If  you buy a knife for 10 cents, and apples for 5 cents;
 how much do you spend?  Ans. 15 cents.   Can you prove that ?
   35.  A  fine little boy asked his  father  for 12 cents to buy a
toy, 2 cents to  buy nuts, 4 cents to buy apples, and 1 cent for
candy; how much did he spend in all? Ans. 19 cents.  Will
 you prove that ?
   36. Jane, setting out for school, asked her sister to give her
2 cents for apples, 3 for nuts, 4 for cakes, and 8 cents for paper;
 how  much did her sister give her in all?  Ans. 17 cents.  Will
you prove that?
   37-  If  you buy a book for 15 cents,  and  paper for 5 cents
 how  much do  the book and  paper come to ?  Ans. 20 cents!
 Will you  prove that ? 
ADDITION.
13
   38. John is 8 years old, and William is 11;  what is the sum
 of their ages ?  Ans. 19 years.   How will you prove that ?
   39. Susan is 9 years old,  and Jane 14  years; what is the
 sum of their ages ? Ans. 23 years.   How will you prove that ?
   40. In a school of young ladies, there are three classes; in one
 class there are 8, in another 6, and in a third 9; how many are
 there in the whole ? Ans. 23.  How can you prove that ?
   41.  A teacher has 5 boys  reading Virgil, 4  boys reading
 Caesar, and 8 reading Sallust;  how many has he in all ? Ans. 17.
 How will you  prove it ?
   42. In a school 9 boys write, 4 cypher, and 5 study grammar;
 how many are there in the school? Ans.  18.  How will you
 prove it ?
   43. If a hat costs 3  dollars, a pair of shoes 2 dollars, and a
 coat 9 dollars;  how much do  they  all cost?  Ans. 14 dollars.
 How do you prove it ?
   44. Mary has 4 tickets for good behaviour, 5 for  good reading,
 and 6 for grammar; how many has she in all? Ans. 15.   How
 will you prove it?
   45.  James has 7 credits for good deportment, 5 for reading,
 and 8 for writing;  how many has he in all ?  Ans. 20.   How
 will you prove it ?
   46.  William says 5  lessons in the morning, and  4 in  the
 afternoon, beside 2 tasks: how many lessons does he say alto-
 gether? Ans. 11.  Will you prove that ?
   47-  Three poor persons asked charity; and a good boy gave
 to the first one 5 cents, to the  second 8, and to the third one 6
 cents, which was all the money he had ; how much did he give
 in all ?  Ans. 19 cents.  How will you  prove it ?
   48.  Jane, on her way to school, meeting  some poor children,
 who asked charity, gave to the first 6 cents, to the second 7cents,
 and to the third 8 cents, which  was all she had; how much did
 she give in all ? Ans. 21  cents.  How will you prove  it?
   49-  A farmer sold a load of hay for 8 dollars, five bushels of
 corn for 3 dollars, and ten bushels of  oats  for 6 dollars; how
 much money did he receive in all? Ans. 17 dollars.  How will
you prove that?
   50.  A merchant sold cloth  for 12 dollars, tea for 3 dollars,
and sugar for  5  dollars; how much did the whole amount to ?
Ans. 20 dollars.   How will you prove  that ?
   When the numbers to be added consist of  two or more places,
you must set them one under the other, observing to set the units
of each number under one another; the tens under  the tens, the
hundreds under the hundreds, &c. as in the following examples:
           b2 
14                  INTRODUCTION.

  51. What is the sum of 425, 146 and 620?     Ans. 1191.,
  Here there are three places of figures ; the first, or right hand,
place is the unit's, the second  is the ten's, and the third is the
hundred's place.
        Put them thus :  425
                       146
                       620
                      1191  the sum of the numbers.
  After you have placed them  under each other, you must add
together the first column, or the figures in the unit's  place, and
observe the sum.   If it is 10, or more ; then set the units in the
sum  or  amount, under the column of units, and add the tens
into  the column of tens ; and proceed in the same way with the
other columns ;  but, in the last  column, you must set down the
whole sum.  In this example, the sum of the first column is 11;
which is one ten,  and one  unit, or 1; set the 1 under the line
of units, and add the 1, which is 1 ten, into  the line  or column
of tens.  This  line amounts to 9 only;  which you must place
under the column of tens; and  because it does  not amount to
ten,  there is none to be added to the next line.  The sum of the
third column, or line of hundreds, is 11;  the whole of which you
must set down, as  in the example.  In the same way, you  must
proceed in the following questions.
  52. What is the sum of 621, 542 and 134?      Ans.  1297
  53. What is the sum of 46, 74, and 92 ?          Ans. 212
  54. What is the sum of 942, 356 and 372 ?      Ans.  1670
  55. What is the sum of 876, 543 and 321 ?       Ans.  1740
  56. What is the sum of 1334, 2467, and 4321 ?   Ans.  8122
  57. What is the sum of 2468,1357 and 7892?  Ans. 11715
  58. What is the sum of 1976, 2543 and 9045 ?  Ans. 13564
  59. What is the sum of 937, 578 and 679?      Ans.  2244
  60. What is the sum of 104, 560 and 809 ?       Ans.  1473
  61. What is the sum of 859, 789 and 999 ?       Ans. 2647
  62. What is the sum of 333, 645 and 777?       Ans.  1755
  63. What is the sum of 444, 555 and  888?       Ans.  1887
   64. What is the sum of 540, 372 and 421 ?       Ans.  1233
   65. What is the sum of 842,573 and 373?       Ans.  1780
   66. James bought a grammar for 4:3 cents, a slate for 18 cents,
 and  an English Reader for 65 cents; how much did they all
 cost?                                     Ans. 128 cents.
  67. A farmer has  34  acres of land in one field, 42 acres in
 another, and 26 acres in another; how many acres has  he in
 aU?                                            Ans. 102. 
ADDITION.
15
  68. A gentleman bought a horse  for 145  dollars, a carriage
for 275,  and a harness for 45 dollars; how much did they all
cost him ?                                 Ans. 465 dollars.
  69. In the year 1820 there were in the city of New York
123,700 inhabitants, in the city of Philadelphia 108,100 inha-
bitants, and in the city of Baltimore 62,700 ;  how many were
there in the three cities ?                       Ans.  294,500.
  70. At the same time there were in the city of Boston 43,300
inhabitants,  in   New  Orleans  27,200   inhabitants,  and in
Charleston  24,800 inhabitants; how many were there in the
three cities?                                   Ans. 95,300.
   71.  A merchant bought four pieces of cloth;  the first mea-
sured 45 yards,  the second  36  yards, the third 29 yards, and
the fourth 56 yards; how many yards were there in all ?
                                                  Ans. 166.
   72. A farmer has in one field 46 sheep, in another 75 sheep,
in a third 6l sheep,  and in a fourth 37; how many has he in
all?                                              Ans. 219-
   73. A mechanic owes  to  one person  2,45 cents, to another
 3,25 cents, to a third 1,87 cents, and to a fourth 96 cents;  how
 much does he owe in all ?                    Ans. 8,63 cents.
   74.   A merchant sent  his clerk to collect  money.  Of one
 person he received 147 dollars, of a second 59 dollars, and of a
 third he received 226 dollars; how much did he  collect in all?
                                                  Ans. 432.
   75.  Jane received, as a new-year's present, from  her father
 1,37 cents, from her mother 87 cents, fiom  her aunt 56 cents,
 from her  uncle 75 cents, and  from her brother 25 cents;  how
 much did she receive in all ?                 Ans. 3,80 cents.
   76.  A person bought a coat for 27 dollars, a vest for 4 dollars,
 a hat for 7 dollars, and a pair of boots for 6 dollars ;  how much
 did they all cost him ?                       Ans. 44 dollars.
   77.  A person travels  the first day 28  miles, the second day
 39 miles, the third day 45 miles, and the fourth  day 52 miles;
 how many miles has he travelled in all?       Ans. 164 miles.
   78.  A young lady bought a  Latin grammar for 68 cents, a
 copy of Virgil for 1,85 cents, and a treatise on algebra for 1,56
 cents ;  how much did they all cost ?           Ans. 4,09 cents.
   79.  A lady has three  daughters  at school; for the first she
 pays 16,50 cents a quarter,  for the second  12,50 cents, and for
 the third 8,75 cents; how much does she pay in  all ?
                                            Ans. 37,75 cents.
    80.  A gentleman has three  sons at a boarding school, and
 one at college ; for the one at college he pays 168  dollars a year ; 
16
INTRODUCTION.
 for the  oldest at the boarding school he pays 156 dollars ; for
 the second 135, and for the youngest he pays 124 dollars; how
 much does he pay for all of them ?          Ans. 583 dollars.
   81.  A person deposited in the Bank of the United States the
 following  sums—345 dollars, 765  dollars, 1142  dollars, and
 127 dollars; how much has he in the bank altogether ? Ans. 2379.
   82.  The income  of a lady is derived from real estate and
 public  funds; from the rent of a farm she receives 246 dollars,
 from house rent 574 dollars, and from stock in the funds 341
 dollars; how much is her income altogether ? Ans. 1161 dollars.
   83.  The expenses of a gentleman's family for three years were
 as follow; the first year he spent 542 dollars, the  second year
 679 dollars, and the third year 746 dollars; how much did he
 spend altogether ?                        A ns. 1967 dollars.
   84.  A gentleman built a house, which  cost him as follows:
 The brick-maker's bill was 364 dollars, the mason's bill was 476,
 the carpenter's bill  was 625 dollars; and the  painter and gla-
 zier's  bills were 254 dollars; what did the house cost him?
                                        Ans. 1719 dollars.
   85. A servant went  to market, and  bought a turkey for 87
 cents, a pair of chickens for 52 cents, vegetables for 44 cents,
 and beef for 1,49 cents; how much did he spend ?
                                          Ans. 3,32  cents.
   In placing the following numbers for addition, the learner must
 observe to set units  under units, tens under lens, hundreds under
 hundreds,  &c, as above directed.

   86. Add together 4621,3746,2154,1405,3246. Ans. 15172.
   87- Add together 74521, 2351, 52432, 20461, 13456.
                                            Ans. 163221.
   88. Add together 87654, 43210, 14532, 10246, 32145.
                                            Ans. 187787.
   89. Add together 32465, 31524, 39982,99210, 21031.
   ~«   a ■,     1                             Ans- 224212.
   90.  Add together 54216, 39495, 21746, 21543, 21467.
   nt   ....    ,       „                      Ans. 158467.
   91.  Add together 21462, 35261, 42574, 31798, 12345.
   ««   a j j     ,                             Ans. 143440.
   92.  Add together 74521, 14752, 31462, 54671, 34576.
  no   a    11       1 •  ,       .             Ans- 209982.
  93.  A miller ground in the month of September 5246 bushels
of wheat, in October 6425, in November 3674, and in Decern
ber 5725;  how many bushels did he grind in the whole ?
                                      Ans. 21070 bushels. 
ADDITION.
17
   94. A merchant sold in the month of March 7461 barrels of
 flour, in April 3246 barrels, and in May he sold 10741 barrels;
 how many did he sell in the whole ?       Ans. 21448 barrels.
   95. The amount  of goods sold by a merchant, in the month
 of October was 4746 dollars, in  the month of November 5796
 dollars, and in the month of December 7894 dollars; how much
 did he sell  in all ?                       Ans. 18436 dollars.
   96. How many days are in the six first months of the year,
 allowing 28 days for February, January having 31, March 31,
 April 30, May 31, and June 30 ?             Ans. 181 days.
   97. How many days are in the six  last months  of the year,
 July having 31 days, August 31, September 30, October  31,
 November  30, and December 31 ?                 Ans. 184.
   98. Suppose,  in  the four principal streets of Philadelphia,
 there are 827 houses in the first, 685 in  the  second, 597 in the
 third, and 546 in the fourth; how  many houses are  there in
 all ?                                     Ans. 2655 houses.

   Questions. 1. What is addition ?—2. How  do you place
 the numbers before you add them together?—3. If the numbers
 consist  of one place only, how do you set them?—4. If there
 are two places,  how  do  you set them ?—5.  If the numbers
 consist of three or more places, how do you set them ?
   6.  What is the first place called ?—7. Why is it so called ?
   8.  What is the second place called?—9. Why is it so called?
 10.  What is the third place called ?—11. Why is it so called ?
 12.  What is the fourth place called?—13. Why is it so called?
 14. What is the fifth  place called ?—15.  Why is it so called ?
 16, Do the other places, to the left hand, increase in the same
proportion ?
 17. How do you proceed  in adding up the columns ?
 18. What do you set down under the first column ?
 19. What do you  do with the tens that remain, after setting
down the units?
 20. What do you do with the second column,  or column  of tens ?
 21. Do you set down the right hand figure  of the  sum under
the column, and  add the  remaining figure or  figures  into  the
next column or hundred's place?
 22. Do you proceed in this way to the last column ?
 23. What do you do with it ?
 24. Will you explain this process by an example ? 
18
INTRODUCTION.
                 SUBTRACTION.

  Subtraction is the operation of taking a less number from a
greater, to find the difference or remainder.
  The less number  must be  placed under the greater, so that
units may stand under units, tens  under tens,  hundreds under
hundreds, as in  addition.
  The upper number is called the  minuend or  number to be
lessened ; the less number is called  the subtrahend or number
to be subtracted; and the difference of the two numbers is called
the remainder.
  The remainder added to the lower number will equal the
upper number :  this proves the operation.

              SUBTRACTION  TABLE.
1 from 1 and 0 remains		2 from 2 and 0 remains			3 from 3 ai		d 0 remains
1 2	1	2	3	1	3	4	1
1 3	2 remain	2	4	2 remain	3	5	2 remain
1 4	3	2	5	3	3	6	3
1 6	4	2	6	4	3	7	4
1 6	5	2	7	5	3	8	5
1 7	6	2	8	6	3	9	6
1 8	7	2	9	7	3	10	7
1 9	8	2	10	8	3	11	8
1 10	9	2	12	10	3	12	9
4 from 4 and 0 remains		5 from 5 and 0 remains			6 from 6 and 0 remains		
4 5	1	5	6	1	6	7	1
4 6	2 remain	5	7	2 remain	6	8	2 remain
4 7	3	5	8	3	6	9	3
4 8	4	5	9	4	6	10	4
4 9	5	5	10	5	6	11	5
4 10	6	5	11	6	6	12	6
4 11	7	5	12	7	6	13	7
4 12	8	5	13	8	6	14	8
4 13	9	5	14	9	6	15	9
7 from 7 and 0 remains		8 from 8 and 0 remains			9 from 9 and 0 remains		
7 8	1	8	9	1	9	10	1
7 9	2 remain	8	10	2 remain	9	11	2 remain
7 10	3	8	11	3	9	12	3
7 U	4	8	12	4	9	13	4
7 12	5	8	13	5	9	14	5
7 13	6	8	14	6	9	Id	6
7 14	7	8	15	7	9	16	7
7 15	8	8	16	8	9	17	8
7 16	9	8	17	9	9	18	9
 
SUBTRACTION.
19
  Questions upon the Table to be answered by the learner.
 1. Take 2 from 5, and how many will remain?
 2. Take 2 from 7, and how many will remain ?
 3. Take 3 from 7, and how many will remain ?
 4. Take 3 from 9, and how many will remain ?
 5. Take 3 from 11, and how many will remain?
 6. Take 4 from 7, and how many will remain ?
 7. Take 4 from 9, and how many will remain ?
 8. Take 5 from 9, and how many will remain ?
 9. Take 5 from 11, and how many will remain?
 10. Take 5 from 13, and how many will remain?
 11. Take 6 from 9, and how many will remain ?
 12. Take 6 from 13, and how many will remain ?
 13. Take 6 from 15, and how many will remain ?
 14. Take 7 from 11, and how many will remain?
 15. Take 7 from 16, and how many will remain?
 16. Take 7 from 13, and how many will remain?
 17. Take 8 from 11, and how many will remain?
 18. Take 8 from 17, and how many will remain ?
 19. Take 8 from 15, and how many will remain ?
 20. Take 9 from 15, and how many will remain?
 21. Take 9 from 17, and how many will remain ?
 22. Take 8 from 15, and how many will remain?
 23. Take 8 from 13, and how many will remain ?
 24. Take 7 from 15, and how many will remain ?
 25. Take 6 from 16, and how many will remain ?
  After this manner, the teacher may require the pupil to subtract
any other numbers, that may be in the  table.  This will be
found a very useful  exercise.

                 PRACTICAL EXERCISES.
  1. If you have 5 apples, and give your brother 2; how many
will you have remaining? Ans. 3.  Why so?
  2. If you have 8 cents, and give 3 of them for marbles; how
many will you have remaining? Ans. 5.  Why so ?
  3. Jane has 11 nuts, and gives her sister 4; how many has
she left? Ans. 7- Why so ?
  4. James has  16 cents, arid buys a book  for 12 cents; how
many cents has he left ? Ans. 4.  Why so ?
  5. Mary had  10 cents, but on her  way to school she spent
4 for cakes ; how many has she left? Ans. 6. Why so?
  6. A person borrowed 15 dollars,  and paid 9; how many
dollars remain unpaid ?  Ans. 6.  Why so ? 
20
INTRODUCTION.
   7. A little boy on his way to school, having 14 cents in his
 pocket, gave 6 cents to a poor woman asking charity;  how
 many had he left?  Ans. 8.  Why so?
   8. Susan gave 25 cents for a book,  the price  of which  was
 only 14 cents; how much change did  she  receive from the
 bookseller? Ans. 11.  Why so?
   9. A merchant bought a lot of goods for 20 dollars, and  sold
 them again for 27 dollars; how much did be gain by the  bar-
 gain ? Ans. 7-   Why so?
   10. If you have a 10 dollar bill, and you buy books and paper
 to the amount of 6 dollars; how much will you have left? Ans.
 4 dollars.  Why so ?
   11. A person bought a barrel of apples for 2 dollars,  and
 gave the seller a five dollar  bill;  how much change must he
 receive? Ans. 3.   Why so?
   12. A person is  travelling to a place 27 miles distant,  and
 after he has travelled 16 miles;  how many miles further has he
 togo?Ans.ll.  Why so ?
   13. James has 25 cents, and  buys an inkstand for 15 cents;
 how much has he left ? Ans. 10.  Why so?
   14. Eliza has 22 cents, and  buys a toy  for 11 cents; how
 much has she remaining? Ans.  11.  Why so?
   15. A farmer sold a cow for 20  dollars, and received in part
 pay 8 dollars; how much remains unpaid? Ans.  12.  Why so?
   16. Jane is 12 years old,  and  her sister is 7 j what  is  the
 difference of their ages ? Ans. 5.  Why so ?
   17. John has  been going to school 7 years, and his brother
2 years; how much longer has  John been at school  than his
brother?  Ans. 5.  Why so?
  18. A lady has two daughters at school;  for one she pays
a quarter 15 dollars, for the other 8 ; how much more does  she
 pay for one than for the other?  Ans. 7.  Why so?
  19. Subtract 25 from 56, how many remain ? —. Why so?
  20. Subtract 16 from 37, how many remain? —. Why so?
  21. Subtract 24 from 48, how many remain ? —. Why so?
  22.  Subtract 64 from 79, how many remain ? —. Why so?
  23. Subtract 125  from 146, how many remain ? —. Why so ?
  24.  Subtract 84 from 99, how many remain ?  —. Why  so?
  25. Subtract 144 from 256, how many remain? —. Why  so?
  26.  Subtract 241  from 472, how many remain?__. Why  so?
  27. Subtract 365 from 676, how many remain ?__. Why  so?
  28. Subtract 526 from 839, how many remain ?__. Why  so?
  29- Subtract 641  from 794, how many remain ?__. Why so?
  30. Subtract 402 from 578, how many remain ?__. Why  so? 
SUBTRACTION.
21
  31. Subtract 528 from 649, how many remain? — Why so?
  32. Subtract 1462 from 2762, how many remain ? — Why
so?
  33. Subtract 569 from 899, how many remain ? — Why so ?
  34. Subtract 4723 from 7924, how many remain ? — Why
so?
  35. Subtract 3465 from 8496, how many remain ? — Why
so?
  36. Subtract 14762  from 65862, how many remain?  —
Why so ?
  37- Subtract 8450 from 9765, how many remain ? — Why so ?
  When the figures in the several places of the upper number,
or minuend, are greater  than the correspondent  figures of the
lower number or subtrahend, the operation is easy.  All  you
have to do  is, to take  the  lower  figure  from the  upper  one,
beginning at the right hand  figure, or unit's place, and set the
difference under its proper  column : and do the same with all
the figures,  and the work is done, as in the preceding examples.
  But if the lower figure is  greater than  the upper figure, you
must add 10 to the upper figure, and from the sum take the lower
figure, and  set down the  difference in its proper place; and,
because you added 10 to the upper figure, you must add 1 to
the next lower figure  on the left hand, and take their sum from
the upper figure, and set down the difference in its proper place.
But, if it should so happen that this sum is greater than the upper
figure,  you  must  add 10 to  the upper figure as you did before,
and from this sum take the subtrahend, and  set down the differ-
ence in  its proper place; and so proceed till the work is done.
  The reason for adding 10 to the upper figure, and only 1 to
the under one or subtrahend, is, that 1 in the ten's place is equal
to 10 in the unit's place;  and 1 in  the hundred's place is equal
to 10 in  the ten's place ;  and  1 in the thousand's place is equal to
10 in the hundred's place;  and so on, increasing in a tenfold
proportion in every place  toward the left hand.
  This  operation does not affect the difference of the numbers
or remainder ; for it is evident, that if you add ten to the upper
number, and 10 to the lower number, the difference will be the
same as if nothing had been added to either.
  38. Subtract 1764 from 3456, how many will remain ?
                                              Ans. 1692.
Place the numbers under each other thus :  3456 the minuend.
                                     1764 the subtrahend,
                                     1692 the remainder.
         C 
INTRODUCTION.
  Begin with the right hand figure;  and because 4 is less than
6, you set down the 2, the difference, in the unit's place under
the line.  In the ten's place, the lower figure is greater than the
upper one, you must therefore add 10 to the upper figure, which
will make the  sum 15 ;  from  this take 6, the lower figure,
and set  the difference or remainder to the left hand of the 2
already found;  and because you added 10 to the upper figure, to
preserve the equality of the numbers,  you must add 1 to the
next place, on  the left  hand of the lower number, which, with
the figure  7, will make 8.  This you  cannot take from the
upper figure because it is less than 8; you must therefore add 10
to it,  which will make the sum 14; from this you can take 8,
and the remainder will be 6; which set down to the left hand of
the 9 which you previously found.   Again, because  you added
10 to the upper figure  in hundred's place, you must add 1 to the
next  lower figure in thousand's place, and take this sum from
the upper figure, and set the difference to  the left hand of the 6
already  found;  and the  operation  is done.  In like manner
proceed to find the difference of any other numbers, when any
one of the lower figures is greater than the upper one.

  39. Subtract 4625 from 8467 ; how  many remain ?
                                             Ans. 3842.
  40. Subtract 3479 from 5627 ;  how  many remain ?
                                             Ans. 2148.
  41. Subtract 17845 from 34762; how many remain?
                                            Ans. 16917.
  42.  Subtract 35276 from 70425 ; how many remain ?
                                            Ans. 35149.
  43.  Subtract 2576 from 9405 ; how  many remain ?
                                              Ans. 6829.
  44. Subtract 1748 from 3740; how  many remain?
                                             Ans. 1992.
  45.  Subtract 9547 from 10456 ; how many remain ?
                                              Ans. 909.
  46. Subtract 8205 from 9128 ;  how many remain ? Ans. 923.
  47.  Subtract 30581 from 45760 ; how many remain ?
                                            Ans. 15179.
  48. Subtract 2469 from 5972;  how many remain ?
                                              Ans. 3503.
  49.  Subtract 3699 from 7422 ; how many remain ?

  50.  Subtract 7426 from 9104; how  many remain?
                                              Ans. 1678. 
SUBTRACTION.
23
  51. Subtract 13254 from 25203 ; how many remain?
                                            Ans. 11949-
  52. Subtract 72064 from  112531; how many remain ?
                                            Ans. 40467.
  53. Subtract 474681 from 742610; how many remain ?
                                           Ans. 267929.
  54. Subtract 142651 from 572103 ;  how many remain ?
                                           Ans. 429452.
  55. Subtract 94721 from  123214; how many remain ?
                                            Ans. 28493.
  56. The distance from Philadelphia to New York is 96 miles,
and to Trenton, on the same road, is 38 miles; how far is the
distance from Trenton to New York ?          Ans. 58 miles.
  57. The  distance from  Philadelphia  to  Pittsburg is  310
miles, and to Lancaster, on the same road, is 66 miles;  how far
is the distance from Lancaster to Pittsburg ?   Ans.  244 miles.
  58. A lady has an income of 575 dollars a year, and spends
out  of it 426 dollars; how much does she save a year ?
                                         Ans. 149 dollars.
  59- A gentleman has an income of 1200 dollars a year, and
his family expenses are 846 dollars; how much does he save of
his  income?                              Ans. 354 dollars.
  60. The  income of a school is 3240 dollars, but the teacher
pays for house rent and assistance 1416 dollars;  how  much
does he save for himself?                  Ans. 1824 dollars.
  6l. A merchant bought a quantity of goods for 4216 dollars,
and sold to  the  amount of 2542  dollars;  what amount has he
remaining on hand?                      Ans. 1674 dollars.
  62. A grocer bought a quantity of  tea for 426 dollars, and
sold the same for 512 dollars;  how much  did he gain by the
bargain ?                                 Ans.  86 dollars.
  63. A gentleman has a tract of  land containing 2427 acres,
of which he sold 749 acres ;  how many acres has he remaining ?
                                         Ans. 1678 acres.
  64. A merchant deposited  in the Bank of the United States
1260 dollars, and afterward drew out 746 dollars ;  how much
has he remaining?                         Ans. 514 dollars.
  65. Baltimore contains 74625 inhabitants, and the District
of Columbia 31472; how many more inhabitants are in Balti-
more than in the District of Columbia?          Ans.,43153.
  66. A mason built two houses of brick ;  in one he put 47642,
in the other 27875; how  many more bricks  did he put in the
one than in the other?                         Ans. 19767.
  67. There are 365 days in a year, and the month of December 
 24                  INTRODUCTION.

 contains 31 days;  how many days do all the other months of
 the year contain ?                           Ans. 334 days.
   68. The present year is  1830, and America was discovered
 by Columbus  in the year 1492 ; how many years have passed
 since it was discovered ?                     Ans. 338 years.
   69. The declaration of American Independence was signed
 on the 4th day of July, 1776 ; how many years will have passed
 by the 4th day of July, 1832 ?                 Ans. 56 years.
   70. The war of our  independence  commenced in the year
 1775, and a  general  peace was made in 1783 ; how many
 years did it continue ?                         Ans. 8 years.
   71. General Washington  was born  in  the year 1732, and
 died in the year 1799  ; how old was he at his death ?
                                           Ans.  67 years.
   72. Constantine the Great was converted to Christianity in
 the year  312 of the Christian era ; how many years have passed
 to the present  year 1830 ?                  Ans. 1518 years.
   73. The victory at New Orleans was gained on the 8th day
 of January 1815 ;  how many years  will have passed on the 8th
 of January 1831 ?                           Ans.  16 years.
   74. The first settlers of New England landed at Plymouth in
 December 1620; how many years will have passed by Decem-
 ber 1831 ?                                Ans. 211 years.
   75. Twelve  lunations make 354 days;  how many  days less
 are 12 lunations than 12 months ?              Ans. 11 days.

   Questions.  1. What do you understand by subtraction ?
   2. How do you place the  numbers to be subtracted  ?      1
   3. What is the upper number called ?
   4. What is the lower number called ?
   5. What is the difference of the two numbers called ?
  6. How do you prove the operation ?
  7. When you have  placed the numbers in proper  order for
subtraction, how do you proceed, if the lower figure is  less than
the upper one ?
  8.  When the lower figure is greater than the  upper one,
what do you do ?
  9.  Why do you add 10 to the upper figure, and only 1 to the
lower one t
  10. How does the value of the figures increase from the right
hand or unit's place ?
  11. Is  1 added to any figure in a higher place,  equal to 10
added to  a figure in the next lower place ?
  12. Is  this the principle on which  the process is founded ? 
MULTIPLICATION.
25
  13. Is the difference of the two numbers any way changed by
adding 10 to the upper number in an rnferfbr place, and only 1 in
the next superior place of the lower number ?
  14. Can  you prove this position, or make it evident  by an
example?
                MULTIPLICATION.

  The operation of finding the sum of any given number, taken
as often as there are units in  any other given number, is called
Multiplication.
  The number to be multiplied is called the multiplicand, aud
is commonly the larger number.
  The number by which you multiply, is called the multiplier,
and is commonly the less number.
  The result  of the operation is called the product of the two
numbers.
  The multiplicand and multiplier are sometimes called  factors,
because by them the operation is carried on, and the result
produced.

             MULTIPLICATION  TABLE.
twice 1 make  2
2	4
3	6
4	8
5	10
6	12
7	14
8	16
9	18
10	20
11	22
12	24
6 times 1 make 6
6	2	12
6	3	18
6	4	24
6	5	30
6	6	36
6	7	42
6	8	48
6	9	54
6	10	60
6	11	66
6	12	72
.n 9
3 times 1 make 3
3 ,	2	., 6
3	3	- 9
3	4	12
3	5	15
3	6	18
3	7	21
3	8	24
3	9	27
3	10	30
3	11	33
3	12	36
7	2	14
7	3	21
7	4	28
7	5	35
7	6	42
7	7	49
7	8	56
7	9	63
7	10	70
7	11	77
7	12	84
4 times 1 make 4	
4,2	.' 8
4 3	, 12
4 4	., 16
4 5	.1 20
4 6	24
4 7	28
4 8	32
4 9	36
4 10	40
4 11	44
4 "• 12	48
8 times 1 make 8	
8	2	16
8	3	24
8	4	32
8	5	40
8	6	48
8	7	56
8	8	64
8	9	72
8	10	80
8	11	88
8	12	96
5 times 1 make 5
5	,. 2	10
5	, 3	15
5	4	20
5	\ 5	25
5	\< 6	30
5	« 7	35
5	8	40
5	9	45
5	10	50
5	11	55
5	12	60
9 times 1 make 9	
9 2	18
9 3	27
9 4	36
9 5	45
9 6	54
9 7	63
9 8	72
9 9	81
9 10	90
9 11	99
9 12	108
 
26
INTRODUCTION.
10 times  1 make 10
10
10
10
10
10
10
10
10
10
10
10
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
 20
 30
 40
 50
 CO
 70
 80
 90
100
110
120
11 times  1 make  11 I 12 times  1 make  12
 3
 4
 5
 6
 7
 8
 9
10
11
12
22;
33
44 12
55 12
66 | 12
77 i 12
88 12
99 12
110
121
132
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
 2-1
 36
 48
 60
 72
 84
 96
108
120
132
144
 9
10
11
12
13
14
15
16
17
Questions upon
 How many are
 How many are
 How many are
 How many are
 How many are
 How many are
 How many are
 How many are
 How many are
 How many are
 How many are
 How many are
 How many are
 How many are
 How many are
 How many are
 How many are I
the Table to be
3 times 4
4 times 5
4 times 7
5 times 6
5 times 8
5 times 9
6 times 4
6 times 7
6 times 9
7 times 4
7 times 7
7 times 9
8 times 4
8 times 5
8 times 6
8 times 8
! times 11
18.
19.
20.
21.
22.
23.
24.
25.
26.
27-
28.
29.
30.
31.
32.
33.
34.
answered by the learner.
 How many are 8 times 12
 How many are 9 times 3 ?
 How many are 9 times 5 ?
 How many are 9 times 8 ?
 How many are 9 times 11
 How many are 9 times 12
 How many are 10 times 5
 How many are 10 times 8
 How many are 11 times 5
 How many are 11 times 8
 How many are 11 times 11
 How many are 12 times 4
 How many are 12 times 6
 How many are 12 times 8
 How many are 12 times 11
 How many are 12 times 9
 How many are 12 times 12
  After  this manner, the teacher may  require  the learner to
multiply any other numbers in the table.  This will be found a
very useful exercise.
                 PRACTICAL EXERCISES.
  1.  If you buy one orange for 5 cents,  what will 3 oranges
cost ? Ans. 15 cents.   Why so ?
  2.  James bought a penknife, and  gave 22 cents for it; how
much would 4 cost at the  same  price ? Ans. 88 cents.  Can
you prove that ?
  3.  Jane bought 4 oranges, and gave 6 cents a piece for them ;
how much did they all cost ? Ans. 24 cents.   How will you
prove that ? 
MULTIPLICATION.
27
  4. If one yard of ribbon  cost 3 cents, what will  12 yards
cost? Ans. 36 cents.  How will you prove that ?
  5. A lady has 4 daughters, and on a Christmas morning, gave
each of them 9 cents;  how  many  cents did she  give  in all ?
Ans. 36 cents.  How will you prove that ?
  6. A person  can  earn  4 dollars  a week;  how many dollars
can he earn in 5 weeks ? Ans. 20 dollars. Will you prove it ?
  7. Jane spends 6 cents a day for cakes and nuts ; how many
cents will she  spend, at that rate, in 6 days? Ans.  36 cents.
Why so?
  8. A cord of wood  sells for 5 dollars; how much  will 12
cords come to ? Ans. 60 dollars.  How will  you prove it ?
  9. A pound  of butter  sells  for 21 cents; how much will  8
pounds come to at that  rate ? Ans. 1,68 cents.  Will you prove
it?
  10.  A spelling book sells for 12 cents;  how much will 11
come to? Ans. 1,32 cents.  V/hy so?
  11.  An English grammar sells for 32 cents; how  much will
3 come to at that rate ? Ans. 96 cents.  Why so?
  12.  A pair  of fowls  sells for 42 cents; what will  4 pair sell
for  at that rate? Ans. 1,68 cents.  How will you  prove it?
   13. A barrel of flour may be bought for 5  dollars; how much
will 9 barrels come  to at that rate ?  Ans. 45 dollars.   Why so ?
   14. A yard  of cloth will cost 4 dollars; what will 9 yards
come to at that rate ? Ans. 36 dollars.  Will you  prove that?
   15. A person can travel 5  miles in an hour; how many
miles  will he  travel at the same  rate in 11 hours?  Ans. 55
miles.   Why  so ?
   16. There are 24 hours in a day; how many hours are there
in two days?  A,ns-  48  hour.  Will you prove it?
   17. If your quarter  bill is 12 dollars, how much  will your
quarter bills amount to in a year?  Ans. 48  dollars.  Will you
prove it ?
   18. If you  pay 2 dollars a barrel for apples, how  much will
 14 barrels amount to?  Ans. 28 dollars.   Why so ?
   19. A lady pays  3 dollars a week for her daughter at a board-
ing school; how  much will she have to pay for 12 weeks ? Ans.
36 dollars.  Will you  prove it ?
   20. There are 3  feet in a yard; how many feet are  there  in
 11  yards? Ans. 33 feet.   How will you prove it?
   21. There are 12 inches  in one foot;  how many  inches are
there in 9 feet? Ans. 108 inches.  Why so?
   22. There are 4  pecks in one bushel; how  many  pecks are
there in 12 bushels? Ans. 48 pecks.   Why so? 
28                  INTRODUCTION.
  23. Ten cents make one dime; how many cents will 8 dimes
make?—How many will 12 muke?
  24. Ten dollars make one eagle;  how many dollars will 10
eagles make?—How many will 9 make?
  25. There are 8 furlongs in one mile; how many furlongs
are there in 9 miles?—How many in 12 miles?
  26. There are 4 roods in an acre; how many roods are there
in 12 acres ?—How many  in 10 acres?
  27. If you divide an apple into 4 equal  parts, how many of
the  like  parts will 8 apples  contain?—How many will 6 apples
contain?—How many will 12 apples contain?
  28. If you multiply 4321 by 3, what will the product be?
  29. If you multiply 632  by 2, what will the product be ?
  30. If you multiply 822  by 4, what will the product be?
  31. If you multiply 9120 by 4, what will the product be ?
  When the product of the multiplier, and any figure of the mul-
tiplicand does not exceed 9, as in the above examples, (except the
last place, where you always set down the whole product) the
operation is easy.   All you have to do,  is, to set the product
down in its proper place.   But when it exceeds 9, you  must
set down the  units in the product under the figure  multiplied,
and add the tens  to the product of the next figure and the mul-
tiplier :  and so proceed till you have multiplied every figure in
the  multiplicand.
  32. Multiply 4536 by 4: what will be the product ?
                                             Ans. 18144.
       Set the multiplicand thus,     4536
      The multiplier thus,             4
                                 18144 the product.
  Here the product of 4 and 6 is 24.   You set down the 4 in
the unit's place, and add the 2, which are two tens, to the product
of 4 and the figure 3 in the ten's place, which make 14.   Here
again you set down the 4, which are four tens or forty, and the 1
which is a hundred, you add to the  product of 4 and 5 in the
hundred's place, which make 21.  You set down the 1, and add
the 2 to the product  of 4  and 4 in the thousand's place, which
make 18.   And because it is the last figure, you set down the
whole  18.   In like manner you must proceed in any other
example, whatever may be the number of figures, or places in
the multiplicand.
  33. What will be the product of 64354, multiplied by 5 ?
                                            Ans. 321770.
  34. What will be the product of 74216, multiplied by 6?
                                            Ans. 445296. 
                   MULTIPLICATION.                  29

  35. What will be the product of 37992, multiplied by 4?
                                            Ans. 151968.
  36. What will be the product of 546201, multiplied by 5 ?
                                           Ans. 2731005.
  37- What will be the product of 62174, multiplied by 6 ?
                                            Ans. 373044.
  38. What will be the product of 54327, multiplied by 7 ?
                                            Ans. 380289.
  39. What will be the product of 46284, multiplied by 8?
                                            Ans. 370272.
  40. What will be the product of 57405, multiplied by 8 ?
                                            Ans. 459240.
  41. What will be the product  of 64217, multiplied by 7?
                                            Ans. 449519.
  42. What will be the product of 7846, multiplied by 9 ?
                                             Ans. 70614.
  43. What will be the product  of 8476, multiplied by 8 ?
                                             Ans. 67808.
  44. What will be the product of 54764, multiplied by 9 ?
                                            Ans. 492876.
  45. What will be the product of 32146, multiplied by 10 ?
                                            Ans. 321460.
  46. What will be the product of 54721, multiplied by 11 ?
                                            Ans. 601931.
  47. What will be the product  of 721401, multiplied by 11 ?
                                           Ans. 7935411.
  48. What will be the product of 47426, multiplied by 12 ?
                                            Ans. 569112.
  49. What will be the product of 532106, multiplied by 12 ?
                                           Ans. 6385272.
  50. What will be the product  of 42176, multiplied by 8 ?
                                            Ans. 337408.
  51. What is the product of 74861, multiplied by 9 ?
                                            Ans. 673749.
  52. What will be the product of 72161, multiplied by 11 ?
                                            Ans. 793771.
  53. What will be the product of 8421, multiplied by 12 ?
                                            Ans. 101052.
  54. If one barrel of flour sells for  5,55 cents, what will 6
barrels come to ?                          Ans. 33,30 cents.
  55. If one load of hay costs 8,75 cents, what will  7 loads
come to ?                                 Ans. 61,25 cents.
  56. If one cord  of wood costs 4,86  cents, what will 8 cords
cost ?                                    Ans. 38,88 cents. 
30
INTRODUCTION.
   57-  If one yard of broad-cloth costs 7,65 cents, what will a.
yards come to ?                           Ans. 68,85 cents.
   58.  A lady purchased 54 yards of muslin at 12 cents a yard;
what was the cost of the whole ?             Ans.  6,48 cents.
   59.  A farmer sold 49 acres  of land at 11 dollars an acre;
what did the whole amount to?             Ans. 539 dollars.
   60.  A carpenter worked 10  days, at 1,46 cents a day ;  how
much did his wages amount to ?            Ans. 14,60 cents.
   61.  What will 764 pounds of pork come to, at 8 cents a
pound?                                   Ans. 61,12 cents.
   62.  A person spends 9 dollars a week;  how  much  will he
spend in 46 weeks ?                        Ans. 414 dollars.
   63.  If you save 12 cents a week, how much will you save in
52 weeks, or one year ? .                    Ans.  6,24 cents.
   64.  James spends for nuts and cakes 8 cents a day;  how
much will he spend in a year or 365 days ?    Ans. 29,20 cents.
   65.  Jane  spends for cakes, nuts, and apples, 6 cents a day;
how much will she spend in 2l6 days ?      Ans. 12,96 cents.
   66.  Susan pays 87 cents for  a pair of shoes, and wears out 7
pair in a year; how much do they all come to?  Ans. 6,09 cents.
   67.  How much will 11 pair of hose  come to at 75 cents a
pair ?                                     Ans.  8,25 cents.
   68.  If you give 9 cents for one orange; what  will 248 come
to at that rate ?                           Ans. 22,32 cents.
   69.  What will  675 pounds  of  cheese come to at 11 cents a
Pound ?                                  Ans. 74,25 cents.
   70.  How  much will a keg of raisins, containing 246 pounds,
come to, at 12 cents a pound ?              Ans. 29,52 cents.
   When the multiplier is more  than 12, you must multiply by
each figure separately, and add  together the several products[for
the products required.  But you must observe to place the  pro-
duct of the first  figure of the multiplicand, and  any figure of the
multiplier under its own proper multiplier ; as in the following
example.
   71. What is the product of 456, multiplied by 24?
                                              Ans. 10944.
             456 the multiplicand.
               24 the multiplier.

            1S24 product of the units of the multiplier.
            912   product of the tens of the multiplier.
10944  the product required. 
                   MULTIPLICATION.                   31

  When the multiplier consists of two or more places, you must
put it under the multiplicand, so that  units may stand  under
units, tens under tens, &c. as in this example.  Then multiply
by the  first  figure or units, and place the product  as  before
instructed, or as in this example.  Then multiply by the figure
in the next or ten's place, and set the product under the former
product; but observe to set it one place higher, or under its own
multiplier.   Proceed in the same manner with each figure of the
multiplier; and add together the several products ; the sum will
be the  product required.
   72.  What will be the product of 475, multiplied by 34?
                                               Ans.  16150.
   73.  What will be the product of 5462, multiplied by 42 ?
                                             Ans. 229404.
   74.  What will be the product of 64215, multiplied by  214 ?
                                           Ans. 13742010.
   75. What will be the product of 7421, multiplied by 36?
                                              Ans. 267156.
   76. What will be the product of 5204, multiplied by 37 ?
                                              Ans. 192548.
   77. What will be the product of 874, multiplied by 52 ?
                                               Ans.  45448.
    78. What will be the product of 1476, multiplied by 324?
                                              Ans. 478224.
    79. What will be the product of 2456, multiplied by 721 ?
                                             Ans. 1770776.
    80. What will be the product of 5723, multiplied by 314 ?
                                             Ans. 1797022.
    81   What will be the product of 7430, multiplied by 452 ?
                                             Ans. 3358360.
    82  What will be the product of 57921, multiplied by 623 ?
                                            Ans. 36084783.
    83  What will be the product of 8432, multiplied by 635 ?
                                             Ans. 5354320.
    84. What will be the product of 999, multiplied by 777 ?
                                               Ans. 776223.
    85. What will be the product of 9040, multiplied by  832 ?
                                              Ans. 7521280.
    86 What will be the product of 20546, multiplied by 369 ?
                                              Ans. 7581474.
    87. What  will 342  yards of cloth  come to at 3,64 cents a
  yardp                                   Ans. 880,88 cents.
    88 What  will  376 barrels of flour come to at 5,26 cents a
  Darrei ?                                 Ans. 1977,76 cents. 
32
INTRODUCTION.
  89. What will 365 acres of land come to at 16,46 cents an
acre ?                                  Ans. 6007,90 cents.
  90. A gentleman bought a farm containing 317 acres of land
at 9,75 cents an acre; what did the whole amount to?
                                       Ans. 3090,75 cents.
  91. A merchant bought 476 barrels  of flour, and gave  5,36
cents a barrel; what did the whole come to ?
                                       Ans. 2551,36 cents.
  "92. A farmer  sold 1246 bushels of wheat at 1,35  cents a
bushel; what did the whole come to?     Ans. 1682,10 cents.
  93. What will 128 barrels of pork come to at 12,25 cents a
barrel?                                     Ans. 1568,50.
  94. There are 60  minutes in an hour; how many minutes
are in 24 hours ?                                Ans. 1440.
  95. There are 24 hours in one day ;  how many hours are in
184 days ?                                Ans. 5416 hours.
  96. How many hours are in  365  days ?     Ans. 8760 hours.
  97. How many hours are in the  month of February, when it
has only 28 days?                         Ans. 672 hours.
  98. In an  acre of land are 160 perches; how many perches
are in 46 acres ?                         Ans. 7360 perches.
  99- A hogshead contains 63 gallons; how many gallons are
in 25 hogsheads?                        Ans.  1575 gallons.
  100.  A bushel contains 32 quarts; how many quarts are in
346 bushels ?                           Ans. 11072 quarts.

  Questions.  1. What is multiplication?—2. What is the
number to be multiplied called?—3. What is  the number, by
which you multiply, called?—4. What is  the result of the ope-
ration called ?
  5. What are the multiplicand and multiplier, taken together,
sometimes called ?
  6. Why are  they  so  called ?—7- How do you  place the
numbers for the operation ?
  8. Having placed them properly, how do you proceed ?
  9. If the product of any figure of the  multiplicand and multi-
plier is less than 10, what do you set down ?
  10. But if it is more than 9, what do  you set  down ?
  11. What do  you  do with  the tens ?—12.  What do you do
with the product of the last figure and the  multiplier ?
  13. When the multiplier consists of two or more places, how
do you place the  figures for the operation ?—14. How  do you
proceed ?
  15. How do you place the  product of  the multiplicand and 
DIVISION.
33
the second figure of the multiplier, or the one in the ten's
place ?
   16. Do you proceed in the same manner with  all the figures
of the multiplier ?
   17. What is the reason for placing the  product of the multi-
plicand  and any figure of the multiplier after the units, one place
higher than the preceding one ?
   18. What do you do with the several products ?
                      DIVISION.

  Division is the operation of finding how often one number is
contained in another.  It is the reverse of multiplication.
  The number to be divided is  called  the dividend.
  The number by which you divide is called the divisor.
  The number of times the divisor is contained in the dividend
is called the quotient.
  The remainder, if there  is any, must be less than the divisor,
and always of the same denomination  with the dividend.
  When the divisor is less  than 12, it is called Short Division :
when more than 12, it is called Long  Division.

                   DIVISION TABLE.

  Note.—The first column contains  the dividend ;  the second
the divisor; and the third  the quotient, or the number of times
the divisors contained in  the dividend.
4 divided by 2 = 1\		6 divided b)		'3=2	8 divided		by4 = 2
6 "	2 = 3	9	a	3 =3	12	(C	4 = 3
8 "	2=4	12	tt	3=4	16	tt	4 =4
10 "	2 = 5	15	tt	3=5	20	ti	4 = 5
12 <«	2 = 6	18	u	3 = 6	24	tt	4 = 6
14 4 16 *	2=7	21	tt	3 = 7	28	tt	4 = 7
	2 = 8	24	n	3 = 8	32	ti	4 = 8
18 "	2 =9	27	tt	3=9	36	it	4 = 9
20 "	2=10	30	tt	3 = 10	40	tt	4=10
22 "	2 = 11	33	tt	3 = 11	44	tt	4=11
24 "	2=12	36	tt	3=12	48	it	4=12
D 
34                   INTRODUCTION.
10 d	ivided	by 5 = 2	12 dividec		by 6 = 2	14 dividec		by7 = 2
15	tt	5=3	18	a	6 = 3	21	tt	7 = 3
20	it	5 = 4	24	it	6 = 4	28	tt	7 = 4
25	it	5 = 5	30	it	6 = 5	35	it	7=5
30	ti	5 = 6	36	tt	6 = 6	42	tt	7 = 6
35	u	5=7	42	it	6 = 7	49	it	7 = 7
40	n	5 = 8	48	tt	6 = 8	56	a	7 = 8
45	tt	5=9	54	it	6 = 9	63	ti	7 = 9
50	it	5 = 10	60	it	6=10	70	n	7=10
55	a	5=11	66	tt	6=11	77	tt	7=11
60	tt	5 = 12	72	a	6=12	84	it	7=12
16 divided		by 8 = 2' 18 divided by 9=2				20 divider		1 by 10=2
24	tt	8=3 27		tt	9=3	30	tt	10=3
32	a	8 =4 36		ti	9=4	40	u	10=4
40	it	8=5' 45		,i	9=5	50	it	10=5
48	tt	8 = 6 54		it	9=6	60	it	10=6
56	a	8 = 7	63	a	9=7	70	a	10=7
64	a	8 = 8	72	u	9=8	80	it	10=8
72	tt	8=9	81	tt	9=9	90	it	10=9
80	a	8 = 10	90	tt	9=10	100	it	10=10
88	it	8 = 11	99	n	9=11	110	ti	10=11
96	tt	8=12	108	u	9=12	120	it	10=12
	22	divided by 11 =		2	24 divided by		12	= 2
	33			3	36 "		12	= 3
	44			4	48 "		12	= 4
	55			5	60 "		12	= 5
	66			6	72 "		12	= 6
	77	" 11 =		7	84 "		12	= 7
	88	" 11 =		8	96 "		12	= 8
	99	" 11 =		9	108 "		lM	«= 9
	110	" 11 =		10	120 «		if	= 10
	121	" 11 =		11	132 "		12	= 11
	132		11 =	12	144	it \	12	= 12
  The divisor and quotient multiplied togethefwill always equal
the dividend, if there is no remainder; and if there is a remain-
der, add it to the product, and the sum will equal the dwidend.
  Parallel lines (thus =) express equality, and implyThat the
numbers between which they placed are equal.  In the table
for example, 4 divided  by 2 = 2.  They imply that the quotient
of 4 divided by  2 is equal to 2,  that is, it is  2, and so in all
other cases. 
DIVISION.                      35
   Questions upon the Table to be answered by the learner.

  1. If you divide 6 by 3, what is the quotient? —. Why so?
  2. If you divide  12 by 4, what is the quotient? —. Why so?
  3. If you divide 15 by 5, what is the quotient ? —. Why so?
  4. If you divide 24 by 6, what is the quotient? —. Why so?
  5. If you divide 35 by 7, what is the quotient? —. Why so?
  6. If you divide 48 by 6, what is the quotient? —. Why so?
  7. If you divide 56 by 8, what is the quotient? —. Why so?
  8. If you divide  63 by 9, what is the quotient? —. Why so?
  9. If you divide 72 by 8, what is the quotient? —. Why so?
  10. If you divide 81 by 9, what is the quotient? —. Why so?
  11. If you divide 96 by  8, what will the quotient be? —.
Why so ?
  12. If you divide  99 by 11, what  will the quotient be? —.
Why so?
  13. If you divide 108 by 9, what  will the quotient be? —.
Why so?
  14.  If you divide 121 by 11, what will the quotient be? —.
Why so ?
  15. If you divide 132 by 11, what will the quotient be ? —.
Why so ?
  16.  If you divide 144 by 12, what will the quotient be ? —.
Why so ?
  17. If you divide 77 by 7, what will the quotient be ? —.
Why so ?
  18.  If you divide 84 by 7, what  will the quotient be ? —.
Why so ?
  19. If you divide 55 by 11, what  will the quotient be? —.
Why so ?
  20.  If you divide 36 by 12, what will the quotient be? —.
Why so ?
  21. If you divide 48 by 12, what will the quotient be ? —.
Why so ?
  22.  If you divide  66 by 11, what will the quotient be? —.
Why so ?
  23.  If you divide  72 by 12, what will the quotient be? —.
Why so ?
  24.  If you divide 96 by 12, what will the quotient be? —.
Why so ?

  In like manner,  the  teacher may require the pupil to divide
any other numbers in  the table,  by the given divisors.  It is
important that the  scholar should perfectly  understand this, and 
36                   INTRODUCTION
the foregoing tables, and be able to repeat them all without hesi-
tation ; otherwise his future progress will be slow.

                  PRACTICAL EXERCISES.

   1. If you have 25 cents and divide them into 5  parts, how
many will each part contain ?  Ans. 5.  Why so?
   2 If 6 yards of cloth cost 30 dollars, what will one yard cost ?
Ans. 5 dollars.   Why so?
   3. If 8 barrels of flour cost 40 dollars, what will one barrel
cost? Ans. 5 dollars.  Why so?
   4.  If 9 pounds of butter cost 2,70 cents, what will one pound
cost ? Ans. 30 cents.   Why so ?
   5.  If 12 yards of muslin cost 1,44  cents, what will one yard
cost? Ans. 12 cents.  Why so?
   6.  If 8 barrels of apples cost 24 dollars, what will one barrel
cost ? Ans. 3 dollars.  Why so ?
   7.  If you  buy a spelling  book for  11 cents,  how many can
you buy for 132 cents?  Ans. 12.  Why so?
   8.  If you buy one cord of wood for 6 dollars, how many can
you buy for 54 dollars ? Ans. 9.   Why so ?
   9.  If you buy one ton  of coal for 7 dollars, how many tons
can you buy for 84 dollars ? Ans. 12 tons.  Why so ?
   10. A  farmer sold 11 fat cattle for  121 dollars ; how much
was the price of one ? Ans. 11 dollars.  Why so ?
   11. A  merchant sold 7  yards  of broad-cloth for 49 dol-
lars ; how much was the price of one yard ?  Ans. 7 dollars.
Why so ?
   12. If you buy a quire of paper for 12 cents, how many quires
can you buy for 108 cents ?  Ans. 9.   Why so ?
   13. If you  buy a slate for 9 cents,  how many can you buy
for 99 cents? Ans. 11.  Why so ?
   14. If you  buy an inkstand for 8 cents, how many can you
buy for 72 cents ?  Ans. 9.   Why so ?
   15. If you buy 9 oranges for 63 cents, how much is the price
of one? Ans. 7 cents.   Why so?
   16. A person bought  8 pounds of sugar for 80 cents; how
much was it a pound ? Ans.  10 cents.  Why so ?
   17- The stage fare is commonly 6 cents a  mile ; how many
miles may a person travel for 66 cents ? Ans. 11 miles.  Why
so?
   18. A  gentleman and his wife pay for boarding, 7 dollars a
week; how many weeks can they board for 56 dollars ?  Ans. 8
weeks.   Why so ? 
DIVISION.
37
   19. A horse travels at  the rate of 6 miles an hour; in how
many hours will he travel  42 miles ? Ans. 7 hours.  Why so ?
   20. In  5 days a man  travels  100 miles ; how many miles
does he travel in one day ? Ans. 20 miles.  Why so ?

   When the dividend consists of two or more places of figures,
and each  figure contains the divisor one or more times without
a  remainder, you must find the  quotient (or  the number  of
times the dividend contains the  divisor)  and set it under the
figure divided,  beginning  at the left hand; as in the following
examples.

   21. If you divide 246 by 2, what will the quotient be ?
                                                 Ans. 123.
                 divisor 2)246 the dividend.
                          123 the quotient.

   23. If you divide 3696 by 3,  what will the quotient be ?
                                                Ans. 1232.
            divisor 3)3696 the dividend.
                     1232 the quotient.

   24. If you divide 4804 by 4,  what will the quotient be ?
                                                Ans. 1201.
   25. If you divide 39630 by 3, what will the quotient be ?
                                               Ans. 13210.
   26. A merchant bought a quantity of pork for 484 dollars,
and paid 4 dollars a hundred pounds; how many hundreds were
there?                                           Ans. 121.
   27. A farmer bought a flock of  sheep, for which he paid 246
dollars ; how many were  there in the whole, provided he paid
2 dollars a piece ?                                 Ans. 123.

   But if there  is a remainder  after division, except in the last
figure, or unit's place, such remainder is to be considered as so
many tens, (because 1 in  a higher place is equal to ten  in the
next lower place)  and added to the next figure for a dividend :
the sum you must  divide by the  divisor,  and place the quo-
tient,  as above directed, and so  proceed  till the operation is
finished.

   28- What will be the quotient of 2568, divided by 2 ?
                                                Ans.  1284.
                divisor 2)2568 dividend.
                         1284 quotient.
           d2 
38
INTRODUCTION.
  Here the first figure contains the divisor once without a  re-
mainder ; but 5 divided by 2 leaves 1, which is one ten (or ten
times as much, as if it stood in the next lower place).  To this
add the6 in the next lower place; and the sum 16 divide by 2,
and place the quotient under the 6.  There being no remainder,
divide the 8 by 2, and the operation is finished.

  29. What will be the quotient of 1455, divided by 3 ?
                                                Ans. 485.
                divisor 3)1455  dividend.
                          485  quotient.

  Here the first figure in the dividend will not contain the divi-
sor; you must consider  it 10, and add  it to the 4 in the next
place, making 14, which contains the divisor 4 times, leaving 2
a remainder.   This 2 is two tens or 20, and, added to the next
lower figure,  becomes 25.  This contains the divisor 8 times,
and  1 remains; which is  1 ten,  and  added to the  5 in the last
place, makes 15.  This sum contains the divisor 5 times,  and
no remainder.  In this manner proceed in the following exam-
ples.   When there is a remainder after dividing  the last figure,
set it to the right of the quotient with this mark or character +
before it, which signifies that so many are over or remain.

  30. What will be the quotient of 4563, divided by 3 ?
                                            Ans. 1522+2.
  31. What will be the quotient of 54631, divided  by 4 ?
                                           Ans. 13657+3.
  32. What will be the quotient of 62074, divided  by 4 ?
                                           Ans. 155184-2.
  33. What will be the quotient of 5374, divided by 5 ?
                                            Ans. 10744-4.
  34. What will be the quotient of 7856, divided by 6 ?
                                           Ans. 1309+2.
  35. What will be the quotient of 85362, divided  by 7 ?
                                           Ans. 12194+4.
  36. What will be the quotient of 94621, divided  by 7 ?
                                           Ans. 13517+2.
  37- What will be the quotient of 85643, divided by 8 ?
                                           Ans. 10705+3.
  38. What  will be the quotient of 76543, divided by 9 ?
                                             Ans. 8504+7-
  39- What will be the quotient of 68954, divided by 10 ?
                                            Ans. 6895+4. 
DIVISION.                     39
  40. What will be the quotient of 54985, divided by 11 ?
                                           Ans. 4998+7-
  41. What will be the quotient of 47652, divided by 12?
                                              Ans. 3971.
  42 What will be the quotient of 8261, divided by 7 ?
                                           Ans. 1180+1.
  43. What will be the quotient of 87546, divided by 8 ?
                                          Ans. 10943+2.
  44. What will be the quotient of 15431, divided by 9 ?
                                           Ans. 1714+5.
  45. What will be the quotient of 21452, divided by 11 ?
                                           Ans. 1950+2.
  46. What  will be the quotient of 65241, divided by 12 ?
                                           Ans. 5436+9.
  47. Divide 67458 dollars among 8 persons, how much will
each one have?                             Ans. 8432+2.
  48. Divide 7642 acres of land among 9 persons, how many
acres will each one have?                     Ans. 849+1.
  49. Divide  1496 bushels of wheat into 7 parts, how many
bushels  will there  be in each part ?              Ans. 213 +5.
  50. Divide 1243 cords of wood among 6 persons, how much
will each one have ?                           Ans. 207+1.
  51. Divide  2236 barrels of flour among 11 persons,  how
many barrels will  each one have ?              Ans. 203+3.
  52. Divide 16482 cents among 12 persons,  how many will
each one have ?                              Ans. 1373+6.

   Questions.  1. What do you understand by Division?—2.
Of what is it the reverse ?
   3. What  is the number to  be divided called ?—4. What is
the number divided  by, called ?—5. What is the  number of
times the divisor  is contained in the dividend, called ?
   6. When any number is over after division, what is it called ?
   7. Must it always be less  than  the divisor?—8. Must it
 always  be of the same denomination with  the dividend ?
   9. When the divisor  is less than 12,  what is it called ?—10.
 When the divisor is more than 12, what is it called ?
    11.   How  do you prove the operation?'—12. If there is a
 remainder,  what  do you  do with it?—13. What do  parallel
 lines signify ?—Will you give an example to elucidate this ?
    15.  When each figure of the dividend contains the divisor
 without a remainder, how do you proceed ?—16.  Will you give
 an example? 
40
INTRODUCTION.
  17. When, after dividing any figure, there shall be a remain-
der, what do you do with it r—18.  Will you give an example?
  19. Why do you call each  one  of  the remainder 10?—20.
What do you do then ?—21. Where do you place the quotient
figure ?
  22. If there is a remainder after dividing the last figure of the
dividend, how do you place it ?—23. Will you give an example
to explain your meaning ?

  When the divisor is more than 12, you must find by inspec-
tion or trial, how often it will be contained in as many figures of
the  dividend,  as there are in  the  divisor;  and put that to the
right of the dividend, but separate from it, as the first quotient
figure.  If, however, the first figure on the left hand in the divi-
dend  is less than  the first figure in the divisor, you must take
one place more from the dividend, and find how often the divisor
will be contained in that number, and place it as before.  Then
multiply the divisor by the quotient figure, and set the product
under that part of the dividend, which you had designated. Sub-
tract this from the dividend, and to the right hand of the remain-
der, add, or bring down, the next figure in the dividend.   This
then becomes  the dividend for the next quotient figure,  which
must be placed to the right of the quotient figures already found;
and so proceed till  the operation is  finished.  But if,  after you
have brought down the figure from the  dividend, it will not con-
tain  the divisor, you  must put a cypher in the quotient, and
bring down one or more additional figures, till it will contain the
divisor :  As in the following examples.

  1. Divide 4524 by 21, what will the quotient be?
                                             Ans. 215+9.
     divisor 21)4524(215  quotient.
              42 ••
              ----                      215+9 rem.
               32                        21     divisor.
               21                    -----
               ---                     224
                114                   430
               105                   _____
               ---                   4524 proof.
                  9 remainder.

  2. Divide 157462 by 34, what will the quotient be  ?
                                            Ans.  4631+8. 
DIVISION.                     41
     divisor 34)157462(4631 quotient.
               136---
               ——         quotient 4631 + 8  rem.
                 214                   34     divisor.
                 204              -------
                 -----              18532
                  106             13893
                  102             -------
                  -----    dividend 157462 proof.
                    42
                    34

                      8  remainder.
  Here the first two figures of the dividend will not contain the
divisor; you must therefore take three figures, that is, 157, as a
dividend for the first quotient figure, which is 4.   This place in
the quotient, and multiply the divisor by it, placing the product
under 157, the portion of the dividend set apart: Substract the
product from the dividend, and the remainder is found to be 21.
To this, bring down the next figure of the dividend, and place
it to the right hand of the remainder, making 214.  This now
becomes a dividend for  the second quotient figure; find by
inspection, or trial, how often the divisor is contained in  this
number, and place that in  the quotient to the right hand of the
quotient figure already found: and so proceed till the operation
is finished.

  3. Divide 64251 by 42, what will the quotient be?
                                          Ans. 1529+33.
  4. Divide 10462 by 57, what will the  quotient be?
                                           Ans. 183+31.
  5. Divide 7231 by 62, what will the quotient be?
                                           Ans. 116+39.
  6. Divide 84031 by 74, what will the quotient be ?
                                          Ans. 1135+41.
  7, Divide 92054 by 84, what will the quotient be ?
                                          Ans. 1095+74.
  8. Divide 104701 by 92, what will the quotient be ?
                                           Ans. 1138+5.
  9. Divide 12474 by 99, what will the quotient be? Ans. 126.
 10. Divide 46215 by 124, what will the quotient be?
                                           Ans. 372 + 87-
 11. Divide 34762 by 216, what will the quotient be ?
                                          Ans. 160+202. 
42
INTRODUCTION.
  12. Divide 24374 by 136, what will the quotient  be ?
                                           Ans. 179 + 30.
  13. Divide 45261 by 144, what will the quotient be ?
                                           Ans. 314+45.
  14. Divide 52135 by 223, what will the quotient be ?
                                          Ans. 233 + 176.
  15. Divide 3467 by 57, what will the quotient be ?
                                            Ans. 60+47.
  16. If you divide  4746 dollars among 26 men, how much
will each one have ?                      Ans. $182,38,4+
  17. Forty-eight  men engage to perform a piece of work for
the sum of 6472 dollars; how much will each man have ?
                                      Ans. 134,83+ cents.
  18. The  President of the  United States has a  salary  of
25,000 dollars a year ; how much has he a day ?
                                      Ans. 68,49+ cents.
  19- If a person has a salary of 7500 dollars a year, how much
has he a day ?                          Ans. 20,54+ cents.
  20. If a person spends 1240 dollars a year, how much does
he spend one day with another?              Ans. $3,39,7 +
  21. A company of 48 persons built a bridge which cost 9765
dollars; how much did each one pay ?   Ans. 203,41+ cents.
  22. A farmer planted 4320 trees in  48 rows;  how many
trees were there in each row ?                  Ans. 90 trees.
  23. If you divide  478,97 cents among 56 men, how many
will be the share of each?                Ans. 855,3+ mills.
  24. There is a tract of land containing 42475 acres; how
many sections of 640 acres each does it contain, and what num-
ber of acres remain ?       Ans. 66 sections, 235 acres remain.
  25. The sum of 18756 dollars is to be paid by a company of
men, in 24 equal payments ; how much is each payment ?
                                       Ans. 781,50 cents.
  26. A lady has  an annual  income of 1210,50 cents; how
much is  that in a day ?                    Ans. $3,31,7,8+
  27- A farmer sold 36 barrels of pork,  and received for the
same 456 dollars;  what was it a barrel ?   Ans. 12,660 cents.
  28. Seventy-five persons associated for the purpose of estab-
lishing a bank, and  they  contributed  altogether  the  sum of
142675 dollars; how much was each  one's share,  supposing
they all contributed equally ?                Ans. #1902,33$.
  29. A company of 32 persons purchased several tickets in a
lottery, and they drew altogether the sum of 25462 dollars, after
all deductions were made; how much was the share of each one?
                                       Ans. $795,68,7+ 
FEDERAL CURRENCY.
43
  30. A merchant purchased 244 yards of cloth for  874,50
cents ; how much was the price of a yard ?      Ans. $3 58 4.
  31. A person travelled  in 44 days 1045 miles; how'many
miles did he travel a day on an average ?     Ans. 53 m. 5 fu.

  Questions.  1. What is the difference between long  and
short division ?
  2. Where do you  place  the quotient in long division ?—3.
Where do you place the divisor ?
  4. How do you proceed to find the first quotient figure ?
  5. What do you do after that ?—6.  Where do you place the
product of the divisor and quotient figure ?
  7. What do you add to the remainder ?—8. How do you pro-
ceed after you have brought down the next  figure  of the divi-
dend ?                    •
  9. But, if it will not then contain the divisor, what do you do ?
  10. After you have put a cypher in the quotient to the right
hand of the figure last found, what do you do ?
  11. Do you proceed in this way  till the operation is finished ?
  12. Will you illustrate this by an example ?
  13. How do you prove the work ?—14. If there is a remain-
der after the division, what do you  do with it ?
              FEDERAL CURRENCY.

  This currency was established by the congress of the United
States, in the year 1786; and therefore it is called Federal Cur-
rency.   It increases in a tenfold proportion, from a  mill, the
lowest denomination, to an  eagle, which is the highest.  It is
more simple than any currency in the world.
TABLE.
10 mills make 1 cent.
10 cents make 1 dime.
10 dimes make 1 dollar.
10 dollars make 1 eagle.
1000 mills=10dimes=l dollar.
 I00mills=  1 dime  =^ doll.
 7.5 mills=^ a cent.
   5 mills=£ a cent.
 2.5 mills=4t a cent.
                  Questions on the Table.

  1. How many  mills are  in  5 cents?—2. How many in 8
cents ?—3. How many in 9 cents ?—4. How many in 6 cents ? 
44
INTRODUCTION.
  5. How many cents in 4 dimes ?—6. How many in 8 dimes ?
—7. How many in 9 dimes ?—8.  How many in 7 dimes ?
  9. How many dimes in 3 dollars ?—10. How many in 6 dol-
lars ?—11. How many in 8 dollars ?
  12. How many dollars in 4 eagles ?—13.  How many in  6
eagles?—14.  How many in  9 eagles?—15. How many in
10 eagles ?
  16. How many mills in 5 dollars?—17- How many in 8 dol-
lars?—18. How many in 9 dollars?
  19. How many  mills in  7 dimes ?—20.  How many in  4
dimes ?
  21.  How many in 8 dimes?—22, How many in9 dimes?
  23. How many mills make 1 dollar ?—24.  How many make
1 dime ?—25. How many make £ a cent.
  Any sum of money in eagles, dollars, dimes, cents and mills,
may be written as so many mills, thus :  7 eagles, 4 dollars,  5
dimes, 6 cents and 4 mills, may be written, 74564 mills, mak-
ing seventy-four thousand, five hundred  and sixty-four mills.
The 4 in unit's place, represents 4 mills, the  6 in ten's place,
represents 60 mills or 6 cents; the 5 in hundred's place, repre-
sents 500 mills or 5 dimes; the 4 in thousand's place,represents
4000 mills or 4 dollars; and the7 in ten-thousand's place, repre-
sents 70,000 mills or 70 dollars :  the whole therefore is 74 dol-
lars, 56 cents and 4 mills.   But the usual way of setting down
any sum of money in dollars, cents, and mills, is, to separate
the first place by a comma, or decimal point  for mills: the two
next for cents, and the remainder will be dollars, whatever may
be the number of places, thus : 74, 56, 4, which  are 74 dollars,
56  cents, and 4 mills, as before.
  1. How many dollars, cents and mills  are in 36452 mills?
  2.  How many dollars, cents and mills are in 732651 mills?
  3. How many dollars, cents and mills are in 125046 mills?
  4.  How many dollars, cents and mills are in 847526 mills?
  5.  How many dollars, cents and mills are  in 3452601 mills?
  6.  How many dollars, cents and mills are  in 140754 mills ?

                  practical exercises.
   1.  If you buy a book for one dollar, how  many  dimes must
 you give for it ? Ans. 10.   Why  so ?
   2. If you  buy  a hat for  4 dollars, how many dimes or ten
 cent pieces must you give for  it ?  Ans. 40.   Why so ?
   3. If you buy an arithmetic for 50 cents, how many dimes or
 ten cent pieces must you give for it ?  Ans. 5.  Why so? 
FEDERAL CURRENCY.                 45
  4. If you purchase a grammar for 30 cents, how many mills
will there be ?  Ans. 300.  Why so ?
  5. A bushel of apples costs 25 cents ; how many mills will
there be ?  Ans. 250.  Why so ?
  6. If you give 12,5 mills for a slate, how many mills will 5
come to?  Ans. 62,5 mills.  Why so?
  7. If a  cord of wood costs 4,25,5 mills,  what will 5 cords
cost? Ans. 21,27,5 mills.   Why so?
  Dollars, cents, and mills are multiplied exactly as whole num-
bers are; only observe, in the product, to place the proper deci-
mal points. They are divided  also as whole numbers are, only
observe to place the proper decimal points, in order to divide the
quotient into dollars, cents, and mills; or into cents and mills, as
the case may be.
  They are added as whole numbers are, only observe to place
the proper decimal points,  and to place dollars under dollars,
cents under cents, and mills under mills.
   8. If your quarter bill is  6,25 cents, how much will your bills
amount to in four quarters ?                 Ans. 25,00 cents.
  9. If you multiply 62,5 mills by 7 mills, what will the pro-
duct be ?                 Ans. 4,37,5  mills, or 4 dols. 37£ cts.
   10. If  you  purchase 24 peaches at 5  mills each, what will
they amount to ?                            Ans. 12,0 mills.
   11. If you  buy 246 quills at  5 mills apiece, what will the
 whole come to ?                            Ans. 1,23,0 mills.
   12. If one yard of muslin costs 12,5 mills, what will 9 yards
come to ?                                  Ans. 1,12,5 mills.
   13. If one yard of broad-cloth  costs  3,37,5 mills, what will 5
yards come to ?                          Ans. 16,87,5 mills.
   14. If 6 persons give in  chanty 1,37,5 mills each, how much
 do they all give ?                          Ans. 8,25 cents.
   15. There  are 57,46,4 mills to be divided among 8 persons;
 how much will each one have ?            Ans. 7,18,3 mills.
   16. A sum of money amounting to 954,62,5 mills is to be
 divided among 5 persons;  how much will each one have ?
                                        Ans. 190,92,5 mills.
   17. A  lady bought 8 yards of silk for 10,37,6;  how much
 was the silk a yard ?                       Ans. 1,29,7 mills.
   18. A gentleman bought 12 yards of cloth for 27,62,4 mills;
 how much was the cloth a yard ?
                    Ans. 2,30,2  mills; or, 2 dols. 30 cts. 2 ms.
    19. If a little boy spends for apples  and cakes 6,2,5 tenths of
 a mill a day; how much will he spend  in 12 days ?
                                            Ans. 75,0 mills. 
46                   INTRODUCTION.
  20. If you save 12,5 mills a day, how much will you save in
30 days, or one month ?                    Ans. 3,75  cents.
  21. If you save 37,5 mills a week, how much will you save
in 52 weeks, or one year?                  Ans. 19,50 cents.
  22. If a person spends 18,7,5 tenths of a mill  a day, how
much will  he spend in a year, or  365 days ?
                            Ans. 68,43,7,5 tenths of a mill.
  23. Add together the following  sums, 5,46,7  mills, 8,65,5
mills, and  9,37,5 mills; what is the sum ?   Ans. 23,49,7 mills.
  24. Add together 14,25,9 mills,  42,75,6 mills, and 75,31,7
mills ; what is  the sum ?                 Ans. 132,33,2 mills.
  25. Add together 216,12,5 mills, 312,18,7 mills, 165,43,8
mills, and  374,87,5 mills;  what  will be the sum ?
                                      Ans. 1068,62,5 mills.
  26. Add together  374,90,4 mills, 104,61,8 mills, 225,05,7
mills, 524,17,6 mills, and 720,21,8 mills ; what is the sum ?
                                     Ans. 1948,97,3 mills.
  27. How much will  49  pounds  of butter  come to, at 16,5
mills a pound?                           Ans. 8,08,5 mills.
  28. How much will 26 pounds of beef come to, at 6,5 mills a
pound?                                    Ans. 1,69 cents.
  29. How much will 12 pair of shoes come to, at  1,37,5 mills
a pair >                                    Ans. 16,50 cents.
  30. If 5 tons of coal come to 36,75,5 mills, how much is it a
tonp                                     Ans. 7,35,1 mills.
  31. If 8 cords of wood  cost  43,87 cents, how  much is it a
C0)cl ?                                  Ans. 5,48,3+ mills.
  32. How much will 16 reams of paper come to, at 2,37,5
mills a ream ?                              Ans. 38,00 cents.

  Questions. 1. Why is  this  currency  called  Federal Cur-
rency ?—2. Is it more simple than any other ?—3. Upon what
principle does  it proceed ?
  4. What are its denominations?—5. May any sum in dollars,
cents, and  mills be set down as so many mills ?
  6. Will  you give an example?—7. How do you divide or
distinguish any given sum in mills, into dollars, cents, and mills?
—8. Will you give an example ?
  9. How are dollars, cents, and mills multiplied ?—10. Will
you give an example ?
  11. How are they divided?—12. Will you give an example?
  13. How are  they added  together?—14.  Will  you give an
example?—15. What must you  observe in all these cases? 
POUNDS, SHILLINGS, AND PENCE.          47
   Before Congress established the Federal currency, the cur-
rency of the United States  was pound, shilling, penny,  and
farthing; and in some of the  States accounts are kept,  and
money reckoned, in that currency, at the present time.

                       TABLE.
               4 farthings make 1 penny, d.
              12 pence make 1 shilling, s.
              20 shillings make 1 pound, £.
            960 grs.=240 d.=20 s.=l<£.
     The parts of a_penny are thus represented :
              J of a penny=1 farthing, marked qr.
              J of a penny=2 farthings.
              § of apenny=3 farthings.
                 Questions on the  Table.
   1. How many farthings are in 3 pence?  —.  Why so ?
   2. How many farthings are in 5 pence ? —.  Why so ?
   3. How many farthings are in 9 pence ? —.   Why so ?
   4. How many farthings are in 11 pence? —.  Why so?
   5. How many pence are in 2 shillings ? —.  Why so ?
   6. How many pence are in 4 shillings ? —.   Why so ?
   7. How many pence are in 7 shillings? —.  Why so ?
   8. How many pence are in 9 shillings? —.   Why so?
   9. How many shillings are in 2 pounds? —.  Why so?
   10. How many are in 5 pounds ? —.   Why so ?—11. How
many are in 6  pounds ?  —.  Why so ?—12. How many are
in 9 pounds ?  —.  Why so ?
   13. How many pence are in  10  farthings, and how many
farthings over ? —.   Why so ?
   14. How many pence are in  13  farthings, and how many
over? —.  Why so?
   15. How many pence are in 15  farthings, and how many
over? —.  Why so?
   16. How many pence are in 19 farthings,  and how many
over ? —.  Why so ?
   17- How many pence are in 20  farthings, and how many
over ? —.   Why so ?
   18. How many shillings are in 19  pence,  and how many
pence over?  —.  Why so?
   19. How many shillings in 27 pence,  and how many pence
over ? —.  Why so ?
   20. How many shillings in 30 pence,  and how many pence
over ? —.  Why so? 
48                  INTRODUCTION

  21. How many shillings in 37 pence,  and how many pence
over ? —.  Why so ?
  22. How many shillings in 45 pence, and  how many pence
over ? —.  Why so ?
  23. How many shillings in 55 pence, and how many pence
over ? —.  Why so ?
  24. How many pounds are in 24 shillings, and how many
shillings over ? —.  Why so?
  25. How many pounds in 30 shillings, and how many shil-
lings over ? —.   Why so ?
  26. How many pounds in 45  shillings, and how many shil-
lings over ? —.  Why so ?
  27. How many pounds in 50 shillings, and how many shil-
lings over? —.   Why so?
  28. If you divide a penny into four parts, what will 1 part
be called ?—How will you represent it?
  29. If you divide it into two parts, what will 1 part be called?
—How  will you represent it ?
  30. If you divide it into four  parts, what will 3 parts be call-
ed ?—How will you represent them ?

  The teacher may ask the pupil any other question of the like
nature, that he may deem expedient.

                  practical exercises.
  1. In 12 pounds, how many shillings ?   Ans. 240 shillings.
  2. In 124 pounds, how many shillings? Ans.  2480 shillings.
  3. In 54 shillings, how many pence ?       Ans. 648 pence-
  4. In 124 shillings, how many pence?    Ans. 1488 pence-
  5. In 75 shillings, how many pence?       Ans. 900 pence-
  6. In 21 pence, how many farthings?     Ans. 84 farthings-
  7. In 52 pence, how many farthings?   Ans. 208 farthings.
  8. In 17 pence, how many farthings ?     Ans. 68 farthings.
  9. In 144 shillings, how many pounds ? Ans. £7, and 4s. over.
  10. In 75 shillings, how many pounds?
                                   Ans. £3, and 15s. over.
  11. In 150 shillings, how many pounds?
                                   Ans. £7, and 10s. over.
  12. In 66 pence, how many shillings?  Ans. 5s., and 6d. over.
  13. In  116 pence, how many shillings?
                                    Ans. 9s., and 8d. over.
  14. In 136 pence, how many shillings?
                                   Ans. lis., and Ad. over.
  15. In 90 pence, how many shillings ?  Ai\s. 7s., and 6d. over. 
TROY WEIGHT.
49
   16. In 17 farthings, how many pence?
                                  Ans. Ad., and 1 qr. over.
   17. In 27 farthings, how many pence ?
                                  Ans. 6d., and 3 qrs. over.
   18. In 12 farthings, how many pence?            Ans. 3d.
   If you  would bring pounds  to shillings, you must multiply
the given  pounds by 20,  because 20 shillings make a pound.
If you would bring shillings to pence, you must multiply by 12 ;
because 12 pence make a shilling; and if you would bring pence
to farthings, you multiply by 4; because 4 farthings make a penny.
   If you would bring pounds to  pence, you  must multiply first
by 20, and that product by 12.  If you  would bring shillings  to
farthings, multiply the given shillings by 12, and that product
by 4.
   On the other hand,  if  you would bring shillings to pounds,
you must divide the given  shillings by 20; because 20 shillings
make a  pound.
   If you would bring pence to shillings,  you must divide by 12;
because 12 pence make a shilling.
   If you would bring farthings to pence, you must divide by 4;
because 4 farthings make a penny.

   Questions.   1.  What  currency  was  used in the United
States before Congress established the Federal currency?
   2. Is this currency sometimes used at  the present, in some
of the states ?
   3. Can you repeat the table ?—4.  How do you represent one-
fourth of a penny ?—5. How one-half of  a  penny ?—6.  How
three-fourths of a penny?
   7. How do you bring  pounds to shillings?—8. How do you
bring shillings  to pence ?—9. How do  you bring pence to far-
things?—What is the reason for your doing so ?
   10. How do you bring pounds to pence, and farthings ?
   11. How do you bring shillings to farthings?—What reason
can you give for the operation ?
   12. How do you  bring  shillings  to pounds?—13. How do
you bring pence to shillings ?—14. How do you bring farthings
to pence?—What is  the reason for doing so?
                 TROY WEIGHT.
  The denominations are pound, ounce, pennyweight, and grain.
By this weight jewels, and gold and silver are weighed.
           w 9 
50                  INTRODUCTION.
                        TABLE.
         24 grains, (gr.) . make 1 pennyweight, dwt.
         20 pennyweights, make 1 ounce, oz.
         12 ounces,   -   make 1 pound, lb.
      5760 grs. — 240 dwts. =  12 oz. =  1 lb.
                 Questions on the Table.
  1.  How many ounces in 2 pounds? —. 2. How many in 4
pounds? —. 3. How many in 7 pounds? —. 4. Ilow many in
10 pounds?  Why so?
  5.  How many ounces in 1 and a half pound ? —. Why so?
  6.  How many ounces in 3 and a half pounds ? —.  Why so?
  7-  How many ounces in 7 pounds and a quarter ? —. Why so?
  8.  How many pennyweights in 3  ounces? —. Why so?
  9-  How many pennyweights in 4  and a half ounces? —.
Why so ?
  10. How many pennyweights in 5 ounces? —. Why so?
  11. How many pennyweights  in 3 and a half ounces? —.
Why so?
  12. How many grains in 2 pennyweights? —.  Why so?
  13. How many grains in 3  and  a half pennyweights? —.
Why so ?
  14. How many grains in 4 and a half pennyweights? —.
Why so?
                  practical exercises.
  1. In 11 pounds of silver plate, how many  ounces?  Ans. 132.
  2. In 5 pounds of gold bullion, how many ounces? Ans. 60.
  3. In 6 ounces of silver, how many pennyweights?  Ans. 120.
  4. In 9 ounces of silver plate, how many pennyweights?
                                               Ans. 180.
  5. In 8 pennyweights of silver, how many grains? Ans. 192.
  6. In 12 pennyweights of gold, how many grains ?  Ans. 288.
  7. The standard weight of a Johannes is 18 pennyweights;
how much is its weight in grains?                  Ans. 432.
  8. If an Eagle  weighs 17  dwls. and 4 grains, what is its
weight in grains ?                                Ans. 412.
  Here you multiply the pennyweights by 24, because 24 grains
make a pennyweight, and to the product add the 4 grains, and
the sum will be the number required : and so in all cases, in which
there are two or more denominations.
  9. A French crown weighs  19 dwls., what is  its  weight in
grains?                                        Ans. 456.
   10.  A doubloon weighs 16 dwts.  and 21 gr.) what is its
weight in grains ?                                Ans. 405. 
TROY WEIGHT.
51
  11. A Spanish dollar weighs 17 dwts. and 6 grains; what is
its weight in grains?                              Ans. 314.
  12. An English guinea weighs 5 dwts. and 6 grains;  what
is its weight in grains?                           Ans. 126.
   13. In 3 pounds of of silver, how many grains? Ans. 17280.
  14. In 4 pounds 5 ounces of silver, how many grains?
                                             Ans. 25440.
  Here you multiply the pounds by 12, to bring them to ounces,
and to the product add the 5 ounces, and multiply the sum by
20, to bring them to pennyweights ; and this product by 24, to
bring them to grains.   If there had been pennyweights also, as
in the following examples, these must have  been added in the
proper place,  and the sum multiplied by 24; and if  there had
been grains also, these must have been added to the last product,
and the sum would have been the number required.
   15.  How many grains are in 6 pounds, 3 ounces, 4 penny-
weights of silver plate?                        Ans. 36096.
   16. How  many grains in 2 pounds, 7 ounces, 10 penny-
weights, and 12 grains of silver plate?           Ans. 15132.
   To bring ounces to pounds, you must divide by 12, because
12 ounces make  a pound;  if you would bring pennyweights
to pounds, you must divide, in the tirst place, by 20; because
20 pennyweights make an ounce;  and then divide these ounces
by 12, to bring them to  pounds.   If grains are given  to
be brought to pounds, divide them  by 24, in  the first place,
because 24 grains make  a pennyweight; then  bring these to
ounces, and these again to pounds, as above directed.
   17-  In 256 ounces, how many pounds?
                               Ans. 21 lbs., and 4 oz. over.
   18.  In  364 ounces, how many pounds?
                               Ans. 30 lbs., and 4 oz. over.
   19.  In  180 pennyweights how many ounces ?     Ans. 9 oz.
   20.  In  396 pennyweights, how many ounces ?
                            Ans. 19  oz., and 16. dwts. over.
   21.  In  405 grains, how many pennyweights?
                            Ans. 16 dwts., and 21 grs. over.
   22.  In  1142 grains, how many pennyweights ?
                            Ans. 47 dwls., and 14 gr. over.
   23.  In 17280 grains, how many pounds ?        Ans. 3 lbs.
   24.  In 36096 grains, how many pounds ?
                         Ans. 6 /6s., and 3 oz. A dwts. over.
   25.  In 25440 grains, how many pounds ?
                                Ans. 4 lbs., and 5 oz. over. 
52                  INTRODUCTION.
  Questions.   1. What articles are weighed by troy weight?—
2. What are its denominations?—3. Will you repeat the table?
  4. How do you bring, or reduce pounds to ounces ?—Why so ?
  5. How do you bring, or reduce  ounces to pennyweights?—
Why so ?
  6. How do you bring, or reduce  pennyweights to grains ?—
Why so ?
  7. When ounces are given along with pounds, what do you do ?
  8. When pennyweights are given along with ounces, what do
you do ?
  9. When grains are given along with pennyweights, what do
you do ?
  10. When pounds, ounces,  pennyweights, and  grains, are
given to be  brought or reduced to grains, how do you proceed?
  11. How do you bring ounces to pounds ?—Why so}—12. How
do you bring pennyweights to ounces ?—Why so ?—13. How do
you bring grains to pennyweights?—Why so?
  14. How do you bring pennyweights to pounds?—Why so?
  15. How do you bring grains to ounces?—Why so?
  16. How do you bring grains to pounds?—What reason can
you give for the operation ?
            AVOIRDUPOIS WEIGHT.

  By this weight such commodities as are coarse and subject to
waste, and all metals, except gold and silver, are weighed.
  The denominations are, ton, hundredweight, quarter, pound,
ounce, and dram.

                        TABLE.
      16  drams,  (dr) make 1 ounce, oz.
      16 ounces      -     1 pound, lb.
      28 pounds,     -     1 quarter, qr.
        A quarters, or 112 lbs. 1 hundred weight, Cwt.
      20 hundred weight,    1 ton, T.
  573440 dr. = 35840 oz. = 2240 /6s. = 20 Cwt. = 1 T.
                 Questions on the Table.
  1. How many hundred weight are in 4 tons ? —.  Why so ?
—2. How many in 6 tons ? —.   Why so ?—3. How many in 8
tons? —.  Why so?—4. How many in 12 tons ? —. Why so?
  6. How many quarters are in 6 hundred weight ?  —.   Why 
AVOIRDUPOIS WEIGHT.
53
so?—7.  How many in 12? —.   Why so?—8. How many in
16? —. Why so?—9. How many in 22? —. Why so?
  10. How many pounds are in 2 quarters ? —.  Why so ?—
11. How many are in 3 quarters ? —.   Why so ?—12. How
many are in 4 quarters?  —.v Why so?
  13. How many ounces are in 3 pounds?  —.   AVhy so?—
14. How many in 6 pounds ? —.  Why so ?—15. How many in
8 pounds ?  —.  Why so ?—16. How many in 12 pounds ? —.
Why so?—17. How  many in 11 pounds? —.  Why so?
  18. How many drams are in 4 ounces?  —.   Why so ?—
19. How many in 6 ounces?  —.  Why so ?—20. How many
in8ounces? —.  Whyso?—21. How many in lOounces?—,
Why so?—22. How  many in 12 ounces? —.  Whyso?

                 PRACTICAL EXERCISES.
   1. In 6 tons of coal, how many hundred weight ?  Ans. 120.
  2. In 11  tons of hay, how many hundred weight? Ans. 220.
  3. In 5 hundred weight, how many quarters ?      Ans. 20.
  4. In 13  hundred weight, how many quarters?     Ans. 52.
  5. In 3 quarters, how many pounds?            Ans. 84.
  6. In 8 quarters, how many pounds?            Ans. 224.
  7. In 11 quarters, how many pounds ?           Ans. 308.
   8. In 9 pounds, how many ounces ?             Ans. 144.
  9. In 12 pounds, how many ounces?           Ans. 192.
  10. In 17 pounds, how many ounces?           Ans. 282.
  11. In 15 ounces, how many drams?            Ans. 240.
  12. In 9 ounces, how many drams?              Ans. 144.
  13. In 4 tons, 16 cwt. how many cwt. ?           Ans. 96.
  14. In 10 tons, 5 cwt. 3 qrs. how many qrs. ?     Ans. 823.
   Here  you multiply the tons by 20, because 20 cwt.  make  1
ton; and to the product add the  5 cwt., and multiply this sum
by 4, because 4 quarters make 1 cwt., and to the product add
the 3 quarters, and the sum is the number  required.  If there
had been pounds arid  ounces also given, you must have  multi-
plied the quarters by 28, because 28 pounds make one  quarter,
and to the  product, have added the given pounds :  these pounds
you must have multiplied by 16, because 16 ounces  make  1
pound,and have added to the product the given ounces; the sum
would have been the number required.

   15. In 6 cwt., 3.quarters, and 16 pounds;  how many pounds?
                                             Ans. 762.
   16. In 14 quarters, 12 pounds, 10 ounces; how many ounces?
                                            Ans. 6474. 
54                  INTRODUCTION

  17. In 21 pounds, 13 ounces, 5 drams ; how many drams ?
                                              Ans. 5590.
  18. In 2 tons, 8 cwt., 3 qrs., 17 pounds, 11 oz., 12 dr.; how
many drams?                               Ans. 1402300.
  19. In 42 cwt., how many tons? Ans. 2 tons, and 2 cwt. over.
  20. In 19 quarters, how many cwt.?
                               Ans. 4 cwt., and 3 qrs. over.
  21. In 346 pounds, how many quarters ?
                             Ans. 12 qrs., and 10 lbs. over.
  22. In 1476 ounces, how many pounds?
                               Ans. 92 lbs., and 4 oz. over.

  To bring hundredweight to  tons,  you must  divide by 20,
because 20 cwt. make one ton:  to bring quarters to cwt., you
must divide by 4, because 4  qrs. make one cwt.: to bring
pounds  to  quarters, you must divide by 28, because  28 lbs.
make one qr.: to bring ounces to pounds, you must divide by
16, because 16 oz. make one lb., &c.
   If you would bring, for instance, drams to cwt.,  you must first
divide them by 16, which will bring them to ounces : then again,
you must divide by 16, which will bring them to  pounds ; then
again, you must divide by 28, which will bring them to quarters:
and these you must divide by 4 to bring them to cwt., as in the
following examples.

   23. In 47654 drams, how many cwt. ?
                       Ans. 1 cwt., 2 qrs., 18 lbs., 2 oz., 6 dr.
   24. In 65427 oz., how many tons ?
                        Ans. 1 ton, 16 cwt., 2 qr., 1  lb. 3 oz.
   25. In 54621 lbs., how many tons ?
                           Ans. 24 tons, 7 cwt., 2 qr., 21 lb.
   26. In 476 qrs., how  many tons ?       Ans. 5 tons, 19 cwt.

   Questions.   1. What commodities are weighed by avoirdu-
 pois weight ?
   2. What are the  denominations?—3. Will you repeat the
 table ?
   4. How do you bring tons to cwt.?  —. Why so ?—5.  How
 do you bring cwt. to quarters ? —.  Why so ?—6. How do you
 bring quarters to lbs.? —.  Why so  ?
   7. How do you bring pounds to  oz.?  —.   Why so?—
 8. How do you bring ounces to drams ?  —.   Why so ?
   9. How do you bring cwt.,  qrs.,  and lbs. to pounds ? —.
 Why so ? 
              APOTHECARIES WEIGHT.              55

  10.  How do you bring lbs.,  ozs., and drs. to drams ? —.
Why so ?
  11.  Howdoyoubring^cwt.totons? —.  Whyso?—12. How
do you bring qrs. to cwt. ?  —.   Why so ?—13. How do you
bring lbs. to cwt.?  —.   Whyso?—14. How do you bring
drs. to lbs.?  —.   Why so?—15. How do you bring lbs. to
cwt. ? —.   Why so ?—16. How do you bring drs. to grs. ? —.
Why so ?—17. How do you bring ozs. to tons ? —.  Why so ?
          APOTHECARIES WEIGHT.

  By this weight, apothecaries mix their medicines, but buy and
sell by avoirdupois weight.  The pound is  the same with the
pound troy, but the subdivisions are different.
  The denominations  are pound, ounce, dram, scruple, and
grain.
                       TABLE.
         20  grains (gr.) make 1 scruple, .... 3
          3  scruples     =  1 dram,  .....3
          8  drams      =  1 ounce,.....|
         12  ounces      =  1 pound,.....ft»
       5760  gr.=288 scruples=96 dr.=12 oz.=l lb.

                 Questions on the Table.
   1. How many ounces are in 3 pounds ? —.  Why so ?
  2. How many drams are in 7 oz.?  —.  Why so?—3. How
many in  11  oz. ?  —.   Whyso?—4. How many in 13 oz.?
—. Whyso?—5.  How many in 24 oz.? —.  Whyso?
  6. How many  scruples are in  5 drams?  —.   Why so?—
7. How many in 12 drams? —. Why so?—8.  How many in
17 drams? —.  Whyso?
  9. How many grains are in 5 scruples ?   —.   Why so ?—
10. How many grains in 8 scruples? —.  Whyso?—11. How
many in  11?  —.  Whyso?—12. How many in 15? —.
Why so ?
  13. How many grains in 1 scruple and a half?  —. Whyso?
  14. How many scruples in 1 dram and a half?  —. Why so ?
   15. How many drams in 1 oz. and a half? —. Why so?
   16. How many ounces in 1 pound and a half?  —. Why so? 
56                   INTRODUCTION.

                 PRACTICAL EXERCISES.
  1. In 7 pounds, how many ounces?            Ans. 84 oz.
  2. In 34 ounces, how many drams ?          Ans. 272 drs.
  3. In 14 drams, how many scruples ?            Ans. 42 9.
  4. In 13 scruples, how  many grains?         Ans. 260 grs.
  5. In 3 pounds, 5 ounces; how many ounces ?  Ans. 41 oz.
  6. In 11 ounces, 7 drams; how many drams ? Ans. 95 drs.
  7. In 7 drams, 2 scruples; how many scruples ?  Ans. 23 9.
  8. In 2 scruples, 17 grains; how many grains?  Ans. 57grs.
  9. In 3 lbs., 5 ozs., 6 drs.; how many drams ? Ans. 334 drs.
  10. In  11  ozs., 7 drs., 2 scruples, 15 grains; how many
grains ?                                     Ans. 5755 grs.
  11. In 137 ounces, how many pounds ?
                               Ans. 11 lbs. and 7 ozs. over.
  12. In 579 drams, how many ounces?
                               Ans. 72 ozs. and 1 dr. over.
  13. In 462 grs., how many scruples ?
                          Ans. 23 scruples, and 2 grs. over.
  14. In 642 drams, how many lbs. ?
                          Ans. 6 lbs. and 8 ozs., 2 drs. over.
  15. In 7624 grains, how many ounces ?
               Ans. 15 ozs. and 7 drs., 0 scruples, 4 grs. over.

  When pounds, ounces,  drams, scruples, and grains are given
to be brought  to grains; you must multiply the  pounds by
12, and to the product add the ounces ; these multiply by 8, to
bring them to drams, and  to the^product add  the given drams:
this  sum multiply by  3, to bring them to scruples; and to the
product add the given scruples : this sum  multiply by 20, and
to the product add the given grains: the sum  will  be the num-
ber required.
  To bring grains to pounds, the operation is just the reverse.
You must divide  the  given sum  by 20, which will bring the
grains to scruples ; these divide by 3, to bring them to drams:
these again by 8, to bring them to ounces; and these again by
12, to bring those ounces to pounds.

  Questions.  1. By whom is this weight  used?—2.  For
what purpose do they use  it?—3. Do they buy and sell by this
weight ?
  4. What are the denominations ?—5. Will you repeat the
table ?
  6. How  do you bring, or  reduce pounds  to ounces?  —
Why so ? 
LONG MEASURE.
57
7. How do you bring ounces to drams? —.  Why so?
8. How do you,bring scruples to grains? —.  Whyso?
9. How do you bring drams to scruples ?  —.  Why so ?
10.  How do you bring pounds and ounces to ounces?   —.
^hv so ?
                 LONG MEASURE.

  This measure is used for lengths and distances.
  The denominations are, degree, league, mile, furlong, pole
or rod, yard, foot, inch, and barley-corn.

                         TABLE.

      3 barley-corns (6. c.)  make 1 inch, in.
     12 inches   -     -     make 1 foot, ft.
      3 feet     -     -     make 1 yard, yd.
      5£ yards, or 16£ feet  make 1 rod, pole or perch, P.
     40 rods, or 220 vards   make 1 furlong,/ttr.
      8 furlongs, or 176O yards " 1 mile, M.
      3 miles    -     -     make 1 league, L.
     60 geographic miles     make 1 degree, deg.
     69* statute miles  -   make 1 degree, deg.
190180 6. c.=63360 zn.=5280 /*.=1760 yds.=320 rods=
  8fur. = l mile.                          .   ..  . • n  <■
  A hand is 4 inches,  and is used for measuring the height ot

  A fathom is 6 feet, and  is used  for measuring the depth of
water.

                  Questions on the Table.

   1. What  is the difference between a geographic, and a statute

mi2C. Is a geographic mile, therefore,  longer than a statute mile
              F 
58                  INTRODUCTION.
  3. How many geographic miles in 4 degrees ?  —.  Why so ?
—4. How many in 6 degrees ? —.  Why so?—5. How many
in 5 degrees and a half? —.  Why so?
  6. How many furlongs in 3 miles ? —.  Why so ?—7. How
many in 5 miles ? —.  Why so?—8. How many in 8  miles?
—.   Why so ?
  9. How many rods in 3 furlongs ? —. Why  so ?—10. How
many in 5 furlongs ?  —.  Why so ?
  11. How  many yards in two rods ? —.  Whyso?—12. How
many in 3 rods ?  —.   Why so ?—13. How many in 5 rods?
—.   Whyso?
  14. How many feet in 5 yards?  —.  Why  so?—15. How
many in 8 yards? —. Whyso?—16. How many in  12  yards?
—.  Why so ?
  17. How many inches in 4 feet ? —.  Why  so ?—18. How
many in 8  feet? —.  Why so?—19- How many in 11 feet?
—.  Why so?—20. How many in 12 feet? —.  Why so?
  21. How many bailey-corns  in 8 inches? —.  Why so?—
22. How many in 11 inches? —.   Why so?
  23. How many furlongs are in 4 miles and a half?  —. Why
so ?—24. How many in 6 miles and a half? —.  Why so ?
  25. How many inches in 5 feet and a half?   —.  Why so?
—26. How many in 8 feet and a half? —.   Why so?—
  27. How many feet in 4 rods?  —.   Why  so?—28. How
many in 6 rods ? —.  Why so ?
  29. How many rods in 2 miles ? —.  Why  so ?—30. How
many in 4 miles ?  —.  Why so ?

                 PRACTICAL  EXERCISES.

   1. In 6 degrees, how many statute miles ?   Ans. 417 miles.
  2. In 8 degrees, how many geographic miles  ?  Ans.  480 m.
  3. In 30 degrees and a half, how many geographic miles?
                                        Ans. 1830 miles.
  4. In 16 degrees and one fourth, how many geographic miles?
                                          Ans. 975 miles.
  5. In 5 leagues and one third, how many miles ? Ans. 16 m.
  6. In 7 miles, how many furlongs?            Ans. 56 fur.
  7. In 6 miles, how many yards?          Ans. 10560 yds.
  8. In 10 miles and a half, how many furlongs? Ans. 84 fur.
  9. In 56 rods, how many yards ?            Ans. 308 yds.
  10. How many yards in 16 furlongs?        Ans. 3520yds.
  11. How  many feet in 19 yards and a half?  Ans.  58  ft. 6 in.
  12. In 21 feet, how many inches ?             Ans.  252 in. 
LONG MEASURE.
59
 13. In 12 yards and a third, how many feet ?    Ans. 37 feet.
 14. In 11 furlongs and a half, how many rods ? Ans. 460 rods.
 15. In 15 miles and a half, how many yards ? Ans. 27280 yds.
 16. In 12 feet, 5 inches, and 2 barley-corns;  how many bar-
ley-corns ?                                    Ans. 449 b.c.
 17. In 3  miles, 5 furlongs, and 31  rods; how  many rods ?
                                           Ans. 1191 rods.

  When it is required to bring  higher denominations to  lower,
you must multiply, in all cases,  by the number which it takes o
the lower  denomination, to make one of the higher : as  in the
last example, you  multiply the miles  by 8, because it takes 8 of
the lower  denomination (to wit, furlongs) to make one mile.
Again, these are to be reduced, or brought to  rods; you must
therefore multiply by 40, because it. takes 40 of the lower deno-
mination,  (to wit,  rods) to make one  furlong.
   One denomination is said to be higher than another, when it
is of more value, or greater in number or magnitude.  Thus a
dollar is of a higher denomination than a dime,  because it is of
more value; and  a  mile of a higher denomination than a rod,
because it  is greater in length ; and so in all other cases.
  When it is required to bring  lower denominations to higher,
the operation is just the reverse; you  must divide the lower
denomination by  the number which  it takes of that denomina-
tion, to make  one of the higher:   thus,  if inches  are  to be
brought to feet, you must divide by 12, because  it takes 12
inches to make one foot; and so in all other cases.

   18. In 47625 barley-corns, how many feet ?
                               Ans. 1322 ft. 11 inches over.
   19. In 12651 inches, how many yards ?
                              Ans. 351  yds. 1 ft.  3 in. over.
  20. In 5460 yards, how many furlongs ?
                                Ans. 24 fur. 180 yds. over.
  21. In 74210 feet, how many furlongs ?
                            Ans.  112 fur. 96 yds. 2 ft. over.
  22. In 8462 rods, how many miles ?
                               Ans. 26 miles, 142 rods over.
  23. In 532 furlongs, how many leagues ?
                                Ans. 22 L.  and 4 fur. over.
  24. In 1462 miles, how many degrees ?
                              Ans. 24 deg. and 22 M over.
  25. In 47246 yards, how many miles ?
                           Ans. 26 M. and I486 yds. over. 
60                   INTRODUCTION.

  Questions.  J. For what  purpose is long measure used?
  2. What are its denominations ?—3. Will you  repeat the
table ?
  4. How do you bring degrees to miles? —.  Whyso?—
5. How do you bring miles to rods ? —.   Why so ?—6. How
do you bring miles to yards ?' —.   Why so ?—7. How do you
bring leagues to rods ? —.   Why so ?—8.  How do you bring
yards to inches ?  —.   Why so?—9- How do you bring feet to
barley-corns? —.  Why so?—10.  How do you bring miles to
inches?  —.    Why  so?—11. How do you bring fuilongs to
feet ? —.  Why so ?.
  12. How do you bring barley-corns to feet ? —.   Why so ?
  13. How do you bring inches to yards?  —.  Why so?
  14. How do you bring yards to miles ? —.  Why so?
  15. How do you bring furlongs to leagues? —.  Why so?
  16. How do you bring miles to degrees ? —.   Why so ?
  17. How do you bring  feet to degrees?  —.   Whyso?
  18. How doyou bring barley-corns to leagues? —.  Why so?
  19. When higher denominations are to be  brought to  lower,
what do you do ?
  20. When is one denomination said to be higher than another?
  21. When  lower denominations are to be brought to higher,
how do you proceed ?
                CLOTH MEASURE.

  This  is used by merchants and others, to measure cloth,
tapes, &c.
  The denominations are, inch, nail, quarter, yard, and ell.

                        TABLE.

           2\ inches (in.) make 1 nail, na.
           A nails       make 1 quarter of a yard, qr.
           A quarters    make 1 yard, yd.
           3 quarters    make 1 Ell Flemish, E. Fl.
           5 quarters    make 1 Ell English, E. E.
           5 quarters    make 1 El! French, E. Fr.
           2\ quarters   make 1 Ell Hamburg, E. H,
          36 inches=l6 nails=4 quarters=l yard. 
CLOTH MEASURE.                 6l
                 Questions on the Table.
  1. How many quarters in 5 yards ? —.  Why so ?—2.  How
many in 8 yards ? —,   Why so ?—3. How many in 13 yards?
—.  Why so ?
  4. How many quarters in 6 ells English ? —.  Why so ?
  5. How many quarters in 7 ells French ?  —. Why so ?—
6. How many in 12?  —.  Why so?—7-  How many in 15 ?
—.  Why so?—8.  How many in 21 ? —.   Why so?
  9. How many quarters in 7 ells of Hamburg ?  —.  Why so ?
—10. How many in 10? —.   Why so ?—11. How many in
16 ? —.   Why so ?
  12. How many nails in 5 quarters of a yard ? —.   Why so ?
—13. How many in  8 ?- —.  Why so ?—14. How many in
12? —.   Whyso?—15. How many in 22 ? —.   Whyso?
   16. How many nails in 4 yards ? —.   Why so ?—17-  How
many  in  7  yards? —.  Why  so?—18. How many in 12
yards ? —.  Why so ?
   19. How many inches in 6 nails ?  —. Why  so?—20.  How
many in  8 ?  —.   Why so?—21. How many in 12 ?  —.
Why so ?

                  practical exercises.

   1. In 8 Ells Flemish, how many quarters ?     Ans. 24 qrs.
   2. In 19 Ells Flemish ; how many quarters ?  Ans. 57 qrs.
   3. In 11 Ells English, how many quarters?    Ans. 55 qrs.
   4. In 16 Ells English, how many quarters ?    Ans. 80 qrs.
   5. In 12 Ells Hamburg, how many quarters ?  Ans. 30 qrs.
   6. In 15 quarters, how many nails ?           Ans. 60 na.
  7. In 19 quarters, how many nails ?           Ans. 76 na.
   8. In 5 yards, 2 quarters, how many quarters ? Ans. 22 qrs.
   9. In 6 yards, 3 quarters, how many nails ?   Ans. 108 na.
   10. In 12 nails, how many inches ?            Ans. 27 in.
   11. In 15 quarters,  3 nails ; how many nails ?   Ans. 63 na.
   12. In 462 nails, how many quarters  ?
                            Ans. 115 qrs. and 2 na. over.
   13. In 1746 quarters, how many yards?
                            Ans. 436 yds. and  2 qrs. over.
   14. In 1664 nails, how many yards ?        Ans. 104 yds.
   15. In 742 quarters, how many Ells  English ?
                             Ans. 148 E. and  2 qrs.  over.
   16. In 524 quarters, how many Ells French ?
                              Ans.  104 E. and 4 qrs. over.
           f2 
62
INTRODUCTION.
  17- In 375 qrs., how many Ells Flemish ?   Ans. 125 Ells.
  18. In 31415 nails, how many yards ?
                            Ans. 1963 yds. and 7 na. over.
  19. In 7421 nails, how many Ells English ?
                            Ans. 371 Ells, and 1 na. over.

  Questions.   1. For what is this measure  used ?—2. What
are its denominations ?—3. Will you repeat the table ?
  4.  How do you bring  yards to quarters?  —.   Whyso?—
5. How do you bring Ells English loquaiters? —.  Whyso?
  6.  How do  you bring  Ells  Flemish to  quarters ?  —.
Why so ?
  7.  How do you bring Ells French to quarters  ? —. Why so?
  8.  How do  you bring  yards to nails?  —.  Whyso?
  9.  How do you bring Ells English to nails ?  —.   Why so?
  10. How do you bring Ells Flemish to nails ? —.  Why so ?
  11. How do you bring quarters to inches ?  —.   Why so ?
  12. How do you bring  yards to inches ? —.  Whyso?
  13. How do you bring  inches to yards ? —.   Why so ?
  14. How do you bring nails to Ells English ? —.  Whyso?
  15. How do you bring nails to Ells Flemish ? —.  Why so?
  16. How do you bring inches to Ells English? —. Whyso?
                LAND  MEASURE.

  This is used in measuring land, and has respect to length and
breadth.
  The denominations are, yard, perch, rood, and acre.

                       TABLE.

           9 square feet make 1 square yard, yd.
          30^ sq. yards make 1 sq perch, P.
          40 sq. perches make 1 rood, R.
           A roods     make 1 acre, A.
   43560 fcet=4840 yds. = 160 perches=4 roods = l acre.

                Questions on the Table.

  1. How many roods are in 6 acres?  —.  Why so?—2. How
many roods in 12 acres? —.   Why so ?—3,  How many in 15
acres? —.  Why so? 
LAND MEASURE.
63
  4. How many perches are  in 6 roods ? —.   Why so ?—
5. How many in 7 roods ? —.  Why so ?
  6. How many perches are in 3 roods and a half?  —.  Why
so ?—7- How many in 11 roods and a quarter ? —. Why so ?
   8. How many square feet are in 4 sq. yards ? —. Why so?
  9. How many perches  are in  1 acre and 2  roods ?  —.
Why so ?
  10. How many roods are in 120 perches ? —.  Why so ?
  11. How many roods in 175 perches, and how many over?
  12. How many in 212 perches, and how many  over?
  13. How many acres are  in 340 perches, and how many
over ?

                  PRACTICAL EXERCISES.

  1. In 4 acres, 2 roods ; how many roods ?        Ans. 18.
  2. In 7 acres, 3 roods ; how many roods ?         Ans. 31.
  3. In 12 acres, 1 rood; how many roods?         Ans. 49.
  4. In 12 roods, how many perches ?            Ans. 480.
  5. In 15 roods, 20 perches; how many perrhes ?  Ans. 620.
  6. In 7 roods, 35 perches ;  how many perches ?  Ans. 315.
  7- In 7 square yards, how many square feet ?      Ans. 63.
  3. In„13 square yards, how many square feet?    Ans. 117.
  9. In 14 roods, how many acres ?
                              Ans. 3  acres, and 2 R.  over.
  10. In 21 roods, how many acres ? Ans. 5 A. and 1  R.  over.
  11. In 547 perches, how many roods ?
                               Ans. 13 R. and 27 P.  over.
  12. In 617 perches,  how many roods ?
                               Ans. 15 R. and 17 P.  over.
  13. In 1476 perches, how many acres?
                                Ans.  9 A. and 36 P.  over.
  14. In 43560 square feet, how many roods ?        Ans. 4.
  15. In 4 square perchrs, how many y;irds?       Ans. 121.
  16. In 12 square perches, how many yards?      Ans. 363.
  17. In 28 square perches, how many sq. yards  ?  Ans.  847.

  Questions.   1. For what purpose is land measure used ?
  2. What are its denominations ?
  3. Will you  repeat the table ?—4. How do you bring roods
to acres ?
  5. How do you bring  perches to roods?—6.  How do you
bring perches to acres ?
  7. How do you bring square yards to perches ? 
64
INTRODUCTION.
  8. How do you bring acres to roods ?—9. How do you bring
roods to perches?—10. How do you bring perches  to square
yards ?
  11. How do you bring perches to acres?—12.  How do  you
bring square feet to roods ?
  13. How do you bring square feet to  acres?—What  is  the
reason for the operation ?
               LIQUID  MEASURE.

  This is used for  measuring  wine,  brandy,  spirits, beer,
cider, &c.
  The denominations are, gill, pint, quart, and gallon.

                       TABLE.

      4 gills (g.)   -    -    make 1 pint, pt.
      2 pints       -    -    make 1 quart, qt.
      4 quarts     -    -    make 1 gallon, gal.
     3l£ gallons   -    -    make 1 barrel, bar.
     63 gallons     -    -    make 1 hogshead, hhH,
       2  hogsheads, or 126 gal.  make 1 pipe or butt, P. or B.
      A hogsheads, or 252 gal.  make 1 tun, T.
   2016 gills=504 pints=252 quarts=63 gal.=l hhd.

                  Questions on the Table.

  1. How many  hogsheads are in 5 tuns of wine? —.  Why
so?—2.  How  many hogsheads in 11 pipes of brandy? —.
Why so ?
  3. How many gallons are in 3 hogsheads ? —.  Why so ?—
4. How many gallons in 5 hogsheads? —.  Whyso?
  5. How many gallons are in 4  barrels? —.  Whyso?—
6. How many gallons in 6 barrels ? —.  Why so ?
  7. How many quarts  are in 5 gallons?  —.  Why so?—
8. How many in  12  gallons?  —.   Whyso?—9. How many
in 15 gallons?  —.   Whyso?
  10. How  many pints  are in 5 quarts?  —.  Why so?—
11. How many in 16 quarts ? —.   Why so?
  12. How  many  gills  are in 3  quarts?  —.  Whyso?—
13. How many in 13 quarts ? —.   Why so ?
  14. How many gills are in 5 pints? — Why so?—15. How 
LIQUID MEASURE.
65
many in  12 pints?  —.   Why so?—16. How many in 18
pints ? —.  Why so ?
  17. How many tuns are in 24 hogsheads ? —.  Why so ?—
18. How many in 40 hogsheads ? —.   Why so ?
  19. How many pipes are in 16 hogsheads ? —. Whyso?—
20. How many in 24 hog>heads ? —.   Why so ?
  21. How many hogsheads are in 252 gallons? —. Whyso?
  22. How many quarts are in 11 gallons?  —.   Why so ?—
23. How many in 15 gallons ?  —.  Why so ?—24. How many
in 22 gallons ?  —.  Why so?
  25. How many pints are in 7 quarts?  —.  Whyso?—
26. How many in 16 quarts?  —.  Whyso?
  27. How many gills are in 7 pints ?  —. Why so ?—28. How
many in  12 pints?  —.    WThyso?—29. How many in 15
pints ? —.  Why so ?

                 PRACTICAL  EXERCISES.
  1. In 5 hhds. and lOgals. how many gallons ?    Ans.  325.
  2. In 7 tuns and 3 hhds., how many hhds.?       Ans. 31.
  3. In 21 hhds. and 46 gals., how many gals.?   Ans. 1369.
  4. In 29 hhds. how many pipes?
                              Ans. 14 P. and 1 hhds. over.
  5. In 37 gallons, how many  quarts ?             Ans.  148.
  6. In  174 gallons, how many quarts ?            Ans.  696.
  7. In 47 quarts, how many  pinis?           Ans. 94 pints.
  8. In 16 pints, how many gills ?                Ans. 64.
  9. In 8 gallons and 3 quaits, how many pints?    Ans. 70.
  10. In  10 quarts and 1 pint, how many gills?     Ans. 84.
  11. In  15 pints, how many gills ?               Ans. 60.
  12. In  12 pints and 3 gills, how many gills ?      Ans. 51.
  13. In  23 hhds., how many  tuns?
                             Ans. 5 tuns and 3 hhds. over.
  14. In  246 gallons, how many hhds. ?
                               Ans. 3 hhds. 57 gal. over.
  15. In  6 barrels, how many  gallons ?           Ans. 189.
  16. In  10 barrels and 12 gallons, how many gallons?
                                             Ans. 327.
  17. In 627 gallons, how many hhds ?
                               Ans. 9 hhds. 60 gals. over.
  18. In  124 quarts, how many gallons ?           Ans. 31.
  19. In 546 quarts, how many gallons?
                            Ans. 136 gals, and 2 qts. over.
  20. In 742 pints, how many gallons ?
                             Ans. 92 gals, and 6 pts. over. 
66
INTRODUCTION.
  21. In 476 gills, how many pints ?              Ans. 119.
  22. In 1244 gills, how many gallons ?
                        Ans. 38 gals, and 3 qt. 4 gills, over.
  23. In 17456 pints, how many hhds. ?
                           Ans. 34 hhds. and  40 gals. over.
  24. In 7465 quarts, how many tuns ?
                      Ans. 7 T., 1 hhd., 39 gals., 1 qt. over.

  Questions.   1. For what is this measure used ?—What are
its denominations ?—3.  Will you repeal the table ?
  4. How do you bring tuns to hhds. ? —.   Why so ?—5. How
do you bring hhds. to gallons? —.   Why so?
  6. How do you bring barrels to gallons ? —.   Why so ?
  7. How do you bring gallons to quarts ? —.  Why so ?
  8. How do you bring gallons to pints ? --.  Why so?
  9. How do you bring quarts to gills ? —.  Why so ?
  10.  How do you bring gallons to gills ?  —.  Why so ?
  11.  How do you bring hhds. to tuns? —.   Whyso?
  12.  How do you  bring gallons to  hhds.?  —.  Whyso?
  13.  How do you bring quarts to gallons ? —.   Why so ?
  14.  How do you bring pints to gallons ?  —.   Why so ?
  15.  How do you bring gills to  pints? —.   Whyso?
  16.  How do you bring  gills to gallons ?  —  Why so?
  17.  How do you bring quarts to hhds.? —.   Why so?
  18.  How do you bring  gills to hhds. ? —.  Why so ?
                  DRY  MEASURE.

  By this, are measured grain, fruits, salt, &c.
  The denominations are, pint, quart,peck, and bushel.

                        TABLE.

           2 pints (pt.)     make 1 quart, qt.
           8 quarts    -    make 1 peck, P.
           4 pecks    -    make 1 bushel, bu.
          64 pints=32 quarts=4 pecks = l bushel.
  A gallon dry measure, contains 268£ cubic inches,  and is
larger than a gallon wine measure, which contains only 231
cubic inches. 
DRY MEASURE.
67
                 Questions on the Table.
  1. How many pecks are in 3 bushels? —.   Whyso?__
2.  How many in 6 bushels ? —.  Why so ?—3. How many in
11 bushels? —.  Whyso?
  4. How many  quarts  are in 6 pecks ?   —.  Why so?__
5.  How many in 8 pecks ? —.  Why so ?
  6. How many pints are in 8 quarts?  —. Why so?—7. How
many in 14 quarts ? —.   Why so ?
  8. How many bushels are in 12 pecks? —.   Whyso?—
9.  How many in 20 pecks?
  10. How many pecks in 40 quarts? —. Whyso?—11. How
many in 64 quarts ? —.   Why so?
  12. How many quarts are  in 16 pints?  —.   Whyso?__
13. How many in 24 pints?  —.  Why so ?
  14. How many bushels are in 128 quarts ?  —.  Why so?
  15. How many pecks are in 72 pints? —.  Why so?

                  PRACTICAL EXERCISES.
  1. In 15 bushels, how many pecks?              Ans. 60.
  2. In 20 bushels, 3 pecks,  how many pecks?      Ans. 83.
  3. In 17 pecks, how many quarts?              Ans. 136.
  4. In 11 pecks, 7 quarts, how many quarts?      Ans. 95.
  5. In 25 pecks, 5 quarts, how many quarts?      Ans. 205^
  6. In 36 pints, how many quarts?               Ans. 18.
  7. In 42 quarts, how many pints?               Ans. 84.
  8. In 13 pecks, 4 quarts, 1 pint, how many pints?  Ans. 217!
  9. In 32 pecks, how many bushels ?               Ans. 8^
  10. In 56 pocks,  how many bushels?             Ans. 14*.
  11. In 94 pecks,  how many bushels ?
                               Ans.  23 bu., and 2 p. over.
  12. In 49 quarts, how many pecks ? Ans. 6p., and 1 qt. over!
  13. In  174 quarts, how many pecks ?
                               Ans. 21 p., and 6 qts. over.
  14. In 250 quarts, how many bushels?
                            Ans. 7 bu., and 3 p. 2 qt over.
  15. In 156 pints, how many pecks? Ans. 9p.,and 6qts. over.
  16. In 346 quarts, how many bushels?
                          Ans.  10 bu., and 3 p. 6 qts. over.
  Questions.  1.  For what  purpose is this measure used?
  2. What are its denominations ?—3. Will you repeat the table ?
  4. How do you bring bushels to pecks ?—Why so ?—5. How
do you bring pecks to quarts?—Why so?—6.  How do you
bring quarts to pints ? 
68                  INTRODUCTION.

  7. How do you bring bushels and pecks to pecks?—Why so?
  8. How do you bring pecks and quans to quarts ?—Why so?
  9. How do you bring quarts and pints to pints? —Why so?
  10. Howdoyou bring bushels, pecks, and quarts, to quarts?—
Why so ?
  11. How do you  bring pecks to bushels?—Why so?
  12. How do you  bring quarts to bushels ?—Why so ?
  13. How do you  bring pints to pecks?—Why so?
  14. How do you  bring pints to bushels?—Why so?
  15. How many pints  make a bushel?—Why so?
  16. How many cubic inches are in a gallon, dry measure?
  17. What number is in a gallon, wine measure?
                       TIME.

  The denominations are, second,  minute, hour, day, week,
month, and year.  The year is the greatest denomination.  The
others are used to mark the divisions and subdivisions of a year.
                        TABLE.
          60 seconds (sc), make  1 minute, min.
          60 minutes     -       1 hour, h.
          24 hours,      -       1 day, d.
           7 days,       -       1 week, w.
          52 weeks,     -       1 year, y.
          30 days,       -       1 month, m.
          12 months, or 365 days, 1 year, y.

  The names of the  months are, January, 31 days ; February,
28 days, (except  in every fourth year it has 29) ; March, 31
days; April/30 days ;  May, 31 days; June, 30 days ; July, 31
days; August,  31 days;  September,  30  days;  October, 31
davs; November, 30 clays ; and December, 31 days.
  But the  tames of  the months, and the number of days each
one contains, can be more easily remembered from this verse :
           Thirty days have September,
           April, June, and November;
           All the  rest have thirty-one,
           Except  February alone;
           Which has but twenty-eight  days clear,
           And twenty-nine every leap year.
  The solar year contains 365 days, 5 hours, 48 minutes, and 
TIME.
69
55 seconds, and is the exact time in which the earth makes one
revolution around the sun.   This would always keep the seasons
the same.
   The common year consists of 365 days; but being shorter
than the true  year, the seasons  shift, or  fall back 5 hours, 48
minutes, and 55 seconds every year.   To remedy this inconve-
nience, Julius Caesar, some time before the Christian era, directed
that  one day, every fourth  year, should be added to the month
of February, which would make the civil year nearly of the
same length with the solar  year;  hence the year thus corrected
is called the Julian year.   Accordingly the 24th day of Febru-
ary was reckoned  twice (bis),  which being  the sixth  of the
calends of March,  called  in  Latin sextilis, the  day was called
from this circumstance bissextilis, and the year in which it took
place was called bissextile.   The Julian year, for three succes-
sive  ones, contains 365 days, and every  fourth one, 366 days.
This is the year now in common use.
   The year contains 52 weeks, 1 day, and 6 hours; but when
the year is divided into weeks, the day and a quarter is rejected.
   Thirty days are not the twelfth part of a year, but in ordinary
calculations, and in the transactions of life, thirty days are con-
sidered a month : the 5 days  and six  hours being rejected.
   The addition of 6 hours  every year, or  1 day in four years, is
too much ; and, in a series of years, would  affect the seasons, and
occasion a change in the equinoxes;  to remedy this, nearly one
day in every 100 years is to be omitted.

                  Questions on the  Table.

   1. How many months are in 5 years > —. Why so ?—2. How
many in 11 years?  —.  Why so?
  3. How many days  in  4  months? —. Whyso?—4. How
many in 7  months?  —.  Why so?—5. How many  in 10
months?  —. Whyso?
  6.  How many weeks in 2 years ? —. Why so?—7.  How many
in 4 years ? —.  Why so ?
  8.  How many days in  7  weeks? —. Why so?—9.  How
many in  12 weeks? —.  Why  so?—10. How many  in 16
weeks ? —. Why so ?
  11. flow many hours in 3 days? —. Why  so?—12.  How
many in 5  days? —. Why so ?
  13. How many minutes in 5 hours ?  —. Why so?—14. How
many in 9 hours ? —.  Why so ?
             G 
70                 INTRODUCTION.
  15. How many seconds in 3 minutes ? —. Why so?—16. How
many in 7 minutes?  —. Why so?
  17. How many months in 180 days?—. Whyso?—18. How
many in 240 days ? —. Why so ?—How many in 330 days ?—.
Whyso?
  20. How many weeks in 210 days ? —. Why  so?—21. How
many in 112 days?  —.  Why so?—22. How many  in 140
days? —. Whyso?  —.
  23. How many days in 96 hours ? —. Why so ?—24. How
many in 144 hours ? —.  Why so ?
  25. How many minutes in 3 hours? —. Why  so?—26. How
many in 5 hours?  —.  Whyso?—27. How many in 12
hours ?  —.  Why so ?
  28. How many seconds in 4  minutes?  —.  Whyso?—
29.  How many in 7 minutes ? —. Why so ?—30. How many
in 12 minutes ? —.  Why so?

                 PRACTICAL EXERCISES.

  1. In 5 years and 4  months, how many months ?   Ans. 64.
  2. In 7 months and  21 days, how many days ?   Ans. 231.
  3. In 3 years and a  half, how many weeks ?     Ans. 182.
  4. In 8 weeks and 4  days, how  many days ?       Ans. 60.
  5. In 15 weeks and  6 days, how many days ?     Ans. 111.
  6. In 11 days, 4 hours; how many hours?      Ans. 268.
  7. In 17 days, 6 hours ; how many hours ?      Ans. 414.
  8. In 12 hours, how many minutes?            Ans. 720.
  9- In 13 hours, 20 minutes ; how many minutes? Ans. 800.
 10. In 14 minutes, how many seconds ?          Ans. 840.
 11. In 8 minutes, 30 seconds; how many seconds? Ans. 510.
 12. In 5 months, 15 days; how many days ?      Ans. 165.
 13. In 31 weeks, 4 days ; how many days ?      Ans. 221.
 14. Tn 8 days, 7 hours ; how many hours ?        Ans. 199.
 15. In 16 hours, 18 minutes ; how many minutes ? Ans.978.
 16. In 17 min. 14 sec.; how many  seconds ?    Ans. 1034.
 17. In 75 months, how many years ?
                             Ans. 6 years, and 3 m. over.
 18. In 175 days, how many months?
                             Ans. 5 m. and 25 days over.
 19. In 148 days, how many weeks?
                          Ans. 21 weeks, and 1 day over.
 20. In 247 min., how many hours ?  Ans. 4 h. and 7 m. over.
 21. In 374 seconds, how many minutes B
                           Ans. 6 min. and 14 sec. over. 
                        TIME.                       71

22  In 4521 seconds, how many hours ?
                            Ans. 1 h. 15 m. 21 sec. over.
23. In 17461  minutes, how many days ?
                            Ans. 12 d. 3 h. and 1 m. over.
24. In 3471 days, how many months ?
                              Ans. 115 m.and 21 d. over.
25. In 54761 days, how  many years of 365 days ?
                           Ans. 150 years, and lid. over.
26. In 3 m., 5 d., 8 h., 25 m.;  how many  minutes?
           7                             Ans. 137305 m.
  Questions.   1. What are the denominations of time ?
  2. For what purpose are they used ?—3. Will you repeat the

3 4? What are the names of the  months, and the number of
days each one contains ?
  5  What is the length of the solar year ?—6. Is this the true
year ?—7.  Would it always keep the same  seasons  to the same
days of the year?                             .
  8. How  many days does the common year contain r"
  7. Did this difference  between the solar and common years
cause any inconvenience?                             .
  8  What was that inconvenience ?—9. Who corrected it in a
great degree ?—10. What did Julius Caesar order to  be: done ?
   11. What  was the year called, in which February had  29
days, or the 24th dayr, was reckoned twice ?
   12  From what circumstance was it so called r
   13. What  is  the year thus corrected, called ?—14. Do we
 use the  same ?                                  .   .   ,
   15  Is it exactly correct?—16.  Is the correction, that is, the
 addition of a  day in 4 years, two much or two little ?
   17. What is necessary to be done in order to keep the  same
 seasons to the same days of the year ?
   18  Are 52 weeks exactly a year?—19. How much do they
 fall short of it?—20.  Are 30 days  exactly a month or the 12th
 part  of a year?—21. How much do 12 such months fall short
 of a true year ?                                  .     5
   22. How do you bring years to months ?  —  Why so i
   23. How  do you bring months to days ? —.  Whyso:?
   24  How do you  bring weeks  to hours?  —.  Whyso."
   25. How  do you  bring hours to  seconds ?  —.Why  so .
   26. How do you bring days to  minutes ? —.   W ny so r
   27. How  do you bring months to years ? —   Why so.
   28.' How do you bring days to years ? —. Why so :
1 
72
INTRODUCTION.
29. How do you bring weeks to years ? —.  Why so ?
30. How do you bring days to months? —.  Why so ?
31. How do you bring hours to days ? —.  Whyso?
32. How do you bring minutes to hours ? —.  Why so ?
33. How do you bring seconds to hours ? —.   Why so ?
34. How do you bring hours to months? —.  Why so?
35. How do you bring seconds to days? —. Whyso?
36. How do you bring minutes to years ? —. Why so ?
       MOTION, OR  CIRCLE MEASURE.

  This measure is  used by astronomers and navigators.   Its
denominations are second, minute, degree, and sign.

                      TABLE.

         60 seconds (")        make 1 minute,'
         60 minutes      -    make 1 degree, °
         30 degrees       -    make 1 sign, sig.
         12 signs, or 360 degrees make I great circle.

                 Questions on the  Table.

  1. How many degrees are in 6  signs?  —.  Whyso?—
2. How many in 9  signs? —.  Whyso?—3. How many in
11 signs ? —.  Why so ?
  4. How many minutes are in 4 degrees?  —.   Why so?—
5. How many in 7 degrees ? —.  Whyso ?—6. How many in
12 degrees ? —.  Why so ?
  7. How many seconds are in 5 minutes?  —.  Why so?—
8. How many in 8 minutes ?  —. Why so ?—9. How many in
12 minutes ?  —.  Why so ?
  10.  How many signs are  in 90 degrees ?  —.  Why so ?—
11. How  many in 120  degrees ? —.   Why so ?—12. How
many in 240 degrees ? —.   Why so  ?
  13. How many degrees are in 180  minutes?  —.   Why
s0?—14.   How many degrees in 240 minutes ? —.  Why so?
15. How  many in 300 minutes? —.  Why so ?
   16. How many minutes are in 240 seconds ? —.  Why |?? 
                   OWB ,  _4
            MOTION, OR  CIRCLE MEASURE.           73

17. How many in  360 seconds?  —.   Why so ?—18. How
many in 480 seconds ? —.  Why so ?—19. How many in 960
seconds ? —.  Why so ?

                 PRACTICAL EXERCISES.

  1. In  5 signs and7 degrees,how many degrees?   Ans. 157.
  2. In  11 signs, 21 degrees;  how many degrees?   Ans. 351.
  S. In  15 degrees, 6 minutes; how many minutes ? Ans. 906.
  4. In  22 degrees, 12 minutes; how many minutes ?
                                             Ans. 1332.
  5. In  45 degrees, how many seconds ?        Ans. 162000.
  6. In  115 degrees, how many minutes ?         Ans. 6900.
  7. In  72 minutes, how many seconds ?          Ans. 4320.
  8. In  45 degrees, 8 minutes ; how many seconds ?
                                           Ans. 162480.
  9- In  17 minutes, 12 seconds; how many seconds?
                                             Ans. 1032.
  10. In 125 degrees, how many signs?
                              Ans. 4 sig. and 5 deg. over.
  11. In 247 degrees, how many signs ?
                              Ans. 8 sig. and 7 deg. over.
  12. In 462 minutes, how  many degrees ?
                                 Ans. 7 deg. and 42' over.
  13. In 7941  minutes, how many degrees ?
                                 Ans. 132 deg., 21' over.
  14. In 1246  seconds, how many minutes ?
                                 Ans. 20',  and 46" over.
  15. In 5472  seconds, how many minutes?
                                 Ans. 91',  and 12" over.
  16. In 42161 seconds, how many degrees ?
                              Ans. 11 deg., 42', 41" over.
  17. In 3145 minutes, how many signs ?
                            Ans. 1 sig., 22 deg., 25' over.
  18. In 4721  minutes, how many signs?
                            Ans. 2 sig., 18 deg., 41' over.

  Questions.  1. By whom is circle measure used ?—2. What
are its denominations?—3.  Will you repeat the table ?
  4. How do you bring signs to degrees ? —.  Why  so ?
  5. How do you bring degrees to minutes ? —.   Why so ?
  6. How do you bring minutes to seconds ? —.  Why so?
  7. How do you bring signs to  minutes ?  —.  Why so ?
            g2
;•**■   • 
74                 INTRODUCTION.
 8. How do you bring degrees to seconds?  —.   Whyso?
 9. How do you bring minutes to seconds ?  —.   Why so ?
10. How do you  bring degrees to signs ?  —.  Why so ?
11. How do you bring minutes to signs? —.   Whyso?
12. How do you bring seconds to minutes ? —.  Why so ?
13. How do you bring seconds to degrees ? —.  Whyso?
14. How do you  bring minutes to degrees ? —.   Why  so ?
END OF THE INTRODUCTION 
ARITHMETIC.
  Arithmetic is that  part  of Mathematics, which treats of
numbers, explains their properties, and shows how to apply
them to the business of life.
  All the operations of arithmetic are performed by the help of
these characters or figures; 1, 2,3, A, 5, 6,7,8, 9, and 0, cypher
or nought.
  Numeration, Addition, Subtraction, Multiplication, and Divi-
sion, are its primary rules.
                    NUMERATION.

   Numeration is that part of Arithmetic, which treats of the
proper arrangement  of figures, to express  any given number,
and the giving* to each figure its proper value.
   Every figure has an absolute and relative value.   When it
stands alone, it represents so many units or ones of any thing.
This  is its  absolute or simple value.   When it is connected
with one or  more figures, its value is increased.   This is called
its relative or local value.   In every place  from  the right hand
toward the left, it increases in a tenfold proportion ; that is, any
figure represents ten times as  many units in the second place,
as if it stood in  the first place; a hundred times as many in the
third place, as if it stood in the first;  and  ten times as many,
as if it stood in the ten's place;  and so in any other places, as
may be seen in  the following Table. 
76                   ARITHMETIC.
3 2	1	
shc*		
c rt	B	
3 =L	sv	
	en"	
fB ^o T3		
	S3	
	CD 1	=One.
S 2	1	=Twenty-one.
3 2	1 .	=Three hundred and twenty one.
3 2	1 =	=Four thousand 321.
3 2	1 :	= Fifty-four thousand 321.
3 2	1 :	= Six hundred and 54 thousand32l.
3 2	1 :	=Seven millions 654 thousand 321.
3 2	1 =	=87 millions 654 thousand 321.
3 2	1 :	=987 millions 654 thousand 321.
'.  Nine figures or places of figures are sufficient  for common
purposes; but sometimes  it may be necessary to  express a
greater number, as in the following  table, which extends to
thirty places of figures.
  Numbers are also expressed by letters; and hence are called
numeral letters, or Roman numbers.
12   3   4   5  6    7    8    9   10  20    30   40
I. II. III. IV. V. VI.  VII.  VIII. IX.  X. XX. XXX. XL.
50  60    70     80     90  100 500 1000     1830
L. LX. LXX. LXXX. XC.  C.  D.   M. MDCCCXXX.

  A letter of less value standing before one of greater value, is
to be subtracted from it; but  placed after it, is  to be added to
it.  Hence, by combining these letters, various numbers may
be expressed.
9 8 7    6 5 4
 ►-* r-i i>
 O © S

 §i§

 o b -a
 s, » ST
ro^-«
 — P
 to a
 a n
    7
  8 7
9 8 7
M h-l _J
o o a


s- o g
O C B
b u a
W » rv«
pj 3 w
B D-T3
Rr«*ST
C9 O
5
5
5
5
5 
numeratR>n.                   77
                 Questions on the Table.

  1. How many would  4, in the second, or ten's place repre-
sent ? —.  Why so ?
  2. How many would  6 in the third, or hundred's place re-
present ? —.   Why so ?
  3. How many would  3 in the fourth, or thousand's place re-
present ?  —.   Why so ?
  4, How many would  2 in the fifth, or ten thousand's place
represent ? —.  Why so ?
  5. How many would 8 in the sixth place represent? —.
Why so ?
  6. How many would 9 in the seventh place represent? —.
Why so ?
  7. How  many would 5 in the eighth place represent ?  —.
Why so ?
  8. How many would 6 in in the ninth place represent ? —.
Why so?
  9. How many would  5 in the  tenth place represent ? —.
Why so ?
  10. How many would 9 in the eleventh place represent? —.
Why so ?
  11. How many would 7 in the twelfth place represent? —.
Why so ?
  12. How  many would  1  in  the  thirteenth place repre-
sent ? —.  Why so ?
  13. How many would 9  in the first place represent? —.
Why so ?

                  PRACTICAL EXERCISES.

  1. Numerate 4365.   Why is it four thousand, three hundred
and sixty-five ?
  2. Numerate 75400.   Why is it seventy-five thousand, four
hundred ?
  3. Numerate 642100.   Why  is it six hundred  forty-two
thousand, one hundred ?
  4. Numerate 8467521. Why is it so much?
  5. Numerate 97654304. Why is it so much?
  6. Numerate 985406242.  Why is it so much ?
  7. Write down in figures four hundred and seventy-five.
  8. Write down in figures seven thousand and four.
  9. Write down in figures twelve thousand eight hundred and
ninety six.
»*~»r* 
78
ARITHMETIC.
  10. Write down in figures one hundred, fifty-four  thousand,
six hundred, and twenty-four.
  11. Write down in figures one million, one hundred  thou-
sand.
  12. Write down  seven millions, four hundred and forty-five
thousand, three hundred and thirty-nine.
  13. Write down in figures forty-four millions, seven hundred,
thirty-six thousand, eight hundred, and ten.
  14. Write down  in figures five hundred seventy-nine mil-
lions, four hundred twenty-seven thousand, seven hundred, and
twenty-seven.

  The teacher may require the learner to write down any other
numbers, that he may  think  proper.   After being  written
on the slate, let the  pupil  numerate them, and give the reason
for assigning to the figures their respective places.  In this way
he will soon become expert in this part of arithmetic.

  Questions.  1.  Is arithmetic a part of mathematics ?
  2.  What part  is  it ?—3. What do you use  in performing its
operations ?
  4.  What are the primary rules of  arithmetic ?—5. Of what
does numeration treat ?
  6.  Have figures a two-fold value ?—7» What are those values
called ?
  8.  When is a figure said to have a simple, or absolute value?
  9.  When has it a  local, or relative value ?—10.  In what pro-
portion does this increase from the right hand toward the left ?—
17- Will you  explain this more fully  by an example ?
  12. Will you repeat the first table, and give the values  of the
numbers ?
  13. Are numbers sometimes expressed by letters?
  14. What are these letters then called ?—15. What is meant,
when a letter  of less value or signification, is  put before one of
greater value ?
  16. What is meant, when it comes after it ?—17- Will you
repeat the several combinations of letters, and give to each one
its proper value ?
   18. Is this  a very inconvenient way of expressing numbers r 
                       ~"^m- ...
                       ADDITION.                     79


                     ADDITION.

  Addition is the operation of finding the sum of two or more
numbers of the same kind ; and is both simple and compound.
  Simple addition is the operation of finding the sum of two or
more numbers of the same denomination and kind.
  Compound addition is the operation of finding the sum of two
or more numbers of the  same kind, but of different denomina-
tions.
  This character  +  placed between two figures or numbers
implies that they are to  be added together,  thus 4+2 are  6;
that is, 4 added to 2 are 6.
  Note.—For directions for placing numbers to be added, and
for performing the operation, see Introduction, under Addition.
EXAMPLES.
yards.	acres.	inches.	rods.
14567	84236	72154	942315
35823	27143	12673	231423
46254	34555	83426	654272
21024	46623	30237	342734
52433	21352	77434 days.	542735
			
minutes.	seconds.		gallons.
846217	274216	572461	2164534
300304	132123	982705	1254317
299177	725462	212471	3613512
411311	402703	240613	4000251
245124	275461	352417	2546503
431542	123125	641243	1310742
  Addition may be proved by beginning at the top, and adding
downward; and, if the result is the same as when you added up-
ward, the operation is correctly performed.  Or, separate the
upper line of figures, and  add together the remainder.   Then
add the upper line  to the  sum of the remainder; and if they
equal the sum of the whole, then the operation is correctly per-
formed.   For the whole of any sum or number must be equal to
all its parts. 
80
ARITHMETIC.
                   PRACTICAL EXERCISES.

   1. What will be the sum of 54765, 3214, 721,  8970, and
92115?                                      Ans. 159785.
   2. What  will be the sum  of  147462, 90042, 7475, 8210,
72143  and 9214 ?                             Ans. 329546.
   3. What  will be the sum  of  80462, 37472, 3164,98762,
21040, and 376421 ?                          Ans. 617321.
   4. What will be the sum of 1468, 457, 321, 58469, 30000,
and 312?                                     Ans. 91027.
   5. A merchant  made deposits in the bank of the United
States, of the following sums, to wit: 4765 dollars, 2146 dollars,
1041 dollars, and 347 dollars; how much has he in the bank?
                                            Ans. 8299 dol.
   6. A merchant bought 476 barrels of flour for 2474 dollars,
226 barrels for 1116 dollars, 540 barrels for 2526 dollars: how
many  barrels of flour has he  in all; and  how much did it cost
him ?     Ans. 1242 barrels  in all.—$6116 cost of the whole.
   7- A merchant retiring from business has the several items of
property,  to  wit:—In cash 14740 d-ols. 25 cts., in  real estate
27460  dols. 50 cts., in bank stock 10474 dols., and in notes and
bonds  20475, dols. 75 cts.; how much property has he in the
whole ?                            Ans. 73150 dols. 50 cts.
   8. A merchant bought at public sale, the following quantities
of dry  goods, to wit:  746 yards  of broad-cloth, 1749 yards of
muslin, 542  yards of linen, 2164 yards of flannel, and 175 yards
of silks;  how many yards did  he purchase in the whole ?
                                           Ans.  5376yds.
   9. A farmer in a favourable season raised 2146 bushels of
wheat, 4746 bushels  of indian corn, 785 bushels of rye, and
940  bushels  of oats;  how many bushels did he raise in the
whole?                                        Ans. 8617.

   Questions.  1.  What is addition ?—2. How is it divided ?
3. What is  the difference between them ?—4. How  do yoa
place the numbers in order to  add them together ?
   5. At which place do you begin to add ?—6.  When the sum
of the units is above 9, what do you do ?
   7. What do you  do  with the tens ?—8. Why do you add only
1 for every ten ?—9.  Is 1 in a superior place, equal to 10 in
the next lower or inferior place ?
   10.  How do you prove addition ? 
SUBTRACTION.
81
                  SUBTRACTION.

  Subtraction is the operation of taking a less number from a
greater, to find the difference between them.
  Subtraction is both simple and compound.  When the num-
bers are of the same denomination and kind, it is called simple
subtraction; when they are of the same kind, but of different
denominations, it is called compound subtraction.
  The greater number is called the minuend, and the less or
lower number is called the subtrahend.   The difference of the
two is called  the remainder.
  Note.—For  directions for placing the numbers, and for per-
forming the operation, see Introduction, under Subtraction.
  A strait line — placed between two figures or numbers im-
plies, that the latter is to be subtracted from the former: thus,
4—2, and 2 remain;  that is,  2  subtracted  from 4,  and 2
remain.	EXAMPLES.	
miles.	rods.	yards.
From 7468912	46215424	896543204
Take 4321641	26854515	749875165
Rem. 3147271
Proof 7468912
        men.
From 8642004
Take 5368123
 bushels.
54678954
12389768
  acres.
146548910
 74895654
Rem.
  Subtraction may be proved by adding the remainder to the
less number, and  if the sum is equal to the greater number or
minuend, the operation is correctly performed.  For the differ-
ence of two numbers, added to the less, must  be  equal to the
greater.
                  PRACTICAL EXERCISES.
  1  What is the difference between 47698 dollars,  and 3241
, ,1   ?                                      Ans. 44457-
  ^'A merchant has in  goods the value of 27460, and  owes
2976 dollars upon them ;  how much has he paidj  ^^ ^
H 
82                   ARITHMETIC.
  3. A gentleman deposited in the bank of the  United States,
2504 dollars, and drew out afterward by a check 1605 dollars;
how much has he remaining in the bank ?     Ans. S99 dolls.
  4. What is the difference between 10000 and 40? Ans. 9960.
  5. What is the difference between 12000 and 101 ?
                                             Ans. 11899.
  6. What is the difference between 150040 and 100101 ?
                                             Ans. 49939.
  7. A lady has an annual income of 1647 dollars; of which
she spends 1010 dollars ; how much does she save ?
                                           Ans. 637 dolls.
  8. William, Duke of Normandy, afterward called the Con-
queror, overcame Harold, the reigning king of England, in
1066; how many years have  passed, reckoning  to  the year
1830?                                         Ans. 764.
  9. A farmer has a tract of land containing 10140 acres, of
which he sold 526 acres ; how many acres has he remaining?
                                         Ans. 9614 acres.

  Questions.  1. What is subtraction ?—2. How is it divid-
ed ?—3. What is the difference between them ?—4. What is the
greater number called ?—5. What is the less called ?—6. What
is the difference between them called ?
  7. How  do you place the numbers ?—8. How do you per-
form the operation ?—9. When the upper figure is  less than the
lower one, what do you do ?
  10. How do you prove subtraction ?—11. What is the prin-
ciple on which this is founded ?
                M ULTIPLIC ATION.

  Multiplication is the operation of finding the sum of any
given number, taken as often as there are units in another given
number.  It is both simple and compound.
  It is simple, when both the numbers are of the same denomi-
nation and kind.   It is compound, when both the numbers are
of the same kind, but of different denominations.
  The number to be multiplied is called the multiplicand : the
number multiplied by is called the multiplier.   The  result of
the operation is called the product. 
MULTIPLICATION.                  83
  The two numbers, taken together, are sometimes called fac-
tors, because by them, the operation is carried on, and the result
produced.
  This character X, placed between two figures or  numbers,
implies that they are to be multiplied together : thus, AX 2 make
8 ; that is, 4 multiplied  by 2 make 8.   It is commonly called
St.  Andrew's cross.
  For the Table, and directions  for placing the  numbers and
performing the operation, see Introduction, under Multiplication.
EXAMPLES.
Multiplicand 746745          546325        8563045
Multiplier  _______6          ______8_        _______9
Product    4480470         4370600        77067405


Multiplicand 567456         346754         4865472
Multiplier   _____11         _____12         ______^_2
Product
  7.  Multiply 5231 by 145.            Product     758495
  8.  Multiply 14063 by 382.            Prod.      5372066
  9.  Multiply 671612 by 114.          Prod.     76563768
 10.  Multiply 7034652 by 144.         Prod.  1012989888
 11.  Multiply 814263 by 75.            Prod-     61069725
 12.  Multiply 7063115 by 96.          Prod.    678059040

  When there are cyphers at the right hand of the multiplier, or
multiplicand, or both,  they  may be  omitted in the operation,
and added to the right hand of the product.
  When  there  is one, or more cyphers between the significant
figures of the multiplier, they are to be omitted, and the product
of the next figure of the multiplier  and multiplicand, put  as
many places higher, as there  are cyphers omitted in the opera-
tion ; as in the following examples.

   13.  Multiply 74621 by 500.               Prod. 37310500

                   Multiplicand 74621
                   Multiplier        500
                   Product    37310500 
84                   ARITHMETIC.

  14. Multiply 65472000 by 65.          Prod. 4255680000

                 Multiplicand 65472000
                 Multiplier      65
                            327360
                           392832
                  Product   4255680000
15.  Multiply 546300 by 4200.          Prod. 2294460000

               Multiplicand 546300
               Multiplier     4200
                          10926
                         21852
             Product     2294460000
16.  Multiply 346754 by 3005.          Prod. 104199577U

               Multiplicand 346754
               Multiplier     3005
                          1733770
                       1040262
1041995770
  17. Multiply 39580 by 2410.           Prod.   95387800
  18. Multiply 190046 by 161500.            30692429000
  19. Multiply 3800920 by 80750.           306924290000
  20. Multiply 3760410 by 4840.             18200384400
  21. Multiply 47001881 by 1140090.      53586374509290
  22. Multiply 62123000 by 130000.        8075990000000

  When the multiplier is  10, 100, 1000,10000, &c. annex the
number of cyphers to the multiplicand, and the operation is
performed.

  23. Multiply 47654 by 10.              Product   476540
  24. Multiply 75476 by 100.                     7547600
  25. Multiply 86427 by 1000.                  86427000

  When the multiplier is equal to the product of any two num-
bers, the operation may sometimes be shortened by multiplying
first by one of those numbers, and the product thence arising by
the other; as  in the following examples. 
Vf -■ -,
                  MULTIPLICATION                  85

26. Multiply 54632 by 24.               Product 1311168
                Multiplicand 54632
                           ______6
                           327792=product by 6.
                          _______4
                          13iTl68== product of 6x4
  Here 24 equals the product of 6 multiplied by 4.  You may
therefore multiply  by 6, and then multiply that product by 4;
or you may multiply by 4, in the first  place, and that product
by 6.   The product will be the same either way.   It will pro-
duce the same result, as  if you multiplied by 24 ;  and so in all
other  cases.
  27.  Multiply 5740632, by 32.        Product   183700224
  28.  Multiply 7063115, by 96.        Prod.     678059040
  29.  Multiply 370731, by 36.        Prod.      "J4**^
  30.  Multiply 43102, by 64.          Prod-       ^758528
  31.  Multiply 12071, by 99-          Prod.        119S029
  Multiplication may be proved in several ways :
  First.   Double the multiplicand, and multiply that sura by
half the multiplier.                                 .
   Secondly.   Take unit from the multiplier, and multiply by
the remainder; to the product add the multiplicand.
   Thirdly.  Divide the product by the multiplier; the quotient
 will be equal to the multiplicand, if the operation is correct.
   Multiplication is the most useful rule in arithmetic.  It per-
 forms, in a short way, what would require many additions.   It
 brings numbers of higher denominations to lower; and, by know-
 ing the value of one thing, we may determine the value  of any
 number of things of the same kind.          / *•'

                   PRACTICAL EXERCISES. ,,
   1  One hundred cents make 1 dollar;  how many cents are
 •  /ok j^iio-oP                            '     Ans.  12500.
 in 125 dollars r                                    ...     .
   2  One thousand mills make 1 dollar; how many mills make
 , .a a iu«J                                  Ans. 146000.
 146 dollarsr                               ,.       ,  _ .c
   3.  Ten dimes  make 1 dollar; how many dimes  make 546
   ..   ?                                        Ans. 5460.
  04?One hundred mills make 1 dime; how many mills make
   **  ,.   -,                                    Ans.  24400.
   5. A person  purchased  147 acres  of land, at  16 dollars.an
 acre; what did the whole come to ?               Ans. m*.
             h2
\ 
86
ARITHMETIC.
  6. A merchant bought 144 pieces of cloth, each containing
29 yards; how many yards were in the whole ?     Ans. 4176.
  7. The floor of a room is 24 feet long and 22 feet wide;  how
many square feet does it contain ?                   Ans. 328.
  8. In a square foot there are 144 inches;  how many inches
are in 25 square feet?                            Ans. 3600.
  9. How many square inches are in 144 square feet?
                                              Ans. 20736.
  10. A  house contains 26 windows, each containing 24 panes
of glass;  how many panes are in the whole?        Ans.  624.
  11. There are 20 shillings in 1 pound; how many are there
in 456 pounds?                                Ans. 9120«.
  12. There are 12  pence in 1 shilling; how  many pence are
in 254 shillings ?                                Ans. 3048.
   13.  There are 24 hours  in  1  day; how many hours are in
151 days?                                      Ans. 3624.
  14. There are 28 pounds in 1 quarter of a Cwt.; how many
pounds are in 124 quarters ?                      Ans. 3472.

  Questions.   1.  What is multiplication?—2.  How  is it
divided ?—3. What is the difference between them ?
  4. What is the number to be multiplied, called ?—5. What is
the number  by  which you multiply, called ?—6. What is the
result of  the operation, called?—7-  What  are the two given
numbers, taken together, sometimes called ?—8. Why are they
called factors ?
  9. What does St. Andrew's cross, placed between two num-
bers, imply ?
  10.  How do you place the numbers for multiplication ?
  11. At what place do you begin to multiply?
  12. When the product of the multiplier, and any figure of the
multiplicand is greater than 9, what do you do?
  13. Why do  you  add only the number of tens to the next
higher place ?
  14. When there are cyphers at the right hand  of the multi-
plier, what do you do with them ?
   15.  If there are  cyphers at the right  hand of the multiplier
and multiplicand, what do you do with them ?
  16. When there is one, or more cyphers between the signi-
ficant figures of the multiplier, what do you do ?
   17. When you multiply by 10, 100, 1000, &c, how is the
operation performed ?
   18.  When the multiplier  is the exact product  of any two
numbers, how may the operation be performed? 
**m
DIVISION
87
  19. Is multiplication a very useful rule ?—20. Why is it so
useful ?
                      DIVISION.

  Division is the operation of finding how often one number is
contained in another.   It is the reverse of multiplication.
  Division is both simple and compound.  It is simple, when
the dividend is a  whole number, or integer: It is compound,
when the dividend consists of two or more denominations.
  The number to be divided, is called the dividend.
  The number by which you divide, is called the divisor.
  The number of times the  divisor  is  contained in the divi-
dend, is called the quotient.
  The remainder, if there is  any, must always be less than the
divisor, and of the same denomination with the dividend.
   Division is also  divided into short and  long.  It is short,
when the divisor does not exceed 12: It is long, when it  is
greater than 12.                                 .
   This character -~  placed between two numbers implies that
the former number is to be  divided  by the latter, thus : 4-^-2
signifies that the 4 is to be divided by the 2.
   For direction in placing the terms, and for proceeding in the
operation, see Introduction, under Division.
           miles.
 Divisor 4)654324
Quotient  163581
EXAMPLES.
      rods.
   6)8574684
     14291U
             yards.
Divisor 10)17869400
Quotient     1736940


     minutes.
12)9476521764
  gallons.
9)7468542
    seconds.
11)546215747
    cents.
8)146237456
   18279682
      mills.
11)564521705
     days.
12)462174684
  The operation may be proved by multiplying the quotient by
the divisor, and adding the remainder, if thrre is any, to the pro-
duct.  If the work is correct, it will equal the dividend.
\ 
88
ARITHMETIC.
  When the divisor i$ more than 12, and not greater than 144;
and, at the same time, is the exact product of 2 numbers, the
operation may be performed by dividing, first, by one of those
numbers,  and then dividing  this quotient  by the other.   But
observe,  if there is a remainder after the last division,  it must
be multiplied by the first divisor, and to the product add the
first remainder, if there is any; the sum will be the true remain-
der. But, if there  is a  remainder after the first division,  and
none after the last, that is the true remainder ; as in the follow-
ing examples. :

  10. Divide 7463521, by  18.  Quotient  414640, and 1  rem.
            6)7463521
            3)1243920 + 1       Here 6 and 3  are the factors,
               414640                 for 6x3 = 18.

   11. Divide 734071, by 72.    Quotient  10195, and 31 rem.
          9)734071
          8) 81563+4 1st. rem.
             10195 + 3x9=27+4=31 true rem.
   Here 9 and 8 are the factors, for 9x8=72.

   Here after the last division, 3 remain, which you multiply by
9 the first divisor,  and to the product add 4, the remainder after
the first division; the sum  is  31, the true  remainder.  In this
manner, the operation, in some cases, may be  considerably
abbreviated.
   12. Divide 20208, by 48.                   Quotient, 421.
   13. Divide 57384659, by 144.   Quotient, 398504, 83 rem.
   14.  Divide 43737, by 21.          Quotient, 2082, 15 rem.
   15. Divide 5704392, by 108.      Quotient,  52818, 48  rem.

   When there are cyphers at the right hand of  the divisor, the
operation may be shortened by separating or cutting them off;
but, observe to separate or cut off as many figures fiom the right
hand of the dividend. These must be  annexed to the remainder
 after division, if there is any; but if there is none, the figures so
 cut off are the true remainder.
   When the divisor is  10, 100, 1000, &c, the operation  may
 be performed, simply by separating the cyphers, and cutting off
 from the dividend  as many  figures, as there  are cyphers so sepa-
 rated ;  the  figures so cut off, will be  the remainder, and the
 others will be the  quotient; as in the following examples: 
    ^"fam^B...

                       DIVISION.                     89

  16. Divide 746524, by 20.     Quotient, 37326, and 4 rem.

                 2,0)74652,4 rem.
                     37326   quotient.

  17. Divide 6543201, by 400.  Quotient, 16358, and 01 rem.

                  4,00)65432,01
                      16358+01 rem.

  18.  Divide 4560700 by 2140. Quotient, 2131, and 360 rem.
  19.  Divide 783567  by 2100.  Quotient, 373, and 267 rem-.
  20. Divide 137000 by 1600.   Quotient, 85, and 1000 rem.
  21.  Divide 4765431 by 5100. Quoiient, 934, and 2031 rem.
  22.  Divide 47652 by 10.        Quotient,  4765, and 2 rem.
  23.  Divide 24675 by 100.       Quotient, 246,  and 75 rem.
  24.  Divide 5798642* by 1000. Quotient, 5798, and 642 rem.
  25.  Divide 14765216 by 3000.  Quotient, 4921, and 2216
remain.
  26.  Divide 3456745 by 4625. Quotient, 747, and 1870 rem.

             4625)3456745(747 quotient.
                   32375'-
                    21924
                    18500
34245
32375
                      1870  rem.

  Having placed your terms for operation, find how often the
divisor is contained in an equal number of figures of the divi-
dend, taken from the left hand, and put it in the quotient; but
if they are less than the divisor,  as in this example, you must
take another figure from the dividend, and put in the quotient
a figure, expressing the number of times the divisor is contained
in that portion of the dividend.  Multiply the divisor by this
figure, and place the product under that part of the dividend so
set  apart.  Subtract this from  the dividend, and to the remain-
der bring down the  next figure  of the  dividend.  This now
becomes the dividend  for  the  next quotient figure.   Find how
often  the  divisor  is  contained in it, and put  the figure to the
right  hand of  the one already in the quoiient.   Multiply the
divisor by the  last quotient figure, and place the product under
the dividend :  find the remainder, and to this, bring down the 
90
ARITHMETIC.
following figure of the dividend; and so proceed till the opera-
tion is finished.
  But, if the  remainder should  be so small, that,  when the
figure from the dividend Ls annexed to it, it will not contain the
divisor, you must put a cypher in the quotient, and bring down
the following figure of the dividend; and, if with this annexed,
it will not contain the  divisor, as may sometimes be the case,
you must put another cypher in the quoiient, and so continue to
do, until it will contain the divisor,  if there are so many figures
remaining  in the dividend ; and, if  there are not, this sum will
be the remainder.

   27. Divide  63125 by 123.           Quot. 513, rem. 26
   28. Divide 5374602 by 671.        Quot. 8009, rem. 5G9
   29- Divide 9736205 by 2507.       Quot- 3883, rem. 1524
   30. Divide 23470525 by 6425.      Quot. 3653
   31. Divide 36737660 by 153694.   Quot. 2315
   32. Divide 101442075 by 4025.    Quot. 25203

                   PRACTICAL exercises.

   1.  The  sum  of 5500 dollars  is  to be  divided among 125
 persons;  how much will each one have ?       Ans. 44 dollars.
   2.  How many shares  of stock  of the  Bank  of the United
 States, can be  purchased with 6552 dollars, at 117 dollars a
 share ?                                            Ans. 56.
   3.  If 45980 pounds of bread be distributed among 2420 per-
 sons  ; how much will be the share of each ?       Ans. 19 lb.
   4. A farmer bought a plantation containing  364  acres for
 8736 dollars; what was the price by the acre ? Ans. 24 dollars.
   5. The earth revolves  on its axis once in 24 hours; how far
 will any point of the equator be carried in an hour ; the circum-
 ference being estimated at 25000 miles ?     Ans. I04l| miles.
   6. How many generations have past since the birth of Christ
 to 1830, allowing 30 years to a generation ?          Ans. 6l.
   7. W'hat number being multiplied by 7969, the product will
 be 1864746?                                     Ans. 234.
    8. A Virginia planter sold  56 hogsheads of tobacco for 6496
 dollars; what did it bring him by the hogshead on an average?
                                           Ans. 116 dollars.
    9. A merchant bought 4104 yards of cloth; how many pieces
  would there be of 36 yards each ?                  Ans. 114.
    10.  Suppose  the distance  to Liverpool in  England to be
  3496 miles, and a ship leaving  Philadelphia arrives in  Liver- 
                  COMPOUND ADDITION.               91

pool in 23 days; how many miles must she  sail  one day with
another?                                  Ans. 152 miles.

   Questions.  1. What is division?—2. Is it  the reverse of
multiplication ?—3. How is it divided ?—4. Has it any other
division ?—5. When is it said  to be short?—6. When is it said
to be long?—7.  When is it said to be simple?—8. When is it
said to be compound ?
   9. What is the number to be divided, called?—What is the
number by which you divide, called ?—11. What is the number
of times the divisor is contained in the dividend, called ?—12. If
there is a remainder after division, is it of the  same denomination
with the dividend ? —13. Is it also less than  the divisor ?
   14. What character  is  that, placed between two numbers,
 implies that the  former is to be divided by the latter?
   15. How is the operation proved?
   16. How  do you proceed when the divisor is less than 12 ?
    17. Will you explain the operation by an example?
    18. When the divisor  is  more than 12, but less than  144,
 and is, at the same time, equal to the product of  2 numbers,
 how may the operation be performed ?
    19.  Will you explain this  by an example?
    20.  When there are cyphers on the right hand of the divisor,
 how may the work be abridged ?
    21.  Will  you explain this  by an example ?
    22.  When the  divisor  is  10,  100, 100, &c. how may the
  work be shortened ?—23. Will you explain  this  by an example?
    24.  What are the figures so cut off from  the dividend?
    25!  When the divisor is more than 12, and is not the product
  of 2 numbers, nor  has cyphers on its right hand, how do you
  proceed  in  the operation ?—26. Will you explain this by an
  example?
            COMPOUND ADDITION.

  Compound Addition is the operation of finding the sum of
two or more numbers of the same kind, but of different denomi-
nations.                                                 ,
  Place the numbers so that the same denominations may stand
under each other.  First add together the right hand column,
or column of the lowest denomination, and divide the sum by
such number, as it takes of that denomination  to  make one ol 
92                      ARITHMETIC.

the next higher; set down the remainder, if there is any, under
the said  column, and add the quotient of  such  division to the
following column, or column  of the next higher denomination;
and so proceed till the operation is finished.
  The work may be proved, in  all cases, as  in addition of
whole numbers, or integers.
                FEDERAL CURRENCY.

  The addition of this currency is, in  every respect, the same
as addition of whole numbers or integers.   The denominations
increase from the lowest to the highest, in a tenfold proportion,
and decrease from the highest to the lowest  in the same propor-
tion ;  that is, ten of every lower denomination make one of the
next higher, and consequently one of  every higher  makes ten
of the next lower.
  For the table, and  its  application, see Introduction, under
Federal Money.
D.	c.	m.	E.	D.	d.	c.	D.	d.	c.	m.
46	25	4	12	4	5	3	1472	7	3	7
54	37	5	16	3	2	4	2141	9	8	4
62	50	2	20	2	7	5	2374	6	3	8
17	46	3	31	4	5	3	1027	3	5	2
24	16	7	51	9	6	7	9423	8	7	4
204	76	l	132	4	7	2	16440	5	8	5
  If mills are given also, you add them together, and because 10
mills make 1 cent, you set down for mills all over  even tens, and
add  these  to the column of cents: you find the sum of  this
column, and set down for cents all over even tens, and add them
to the column of dimes, because  10 cents make 1 dime.  You
add up the column of dimes, and set down for dimes all over
even tens, because 10 dimes make 1 dollar, and add these to the
column of dollars.  If eagles are also giving, you must add up
the  column of dollars, and set down for dollars all over even
tens, and add them to the column of eagles, because 10 dollars
make 1 eagle. But in business, it is usual to consider eagles and
dollars,  all as dollars, and dimes and cents, all as cents; as in
the first example. 
FEDERAL CURRENCY.
93
                   PRACTICAL EXERCISES.

   1. A merchant bought, September 14th, goods amounting to
1246 dollars, 50 cents; Oct. 12th, he bought to the amount
of 1602 dollars, 75  cents; Nov. 2nd, he bought goods to the
amount of 742 dollars, 87£ cents, and Dec. 1st, to the amount
of 1799 dollars, 37J cents; how much was the amount of his
several purchases ?                    Ans.  $5391,50 cents.
   2. A merchant sold 4 lots of goods ; the 1st, for 1012 dollars,
27 cents, and 5 mills; the 2d, for 1250 dollars, 75 cents; the
3d, for 741 dollars,  87 cents, and 5 mills; and the 4th, for 549
dollars, 62 cents, and 5 mills; what did the whole amount  to?
                                         Ans. $3554,52,5.
   3. Four several lots of ground were sold in Philadelphia for
the following sums :  No. 1, for 1574 dollars, 87£ cents; No. 2, for
1241 dollars, 12 J cents, No. 3, for 1165  dollars, 62£ cents, and
No. 4, for 965 dollars, 37£ cents ; for how  much did they all
sell ?                                      Ans. $4947,00.
   4. A lady of fortune made the following deposits in the  bank
of the  United States : At one time, 742 dollars, 50 cents ; at
another time,  1112 dollars, 75 cents;   at a third time, 549
dollars, 37£ cents;  at a fourth,  826 dollars,  37£ cents;  how
much has she in bank ?                      Ans. $3231,00.
   5. A gentleman has the following property : a house valued
at 5216 dollars, 87£ cents; a farm valued at 1650 dollars, 37\
cents;  bank stock to the amount of 7466 dollars, and 25 cents;
and cash 5216  dollars, 62\  cents;  what is the sum of all his
estate ?                                 Ans. $19550,12£.
   6. A person deceased left to his 4 daughters, the following sums:
to the oldest, 2425 dollars, 25 cents; to  the next, 2174 dollars,
37J  cents ;  to the third  he left a house  valued at 2725 dollars,
50 cents; and to the youngest, whose education was not finished,
the sum  of  3216 dollars, 50 cents;  how much did  the whole
amount to?                              Ans. $10541,62^.
   7. A person  has  due to him the following  sums of money,
(viz.) 574 dollars, 06 cents, 2 mills, and 5 tenths of a mill; 425
dollars, 25 cents, and 4  mills ; 175 dollars, 76 cents, 4 mills;
341 dollars,  87  cents,  5 mills;  1021 dollars, 12 cents, 5 mills;
how much is due to  him in all ?           Ans. $2538,08,0,5.

    SUBTRACTION OF FEDERAL CURRENCY.
   The operation  is the same as subtraction of integers, or whole
numbers.
           I
* 
94                    ARITHMETIC.
EXAMPLES.
       D.   c.  m.         D.   c.   m.       D.   c.  m.
From 1746  46   5       1216  37  5        746  62  5
Take 1278  52   4        718  94  l        379  83  2
Rem.
       D.    c.  m.        D.    c.  m.        D.    c.  m.
From 1946   50   0       8476   12  5       5476  75  0
Take  984   62   5       3792   16  8       2593  48  2
Rem.
PRACTICAL EXERCISES.
  1. A person borrowed the sum of 1647 dollars, 25 cents, and
paid 1141 dollars, 12,5 cents; how much does he  still owe?
                                          Ans. $506,12,5.
  2. A merchant purchased goods amounting to 2416 dollars,
87,5 cents, and paid in hand 748 dollars, 62,5 cents; how much
does he owe ?                              Ans. $1668,25.
  3. A gentleman purchased a farm for 7416 dollars,  50 cents,
and paid in hand 2746 dollars, 18,7,5 cents; how much remains
unpaid ?                                Ans. $4670,31,2,5.
  4. A lady has an annual income of 1226 dollars,  80 cents,
and her expenses are 941  dollars, 50 cents ; how much less are
her expenses  than her income?                 Ans.  $285,30.
  5. A young lady had left her by her father the  sum of 6427
dollars, 50 cents, but her education is  estimated  to cost 1189
dollars, 37 cents, 5 mills ;  how much will she have to receive,
when she arrives at the age of 18 years?     Ans. $5238,12,5.
  6. A gamester began to play cards with the sum of 1749 dol-
lars , 62,5 cents, and after  playing 15 games found that he had
onlv 916  dollars,  25 cents remaining ; how much  had he lost?
   '                                       Ans. $833,37,5.
  7. A  lady went a  shopping with the sum of 55  dollars,
55 5 cents;  and her purchases amounted to 49 dollars, 56,2,5
cents ; how much had  she remaining ?        Ans. $5,99,2,5.
   8. Add together the following  sums: 149  dollars, 01 cent;
 75 dollars, 75,5 cents ; 114 dollars, 25 cents; 137 dollars 37,5
 cents- 87,5  cents, and 1144 dollars, 50 cents;  and from the
 amount subtract 1047 dollars, 75 cents;  what will remain ?
                                           Ans. $574,01,5. 
FEDERAL CURRENCY.
95
  9. A merchant purchased a lot of goods for 1894 dollars,
62,5 cents, and sold the same without  delay  for  196l dollars,
55 cents; how much did he gain by the bargain ?
                                            Ans. $66,92,5.
  10.  A gentleman's income is 969 dollars, 80,5 cents a year,
and his expenses are 627  dollars,  87,5 cents; how much does
he save of his income ?                        Ans. $341,93.
  11.  What is the difference between ten dollars, ten dimes, and
ten cents ;  and nine dimes and one cent ?        Ans. $10,19-

   MULTIPLICATION OF  FEDERAL  CURRENCY.

   The operation is the same as multiplication of whole numbers.
   When a sum of money is multiplied by an integer; separate
 the product into dollars, cents, and mills, or into dollars and
 cents, (as the  case may be,)  by pointing off the first place on the
 right hand for mills (if any) and  the two next places for cents :
 the remainder will be dollars.   Or, when any given  number is
 multiplied by dollars, cents, and mills; by  cents only; by cents
 and mills ; or by mills only;  observe the same rule.
EXAMPLES. D. c. m. Mult. 78546 37 5, by 9-9 706917 37 5 product.	D. c. Mult. 4764 50, by 11. 11
D. c. m. Mult. 17645 86 5, by 12. 12	D. c. Mult. 9865 36, by 8. 8
	______
D. c. Mult. 7461,75, by 24. 24	D. c. Mult. 987,46, by 54. 54
        29847,00                       3949,84
       149235,0                       49373,0
     179082,00 Product.               53322^84 Product.


  In the first example, dollars,  cents and mills are given; in
the  product, therefore, separate, as above directed, the right 
96
ARITHMETIC.
hand figure for mills, the two next for cents:  the remainder will
be dollars.
  In the two last examples, only dollars and  cents are given;
separate the two places on the right hand for cents ; the remain-
der will be dollars.  And so in all other examples.
PRACTICAL EXERCISES.
  1. What will 48  peaches come to at  15 mills, or  1 cent, 5
mills?                                        Ans  72 cents.
  2. What  will 25 pounds of beef come to  at 65 mills, or G
cents, 5 mills ?                               Ans. $1,62,5.
  3. What will 12 pounds of butter come to > I 125 mills, or 12
cents, 5 mills ?                                 Ans. $1,50.
  4. What will 5 bushels of apples  come to at 375 mills, or 37
 cents, 5 mills ?                               Ans. $1,87,5.
  5. What will  117 bushels of wheat come to at 875 mills, or
 87  cents, 5 mills?                          Ans. $102,37,5
  6. What will  25 barrels  of flour come to at 4  dollars, 75
 cents a barrel?                               Ans.  $118,75.
  7. What will 45 yards of cloth come to at 125 cents, or 1 dol-
 lar, 25 cents a yard ?                          Anf'.i^56,2i
  8 What will  127 acres of land come  to at  15  dollars,  50
 cents an acre?                             Ans. $1968,50.
  Q What will 48  cords of wood come to at 4 dollars, 87 cents,
 5 mills a cord?                               Ans. $234,00.
   10  What will 18 pair of shoes come to, at 94 cents a pairr
                                              Ans. $16,92.
    11. What will 24 pounds of tea come to, at 1,75 cents,  or 1
 dollar, 75 cents a pound ?                      Ans. $42,00.
    12. If a mechanic earns 125 cents, or 1 doll. 25 cents a day,
 how much will he earn in 28 days ?               Ans. $35.
    13. A gentleman bought a piece of cloth containing 28 yards,
 at  4375  mills, or  4 dolls. 37 cents, 5  mills a yard; what did
 "he whole cost'him?                   ,  ^  Ans $122,50.
    14. A merchant bought 246 pounds of coffee at 155 mills, or
 15 cents, 5 mills, a pound; what did the whole come to ?
                                               Ans. ^»8,U.
    15  A lady bought, for the use of her family, a barrel of sugar,
 weighing 312 pounds,at 115 mills,  or 11 cents,  5 mills a pound;
 whit did the whole come to ?                   Ans  $35, 3.
    16 A gentleman and his wife on a journey, spend 240 cents,
  or 2 dollars,  40 cents, a day;  how much will their expenses be
  for 24 days?                                  Ans. $57,60. 
   ^  fa*
                 FEDERAL CURRENCY.                97

  17. If a person spends 145 cents, or 1 dollar, 45 cents a day ;
how much will he spend in 56 days at that rate ? Ans. $81,20.
  18. How much will a hogshead of wine containing 63 gal-
lons come to, at 224 cents, or 2 dollars, 24 cents a gallon ?
                                           Ans. $141,12.
  19. How much Will 51 yards of linen come to at 42 cents a
yard ?                                       Ans. $21,42.
  20. If a  ton of hay sells for 7 dollars,  50 cents,  or 750
cents, what will 13 tons amount to ?             Ans.  $97,50.
  21. If a person pays  for board 350 cents, or 3  dollars, 50
cents a week, how much will he pay in 26 weeks ?   Ans. $91.
  22. If a ton of coals cost 625 cents, or 6 dollars,  25 cents, how
much will 11 tons come to ?                    Ans.  $68,75.

  When there are mills and tenths of a mill  given, you must
separate in the product the right hand figure for  tenths of a
mill, the next figure for mills, and the cents, as above directed.

  23.  How much will 126 pounds of coffee  come to,  at 18
cents, 7,5 mills a pound ?                    Ans. $23,62,5.
  24.  What will 132 pounds of  beef come to, at 6 cents, 2,5
mills a pound ?                                Ans. $8,25.
  25.  What will 174 yards of silk come to at 93 cents, 7,5 mills
a yard ?                                  Ans. $163,12,5.
  26.  What will 346 bushels  of corn come to, at 56 cents 2,5
mills a bushel ?                            Ans. $194,62,5.
  27.  What will 2164 pounds  of pork come to, at 7 cents, 5
mills a pound ?                             Ans. $162,30.
  28.  What will 127  yards of muslin come to, at  21 cents, a
yard ?                                       Ans. $26,67.
  29.  What will 526 barrels of  flour come to, at 5 dollars, 06
cents a barrel ?                             Ans. $2661,56.


        DIVISION OF FEDERAL CURRENCY.

  The operation  is the  same as division of integers or whole
numbers.   But observe to separate the quotient into dollars and
cents, or into dollars, cents and mills, as the case may be.

                        EXAMPLES.
Divide 647dols.75cts.by5. Divide7684 dols. 87 cts.5  m. by 6.
          .5)675,75                6)7684,87,5
             135,15 quotient.         1280,81,2,5 quotient
       i2 
98
ARITHMETIC.
            Dolls, cts. m.                 Dolls, cts. m.
  3. Divide 14765 62  5  by  11.  Quotient 1342  32 9,5+
  4.  "      1574 25 0  by  12.      "     131  18 7,5
  5.  "      9876 76 5  by   9.      "   1097  41 8,8*
  6.  "       108 25 2  by  36.      "       3  00 7
  7.  "       436 56 0  by 214.      "       2  04 0
  8.  "       673 62  5  by 317.      "       2  12 5
  9.  "       328 42  5  by 755.      "       0  43 5
 10.  "       385 02  0  by 186.      "       2  07 0
 11.  "     10476 31  5  by 144.      "     72  75 2 +
 12.  "     11216 81 2,5 by  75.      "     149  55 7,5

                  PRACTICAL EXERCISES.

  1. A lady bought 7 yards of silk for 11,37,5 ; how much was
it a yard ?                          Ans. 1 doll. 62 cts. 5 m.
  2. A miller bought 378 bushels of wheat for 380 dollars, 26
cents, 8 mills ; how much was it a bushel ?
                                   Ans. 1 doll. 00 cts. 6 m.
  3. A gentleman bought 15 cords of wood for 70 dollars, 75
cents;  how much was it a cord ?             Ans. $4,71,6,0
  4. A person purchased a box of tea, weighing  14 pounds, for
17 dollars, 50 cents ;  how much was it a pound ?  Ans. $1,25.
  5. A grocer bought 350 pounds of coffee for  41 dollars, 62
cents, 5 mills; how much was it a pound ?
                             Ans. 11 cts. 8,9 m.+a pound.
  6. A gentleman bought 8 yards of broad-cloth for  51 dolls.
62 cents, 5 mills ; how much was it a yard ? Ans. $6,45,3,125.
  7« A mechanic performed a piece of work  in 16 days, for
which he received 25 dollars, 76 cents ; how much did he earn
a day?                                        Ans. $l,6l.
  8. A lady bought for her own use a barrel of sugar,  weighing
367 pounds,  for 41 dollars, 87cents, 5 mills; how much was it a
pound?                                 Ans. 11 cts. 4,1 +
  9. A farmer sold 14 tons  of hay for 137 dollars,  50 cents;
how much was it a ton?                   Ans. $9,82,1,4+
  10. If 496 pounds of rice cost 15 dollars, 37  cents, 6 mills;
bow much is it a pound?                   Ans. 3 cts. 1 m.
  11. If 24 pair of shoes cost 35 dollars, 25 cents; how much is
the  price of one pair ?                      Ans. $1,46,8+
  12. A person spent in 22 days the sum of 37 dollars, 50 cts.;
how much did he spend a day ?            Ans. $1,70,4,5+
  13. A person spends in a year 741 dollars, 62  cents, 5 mills;
how much is  it a day, 365 days being a year ? Ans. $2,03,1,8+ 
            POUNDS, SHILLINGS, AND PENCE.         99

  14. The rent of a plantation is 466 dollars, 50 cents a year ;
what is k a day ?                           Ans. $1,27, 8 +
  15. A gentleman rents a house for 650 dollars a year; how
much is it a month ?                        Ans. $54,16,6,0.

  Questions.  1. What is compound addition ?—2. How  do
you place the numbers for the operation ?
  3.  How do you proceed in adding up the several columns ?
  4.  How may the work be  proved ?
  5.  What are the denominations of Federal currency ?
  6.  By whom was  it established ?—7. In what year was it
established ?
   8.  How is addition performed ?—9.  What is the reason that
 it is so  performed?—10. In what proportion or ratio,  do  the
 several denominations increase ?
   11. In the transactions of business, how are eagles and dollars
 considered ?—12. How are dimes and cents considered ?
   13.  How  is subtraction of Federal currency performed ?
   14.  Why is it so performed?—15.  How do you place the
 terms ?—16. How is multiplication of this currency performed?
   17.  What rule do  you  observe in dividing the product  into
 dollars and cents ;  or into dollars, cents, and mills ?
   18.  When mills and tenths of a mill  are given, how do you
 point off the product ?
    19.  How is division of this currency performed ?—20. What
 do you observe in regard to the quotient ?
    21.  How is division proved ?
                      ADDITION
       OF POUNDS, SHILLINGS, PENCE, AND FARTHINGS.

  For the table, and its application, see Introduction.
  When the several denominations do not increase in the same
ratio, or proportion as in  Federal currency,  you must add up
each column separately, beginning at the right hand, or lowest
denomination, and  note the sum: divide this sum by as many
of that denomination as make one, or unit, of the next greater:
set down the remainder, if there is any, under the said column,
and add the quotient to the next higher denomination ; and so
proceed till you come  to  the last denomination, which add up
exactly, as in addition  of whole numbers. 
100
ARITHMETIC.
  This is a general rule, applicable to all cases, whatever may
be the denominations.
  The work may be proved, as in addition of whole numbers,
which see.
			EXAMPLES.			
£.	s.	d.	£. s. d.	£.	8.	d.
487	13	8	8764 12 8	1245	15	5
512	6	4	5421 11 9	3142	12	11
671	11	3	2132 6 4	1452	13	2
764	18	10	4578 16 7	2785	17	8
2436	10	1				
£.	s.	d.	£. s. d.	£.	8.	d.
7A6	10	4*	55 10 6g	254	16	7k
124	16	2	42 7 2	147	15	3
234	12	5*	75 12 A\	241	11	7\
342	6	6	12 13 7	525	4	8
542	7	n?	17 17 6£	317	2	64
125	14	6	17 11 9	742	11	8
421	17	8*	28 14 7\	942	17	93
			________	____		
  In the first example, you add  up the column of pence, and
find  the sum to be 25;  which you divide by 12, because 12
pence make 1  shilling, or 1 of the next higher denomination.
You set 1, the remainder,  under the column of pence, and add
or carry 2, the quotient, to  the column of shillings; this you
also add up, which, with the 2 from the last column, makes
50 shillings.   These you  must divide  by 20, because 20 shil-
lings make 1 pound, or 1 of the next higher denomination;  set
down the remainder or 10, and add or carry  the quotient 2 to
the  column of pounds; which you add up as whole numbers.
If farthings had been given also, as in the last examples, you
must have added  them up,  and divided the sum by 4, because
4 farthings make 1 penny, or one of the next  higher denomina-
tion ; set down the remainder, and added the quotient to  the
column of pence. 
POUNDS, SHILLINGS,  AND PENCE.
                    SUBTRACTION

       OP POUNDS, SHILLINGS, PENCE, AND FARTHINGS.

  When any denomination of the subtrahend is less than that
of the minuend, the operation is performed simply by taking
their difference, and setting it down under its proper column.
That difference will be the  remainder required.  But, if any
term of the  subtrahend is greater than the same term of the
minuend, you must add to the upper number so many of the
said denomination, as will make 1 or unit of the next greater
denomination, and from this sum subtract the subtrahend, and
set the remainder down under its proper column; add or carry
1 or unit to the subtrahend of the  next  higher  denomination ;
and so continue to  do, when the lower number is greater than
the upper one, till you come to the last denomination ; which
subtract as whole numbers.  This is a general rule,  applicable
to all cases, whatever may be the denominations.
  The work may  be proved as subtraction of whole numbers,
which see.

                       EXAMPLES.

           £.    s.  d.      £.   s.  d.
Minuend   450   16  8      365  14   5
Subtrahend 230   12  6      247  15   3
Remainder 220    4  2
         £.   s.  d.      £.    s.   d.
Minuend 756  11  7£     1100    7  6£
Subtrah.  527   8  9J      746   12  3|
Rem.
                  MULTIPLICATION

       OF POUNDS, SHILLINGS, PENCE, AND FARTHINGS.

  When  the product of the multiplier and any term of the
multiplicand is less than the number required of that denomina-
tion to make 1  or unit of the next higher, you set  down the pre-'
duct under its own proper column, and proceed to the next one.
  But, if the product is greater, you must divide it by  the num-
ber required of that denomination to make 1 or unit of the next
^                                 i
 £.   s.   d.
549   12   6
427   16   8
		
£.	s.	d.
A76	15	4f
219	11	8f
		
 
102
ARITHMETIC.
higher;  set down (he remainder, if there is any, under its
own proper column, and add the  quotient to the product of the
multiplier and  the next term of the multiplicand.   If this sum
is greater than the number required of that denomination to
make 1 or unit of the next higher, you must divide, as  before,
set down the remainder, if there is any, and add the quotient to
the next higher denomination; but, if it  should be less, you set
down the product in its proper place.   In like manner you pro-
ceed, till you come to the last denomination, which you multiply,
and set down as whole numbers.
   This is a general rule, applicable to all cases, whatever the
denominations may be.

                         EXAMPLES.

             £    s.  d.               £   s.    d.
Multiplicand 25  12  6,  by 5.        154   15   4, by 8.
Multiplier           5                         8 multiplier
Product   128   2~~6             1238    2~8 product.
            £  s.  d.       £    s.   d.        £   t.   d.
Multiplicand 46  5  6£      65   15   3£      56   10  7\
Multiplier________9_              11         _____12_
Product
                        DIVISION

       OF POUNDS, SHILLINGS, PENCE, AND FARTHINGS.
  When the divisor is  contained exactly  in any term of the
dividend, you have only to divide it, and set down the quotient
under its own proper column, beginning at the left hand, or high-
est denomination.  But, when there is a  reminder, you must
multiply it by the number required of the next lower denomina-
tion to make 1 or unit of the higher, that is, you must reduce itto
the next lower denomination, and add the  numbers standing in
that lower place to it.  This sum divide, and set down the quo-
tient.   The remainder, if there is any, you must reduce to the
next lower  denomination, and add to it the number in  the next
lower place : this sum divide as before; and so proceed till the
work  is done.
  This rule is general, applicable to all cases, whatever may be
the denominations. 
          POUNDS, SHILLINGS AND PENCE.        103

                     EXAMPLES-

Divide .£147 16*. 6d., by 4.    Divide £A6i 12s. Sd., by 6.

         £.    s.  d.                £.    s.  d.
       4)147   16   6              6)465   12   8
 Quotient    36  19   l£     Quotient   77   12  lj+


Divide«£547l5s.7£d.by4.    Divide£ll42 17*. 9Hbv9-
       £79 17*. Ad. by 7.            £124  lis. Ad. by 10.
     £1461 18*. lOd. by 11          £7640  11*. A%d. by 12.

  In the first example, after dividing the pound by 4, there are
3 remaining : which multiply by 20, because 20  shillings make
a pound ;  to the  product add the 16 shilling in  the next lower
place; the sum is 76  shillings.   This divides by 4 without a
remainder : the quotient  19 set  down in  shilling's place ;  and
divide 6 the next lower denomination by 4; the quotient is 1,
which set down in pence place: the remainder 2, you multiply
by 4, because 4 farthings make 1 penny:  divide the product 8,
by the divisor, and nothing remains :  the quotient is  2, or ^ a
penny, that is, 2 farthings ; and the work  is  done.
  In like manner proceed in the other examples.  On the same
principle are all compound numbers divided, whatever may be
their denominations.

   Questions.  1. What is componnd addition?—2. How
many shillings make a pound ?—3. How many pence make a
shilling ?—4.  How many farthings make a penny ?—5. How
do you proceed in the  addition of this money ?—6. How do you
consider the last column, or highest denomination ?
   7. Do you proceed in  a similar way in  the addition of any
other denomination ?
   8. What is compound substraction ?—9- How do you pro-
 ceed in  compound  subtraction ?—10. What do you mean by
 the subtrahend ?—11. What by the minuend ?—12. Is this rule
 applicable to numbers, whatever may be their denominations ?
   13. What is  compound multiplication?—14. How  do you
 proceed  in compound multiplication ?—15.  How do you con-
 sider the last term or highest denomination ?
   16. Is  this rule applicable to numbers  consisting of other
 denominations ?
   17. What is compound division?—18.  With what denomi-
 nation  do you begin?-—19.  How  do  you  proceed  in  the 
104
ARITHMETIC.
operation ?—20. Will this method apply to numbers consisting
of other denominations ?
  21. How do you prove the operation of compound addition ?
  22. Do you prove that  of subtraction, multiplication, and
division in the same way ?
          ADDITION OF TROY WEIGHT.

  For the table, and the denominations, see introduction,
under troy weight.
  For directions to  perform the operation, see addition of
pounds, shillings and  pence.
				EXAMPLES.					
lb.	oz.	dwt.	gr.	lb. oz. dwt.	gr.	lb.	oz.	dwt.	gr-
8	5	10	12	12 7 14	16	48	10	19	23
6	4	11	15	11 9 10	12	56	9	7	00
9	2	4	6	7 8 17	19	26	11	12	20
10	7	6	16	14 7 8	22	19	7	6	5
11	5	14	7	17 8 12	21	12	8	11	15
Sum						—			
        SUBTRACTION OF  TROY WEIGHT.

  For directions to perform the  operation, see subtraction
of pounds, shillings and pence.
EXAMPLES.
     lb.  oz.  dwt. gr.  lb.   oz.  dwt. gr.    lb. oz. dwt. gr.
From 48   6  10  4   17  10  12  16    25  7  17  lb
Take 25   7   6  8     4  11   7  20    12  8  16  19

Rem.
      MULTIPLICATION  OF  TROY  WEIGHT.

  For directions to perform the operation, see multiplication
of pounds, shillings, and pence. 
TROY WEIGHT.                  105
EXAMPLES.
        lb.  oz.  dwt.  gr.                   lb.   oz.  dwt. gr.
Multiply 22   5  10  16           Multiply 46   10  14   6
by____________4_          by____________7
Prod.
         lb.  oz.dwt.gr.                      lb.  oz.dwt.gr.
Multiply 16  8   4  6              Multiply 116  3  2   7
by__________8              by___________9
Prod.
            DIVISION OF  TROY WEIGHT.

  For directions to perform the operation,  see Division of
pounds, shillings, and pence.
       lb.   oz.  dwt.  gr.               lb.  oz.  dwt.  gr.
Divide 36   10   11  12, by 4.  Divide 47  11   16  17, by 6

         lb.  oz. dwt. gr.          lb.   oz. diet.  gr.
       4)36  10  11   12        6)47   11   16  17
Quot.                    "  Quot.


  Divide 18 lb. 7 oz. 12 dwt. 15 gr. by 8.
  Divide 12 lb. 8 oz. 16 dwt. A gr. by 7.
  Divide 55 Z6.10 oz. 14 dwt. 22 gr. by 9.
  Divide 116 lb. 9 oz. 17 dwt. A gr. by 12.

                   PRACTICAL EXERCISES.
   1.  A silversmith purchased 4 ingots of silver; the first weighed
2 lb! 6 oz.  10 dwt. 4 gr. ;  the second, 3 lb. 8 oz. 0 dwt.  16 gr.;
The third, 3 lb.  11 oz. 16 dwt. 22 grains ; and the fourth, 4 lb.
3 oz 7 dwt. 16  gr.;  what  was the weight of the whole ?
                              Ans. 14 lb. 5 oz. 15 dwt. 10 gr.
  2.  A gentleman just married  purchased the following articles
of a*silversmith, (viz.) bowls weighing 4 lb. 6 oz. 11 dwt. 20 gr.;
plates weighing 5 lb. 10 oz. 16 dwt. 8 gr.; spoons weighing 3 lb.
4 oz.  6 dwt. 5gr.; and salts weighing 10 oz.4 dwt. 12 gr.; what
was the weight  of the whole ?   Ans. 14 lb. 7 oz. 18 dwt. 21 gr.
  3. A gentleman presented his daughter, on her marriage day,
with the following articles  of plate, (viz.) spoons weighing 3 lb.
                K 
106                   ARITHMETIC.

11 oz. 6  dwt.; a tea-set weighing 4 lb. 8 oz. 0 dwt. 20 gr.; a
tankard weighing 5 lb. 3 oz. 18 dwt.; and salts weighing 8 oz.
16 dwt. 7^> f what was the weight of the whole ?
                               Ans. 14 lb. 8 oz. 1 dwt. 9 gr.
  4. A silversmith purchased  16 lb. 8 oz. 12 dwt. 20 gr. of sil-
ver, and manufactured 5  lb.  11 oz. 16  dwt. 4 gr. of it;  how
much has he remaining ?      Ans. 10 lb. 8 oz.  16 dwt. l6gr.
  5. A silversmith wrought 24 lb. 5 oz. 16 dwt. of silver bullion
into spoons, plates, salts, and bowls; and  afterward  weighed
the articles, and  found that their weight was  only 23 lb. 11 oz.
18 dwt. and 22 gr.;  how much was the loss  in the whole ?
                                   Ans. 5 oz.  17 dwt. 2 gr.
  6. A silversmith has 6 ingots of silver, each weighing 6 lb.
4 oz. 14  dwt. and 6 grains;  how much is  the  weight of the
whole ?                       Ans. 38 lb. 4 oz. 5 dwt. 12 gr.
  7. A person  has a quantity of silver, in weight 16 lb. 7 oz.
0 dwt. 20  gr., and wishes to  divide it into 5 equal parts;  how
 much will there  be in each part?  Ans. 3  lb.  3 oz. 16 dwt. 4 gr.
   8. A gentleman deceased ordered his plate  to be equally
divided between his three daughters.  Its weight was found to
be 17 lb. 2 oz. 17 dwt. and 16 gr.; what was the share of each
one ?                         Ans. 5 lb. 8  oz. 19 dwt. 5$ gr.
      ADDITION OF AVOIRDUPOIS WEIGHT.

  For the table, and denominations, see Introduction under
avoirdupois weight.
  For directions to perform the operation, see  Addition of
pounds, shillings, and pence.

                       EXAMPLES.
T.	C.	qr.	lb.
16	5	2	16
12	14	3	22
24	6	1	16
54	17	0	22
31	5	3	11
44	16	1	25
C.	qr.	lb.	oz.	Qr.	lb.	oz.	dr.
5	2	21	12	2	16	4	'j
21	1	16	4	1	21	10	10
32	2	24	8	3	4	6	11
41	3	12	5	0	26	12	9
54	1	11	14	3	14	14	14
61	0	22	0	1	11	15	5
 
^%Rt*''-
                AVOIRDUPOIS WEIGHT.              107

   SUBTRACTION OF AVOIRDUPOIS WEIGHT.

  For directions  to perform  the operation, see Subtraction of
pounds, shillings, and pence.             .

                       EXAMPLES.
      T.   C.  qr.  lb.     C.  qr. lb.  oz.  qr.  lb.   oz.  dr.
From 44   16  1  21     56  2  12   8   3   11   12   6
Take 29   17  0  14     41  3  13   4    1   16    8  12
Rem.
 MULTIPLICATION OF AVOIRDUPOIS WEIGHT.
  For directions to perform the  operation, see Multiplication
of pounds, shillings, and pence.

                       EXAMPLES.
         T.  C.  qr.  lb.                C.  qr.  lb.  oz.
    Mult. 7  14  2   10         Mult. 16   2   21   6
    by    ___________5         by    _^_______7
    Prod.                       Prod.
     T.  C.  gr. lb.           T.  C.  qr.  lb," oz.
Mult. 16  7  1  16      Mult. 5  l   24  6  10
by   ___________8       by_____________9
Prod.                     Prod.
      DIVISION OF  AVOIRDUPOIS WEIGHT.

  For directions to perform the operation, see  Division of
pounds, shillings, and pence.

                       EXAMPLES.
         T.  C. qr. lb.                 C. qr. lb. oz.  dr.
  Divide 22  16 2 21, by 5.     Divide 24 3  17 10 6, by 7
      T.  C.  qr.  lb.             C.  qr. lb.  oz.  dr.
    5)22  16   2  21           7)24  3  17  10   6
Quot. ~                   Quot.
Divide 56 T. 12 C. 1 qr. 16 lb., by 8.
Divide 25 T. 7 C. 3 qr. 24 lb., by 10.
Divide 14 C. 1 qr. 22 lb.  7 oz. 10 dr., by 3.
Divide 37 C. 2 qr. 17 lb. 11 oz. 13 dr., by 11. 
108                   ARITHMETIC.
                   PRACTICAL EXERCISES.

  1. A grocer bought 3 hogsheads of sugar;  the weight of the
1st, was 12 C. 3 qr. 16lb.;  of the 2d, was 14 C.  2 qr. 22 lb.;
and of  the 3d, was 15 C. 3 qr. 16 lb.;  how much was there in
the whole ?                          Ans.  43 C. 1 qr. 26 lb.
  2. A planter in Virginia sold 4 hogsheads of tobacco; No. 1,
weighed 8 C. 2 qr. 16 lb.;  No.  2,  weighed  9 C. 3 qr. 20lb.;
No. 3, weighed 10 C. 1 qr. 17 lb,; and No. 4, weighed 12 CO qr.
17 lb.;  what was the weight of the whole ?
                                    Ans.  41 C. Oqr. 14 lb.
  3. A merchant purchased 4 bags of cotton, weighing as fol-
low : No. 1,4 C. lqr. 16 lb.; No. 2, 5 C. 2  qr. 14 lb.; No. 3,
3 C. 3 qr. 22 lb.;  and No. 4, 5 C. 1 qr. 20 lb.; what was the
weight  of the  whole ?                 Ans.  19 C. 1 qr. 16  lb.
  4. A merchant bought 24 C.  3  qr.  16 lb. of rice, and sold
15 C. 1 qr. 22 lb.; how much remains  on hand ?
                                     Ans. 9 C. I qr. 22  lb.
  5. A gentleman bought 21 T.  17 C. 2 qr. of coal, and sold
9 T. 19 C. 3  qrs.; how much has he on hand ?
                                   Ans. 11 T. 17 C. 3  qr.
  6. A merchant purchased  6 hogsheads of sugar, each weighing
16 C. 2 qr. 12 lb.; how much was the weight of the whole ?
                                    Ans. 99 C. 2qr. l61b.
  7. A grocer bought 12 boxes of figs, each weighing 3 qr. 14 lb.
12 oz.; what was the weight of the whole ?   Ans. 42. qr. 9 lb.
  8. A merchant has 4 parcels of iron ;  in each of which there
are 5 C. 3 qr. and 20 lb.; how much.is there of the whole?
                                    Ans. 23 C. 2 qr. 24  lb.
  9. A farmer cut 25 T. 17 C. 3 qrs. of hay ; which he wished
to put  into 3 stacks  for the better convenience of his stock;
how much was in each stack?    Ans. 8 T. 12 C. 2 qr. 9jlb.
  10.  In a certain heap there are 16 T. 15 C. 3 qrs. of coal,
which  it is required  to  divide into  5 equal  parts ; how  much
will be in each part  ?            Ans. 3 T. 7 C 0 qr. l6| lb.
  11.  A merchant has 10  T. 17 C. 2 qr.  of iron, which  he
wishes to divide into 8 equal parts;  how much  will there be in
each?                            Ans. IT. 7 C. Oqr. 21 lb.
    ADDITION  OF APOTHECARIES'  WEIGHT.

  For the table, and the denominations, see Introduction under
apothecaries' weight. 
APOTHECARIES' WEIGHT.
109
  For directions to perform the operation,  see Addition of
pounds, shillings, and pence.
				EXAMPLES.					
lb	I	3	9	gr-	lb	I	7,	A	gr.-
6	7	5	l	10	14	7	6	2	19
10	11	3	2	8	11	11	7	1	14
12	4	1	0	11	12	7	6	2	5
14	8	6	2	4	7	5	2	0	8
7	6	5	1	16	16	9	1	2	16
4	10	7	0	14	21	8	5	1	12
Sum									
 SUBTRACTION OF APOTHECARIES' WEIGHT.

 For directions to perform the operation, see  Subtraction of
pounds, shillings,  and pence.

                     EXAMPLES.

    n    lb  I   3  9  gr.      lb  g   3  9 #..
    From 12   7 6   1  10      18  6   5  1  14
    Take  9  11  5   2  16      11  7   6  2  10
    Rem.
MULTIPLICATION OF APOTHECARIES' WEIGHT.

  For directions to perform the operation, see Multiplication
of pounds, shillings, and pence.

                      EXAMPLES.

         lb  3   3   3  gr.         lb  ?  3  G  gr-
    Mult. 5   10  6   1   12         7  6  7  2  16
      By               5                      8
Prod.
     DIVISION OF APOTHECARIES' WEIGHT.

  For directions to perform  the operation, see Division  of
pounds, shillings, and pence.
         k 2 
110                  ARITHMETIC.
                       EXAMPLES.

      lb   I  3  9  &•■
Divide 14   7  6  1   10, by 6

        lb   I  3  9 gr-
      6)14   7  6   l  10
Quot.
  Divide 6 lb, 8 I, 5 3, 2 3, 10 gr., by 6.
  Divide 11  1, 6 3, 1 3, 19 gr., by 9-
  Divide 11  lb, 10 1, 6 3, 1 3, 19 gr., by 7-
  Divide 24 I, 7 3, 2 3, 16 gr., by 10.

                 PRACTICAL EXERCISES.

  1. A druggist would mix 4 sorts of drugs; the 1st sort weigh-
ing 7 lb, 4 1,6 3,2 3, 16 gr.; the 2dsort, 16 lb, 11 I, 5 3,
0 3, 19 gr.; the 3d, 9 ib, 9 !, 5 3, 2 3,16 gr.;  the 4th, 8 lb,
11 |, 7 3,0  3,12 gr.; what is the weight of the whole ?
                             Ans. 43 ib, 2 I,  1 3,1 3, 3 gr.
  2. An apothecary having mixed 12 ib, 10 |, 5 3, 2 3,16 gr.
of drugs, would take from the mass  4 lb, 5 ^, 63, 1 3,12 gr.;
how much will remain ?         Ans. 8 lb, 4 %, 7 3,1 3,4 gr.
  3. A druggist would  compound 4 several sorts of  drugs in
equal proportions of 5 ft., 10 I, 6 3, 2 3,15 gr., each; what will
be the weight of the whole ?         Ans. 23 lb, 7  1, 3 3, 2 3.
  4. A druggist compounded a quantity of drugs, weighing in
the whole, 16 lb, 10 1,7 3,2 3,16 gr.;  which he would divide
into 6  parcels; how much would there be in each ?
                           Ans. 2 lb, 9 1, 6 3,1 3, 19| gr.
         ADDITION OF LONG MEASURE.

  For the table, and denominations, see Introduction  under
long measure.
  For directions to perform the operation, see Addition of
pounds, shillings, and pence.
       Ib   §   3  3
Divide 18  10   7  2, by 8.

       Ib   3   I  9
     8)18  10   7  2 
LONG MEASURE.
Ill
                 EXAMPLES.
Deg. M.fur. P.               Yds. ft.  in. b. c.
12	42	5	30
16	36	7	27
39	21	4	21
46	25	3	17
21	17	6	33
26	15	2	16
Sum.
246	2	10	1
341	1	11	2
410	2	7	1
114	0	6	0
211	1	5	2
416	2	8	1
			
       SUBTRACTION OF  LONG MEASURE.
  For directions to perform the operation, see Subtraction of
pounds, shillings, and pence.
                      EXAMPLES.
      Deg. m. fur. P.       Yds.  ft. in.     Ft. in. b.c.
From 24  36  5  16       176  2  10      2  7  2
Take 17  41  3  18       127  1  11       19  2
Rem.
     MULTIPLICATION OF LONG MEASURE.
  For directions to perform the operation, see Multiplication of
pounds, shillings, and pence.

                      EXAMPLES.
        Deg.  m.  fur. P.               Yds.  ft.  in.  b.c.
Multiply 10   16  4  21       Multiply 125   2   10  2
by____________6       by____________7
Prod.
        Deg. m. fur. P.                Yds.  ft.  in. b.c.
Multiply 16  4  7  30        Multiply 241  2   11  l
by____________8        by      _____  _____9
Prod.
          DIVISION OF LONG MEASURE.
  For directions to perform the operation, see  Division of
pounds, shillings, and pence. 
112                   ARITHMETIC.
                    EXAMPLES.
Divide 46 deg. 35 m. 5 fur. and 26 P. by 5.
Divide 246 yds. 2 ft. 10 in. and 2 b.c. by 7-

       Deg. m.fur. P.               Yds. ft. in.b.c.
       5)46 35 5 26            7)246 2 10 2
  Divide 160 deg. 52 m. 4 fur. 16 P. by 9-
  Divide 48 deg. 54 m. 7 fur. 24 P. by 11.
  Divide 45 m. 6 fur. 120 yds. 2 ft. by 12.
  Divide 134 m. 7 fur. 175 yds. 1 ft. by 10.

                  PRACTICAL EXERCISES.

  1. What is the sum of 5  deg. 16 m. 5 fur.  36 P.; 10  deg.
54 m. 5 fur. 26 P.; 26 deg. 36 m. 4 fur. 16 P.; 35 deg. 40 m.
7 fur. 36 P.; and 12 deg. 16 m. 2 fur. 25 P. ?
                         Ans. 90 deg.  45 m. 2 fur. 19 poles.
  2. A person set out on a journey;  the first day he went 25 m.
4 fur. 16 P.; the second, 27  m. 7 fur. 36 P.; the third, 30 miles,
5 fur. 35 P.; the fourth, 35  miles, 5 fur. 39 P.; how far did he
travel in the whole ?               Ans. 120 m. 0 fur. 6 poles.
  3. What is the difference between 140 degrees, 40 m. 7 fur.
16 P. and 74 deg. 50 m. 4 fur. 26 P. ?
                          Ans. 65 deg. 50 m. 2 fur. 30 poles.
  4. Suppose the distance from Philadelphia to Trenton, to be
38 m. 5 fur. 36 P., and to New York, 95 m. 7 fur. 16 P.; how
far is the distance from Trenton to New  York ?
                                 Ans. 57 m. 1 fur. 20 poles.
  5. Suppose the distance from Philadelphia to Lancaster to be
65 m. 6 fur. 36 P.; how many miles must a person travel to go
the distance 6 times?             Ans. 395 m. 1 fur. 16 poles.
  6. A gentleman has a field exactly square, and one side  mea-
sures 175  yds. 2 ft. 9 in.; can you tell how many yards will be
in all the sides ?                        Ans. 703 yds. 2 feet.
  7.  A person in 6 days travelled 154 miles, 6 fur. 22 P.; how
far did he travel one day with another ?
                                 Ans. 25 m. 6 fur. 17 poles.
   8.  A farmer has a square field, whose sides altogether  mea-
sure 3 m. 3 fur. 24 P.; what is the length of one side ?
                                       Ans. 6 fur. 36 poles. 
CLOTH MEASURE.                113
         ADDITION OF CLOTH MEASURE.

  For the table, and the denominations, see Introduction under
cloth measure.
  For directions to perform  the operation, see Addition of
pounds, shillings, and pence.
Sum
			EXAMPLES.				
Yds.	qr.	na.	E.E.	qr. na-	E.F.	qr.	na.
125	2	1	54	4 3	41	2	3
216	3	2	62	3 1	62	1	2
171	3	1	41	3 2	34	2	3
55	2	3	25	4 3	21	1	0
16	1	2	16	1 2	49	2	1
142	2	1	17	4 2	52	0	2
SUBTRACTION OF  CLOTH  MEASURE.
  For directions to perform the operation, see  Subtraction of
pounds, shillings, and pence.
			EXAMPLES.			
From Take	Yds. 75 46	qrs. na. 2 3 3 1	E.E.qr. na. 66 3 2 24 4 3	E.F. 124 85	qr. 2 2	na. 2 3
Rem.						
    MULTIPLICATION OF CLOTH MEASURE.

  For directions to perform the operation, see Multiplication of
pounds, shillings, and pence.
                      EXAMPLES.
         Yds. qr. na.      E.E. qr. na.     E.F. qr. na.
Multiply  116  3   2      74 4  1       56  2  1
by       _______6      _______8       _______9
Prod.
         DIVISION OF CLOTH MEASURE.

  For directions to perform the  operation, see  Division of
pounds, shillings, and pence. 
H4                 .  ARITHMETIC.

                       EXAMPLES.

  Divide 175 yds. 3 qrs. and 2 na. by 6.
  Divide 216 yds. 2 qrs. and 1 na. by 8.

           Yds.  qr.  na.               Yds.  qr. na.
         6)175  3   2              8)216  2   1
Quotient
  Divide 76 E. Eng. 4 qrs. and 3 na. by 9.
  Divide 57 E. Eng. 3 qrs. and 2 na. by 7-
  Divide 216 E. Flem. 2 qrs. and 3 na. by 11.
  Divide 124 E. Flem. 1 qr. and 3 na. by 12.

                 PRACTICAL EXERCISES.

  1. A merchant bought 4 pieces of cloth ; No.  1,  containing
25 yds. 3 qrs. 1 na__No. 2, 30 yds. 2 qr. 2 na.—No. 3, 32 yds.
1 qr. 2 na.—No. 4, 36 yds. 3 qrs. 3 na.;  what is the number of
yards in the whole ?                    Ans. 125 yds. 3 qrs.
  2. A merchant sold the following pieces of cloth, viz : No. 1,
containing 12 yds. 3 qrs. 3 na.—No. 2, 16 yds. 2 qrs. 1 na.—
No. 3, 14 yds. 2 qrs. 3 na.—No. 4, 18 yds. 1 qr. 3 na., and
No. 5, 20 yds. 2 qrs. 3 na.; how much did he sell in all ?
                                   Ans. 83 yds. I qr.  1 na,
  3. A person bought a piece of broad-cloth, containing 24 yds.
2 qrs. 3 na., and sold of it 8 yds. 3 qrs.; how much has he on
hand?                            Ans. 15 yds. 3 qrs.  3 na.
  4. A merchant tailor purchased a piece of cloth, containing
42  yds. 3 qrs. 2 na., and  cut from it 5 yds.  1 qr, 3 na.; how
much remains?                      Ans. 37 yds. 1 qr. 3 na.
  5. A merchant  bought 6 pieces of  cloth, each  containing
26 yds. 2 qrs. 1 na.;  how many yards were in the whole ?
                                  Ans. 159 yds. 1 qr.  2 na.
  6. If you buy 18 yds. 2 qrs. 1 na. of cloth, and make use of
8 yds. 2 qrs. 2 na.; how many yards have you remaining ?
                                    Ans. 9 yds. 3 qrs. 3 na.
  7. If you have 42 yds. 1 qr. 3 na. of cloth,  and divide them
nto 5 parts ; how many yards will there be in each ?
                                   Ans. 8 yds. 1 qr. 3| na.
  8. A merchant bought 8 pieces of broad-cloth, containing in
the whole 164 yds. 2 qrs. 3 na.; how many yards were in each
piece, supposing they were all of equal length ?
                                  Ans. 20 yds. 2 qrs. If na.
* 
LAND MEASURE.                115
         ADDITION OF LAND MEASURE.

  For the table, and the denominations, see Introduction under
land measure.
  For directions  to  perform the operation, see Addition of
pounds, shillings, and pence.
                     EXAMPLES.
      A.  R.  P.      A.  R. P.      A.   R.  P,
     120  2  30      27  3  36      196  3  30
     219  3  25      19  2  21      247  2  36
     112   1  26      41  1  00      516  2  11
      74   2  16      35  3  11      172-3  17
      27   2  27      48  2  17      217  1  00
      71   3  36      55  0  36      415  3  33
Sum
       SUBTRACTION OF LAND MEASURE.

  For directions to perform the operation, see Subtraction of
pounds, shillings, and pence.

                     EXAMPLES.
      A. R.  P.      A. R.  P.       A.  R.  P.
From 74  2  16      54  2  17      144  1  36
Take 27  3  32      27  3  12       79  3  27
Rem.
      MULTIPLICATION OF LAND MEASURE.
  For directions to perform the operation, see Multiplication
of pounds, shillings, and pence.
                      EXAMPLES.
         A.  R.  P.                       A.   R. P.
Multiply 47  3  27                Multiply 175  3  21
by        ______6                by     __________8
Prod.     ~~                       Prod.

         A.   R.   P.                       A.  R.  P.
Multiply 214  2   25                Multiply 96  3  36
by_______11                by     ________12
Prod.                              Prod. 
116                   ARITHMETIC.

           DIVISION OF  LAND MEASURE.

  For directions to perform the operation, see Division  of
pounds, shillings, and pence.

                       EXAMPLES.
Divide 172 A. 3 J*. 34 P. by 6.    Divide 68 A. 3 R. 39 P. by 9.

          A.   R.  P.                    A.  R.  P.
       6)172  3   34                  9)68   3   39
Quot.                          Quot.
Div. 240 A. 2 R. 26 P. by 8.     Div 500 A. 2 R. 29 P. by 12.
Div. 360.4. 3 R. 39 P. by 7-    Div. 211 A. 3 R. 2lP. by 11.

                  PRACTICAL EXERCISES.
  1. A farmer has several tracts of land; No. 1. containing 47
acres, 3 roods, 26 perches; No. 2, 27 acres, 2 roods, 26 perches;
No. 3, 36 acres, 3  roods, 29 perches; No. 4,17 acres, 2 roods,
27 perches ; how much has he in all ? Ans. 130 A. 0 R. 28 P.
  2. A planter of Virginia has the following fields under culti-
vation;  No. 1, containing 46 acres, 2 roods, 27 perches ; No.
2, containing 59 acres, 3 roods, 16 perches; No. 8 containing
65 acres, 2 roods, 36 perches ; and No. 4, containing 74 acres,
1 rood,  16 perches; how many acres has he in all ?
                                   Ans. 246 A. 2 R. 15 P.
  3. A gentleman  having a  tract of land containing 527 acres,
3 roods, 36 perches, sold 129 acres, 3 roods, 39 perches ;  how
much has he remaining?             Ans. 397 A. 3 R. 37 P.
  4. A farmer  having 456 acres, 2  roods,  34 perches of land,
set off to his son  on  his marriage day, 175 acres, 2 roods, 36
perches, as his portion of his estate; how much had he remain-
ing?                    Ans. 280 acres, 3 roods, 38 perches.
  5. A person has 4 tracts  of land, each containing 24 acres,
1 rood, and 22 perches; how much has he in all ?
                          Ans. 97 acres,  2 roods, 8  perches.
  6. A gentleman  has 6 fields exactly of the same size, each
containing 12 acres, 2 roods, and 20 perches; how many acres
do they all contain  ?                  Ans. 75 acres, 3 roods.
  7. A landholder  divided 1326 acres, 3 roods, and 36 perches
of land  equally among his 5 sons; how many acres had each
one?                    Ans. 265 acres, 1 rood, 23^ perches.
  8. A farmer  having a  tract of land containing 216 acres, 2
roods, and 16 perches, would divide it into 12 equal lots;  how
many acres must each one contain ?      Ans. 18 A. 0 R. 8 P.
« 
LIQUID MEASURE.
117
        ADDITION OF LIQUID MEASURE.

  For the table, and the denominations, see Introduction, un-
der Liquid Measure.
  For directions to perform  the operation, see Addition of
pounds, shillings, and pence.
Sum
		EXAMPLES.				
T. hhd.	gal.	Hhd.	gal. qt.	Gal	qt.	pt.
12 3	36	18	21 3	56	3	1
10 2	45	25	48 1	47	2	0
9 1	21	36	12 2	75	3	1
16 2	31	75	52 3	27	0	1
22 3	48	81	24 2	36	3	1
SUBTRACTION  OF  LIQUID MEASURE.
  For directions to perform the operation, see Subtraction of
pounds, shillings, and pence.
      T. hhd. gal.
From 27  2  26
Take 18  3  45
Rem.
EXAMPLES.
Hhd. gal. qt.
127 42  1
  59 61  2
Gal. qt. pt.
57  2   0
49  3   1
    MULTIPLICATION OF LIQUID MEASURE.

  For directions to perform the operation, see Multiplication of
pounds, shillings, and pence.
         T. hhd. gal.
Multiply 12  3   25
By________7
Prod.
         Gal. qt. pt.
Multiply  54  2  1
By______1}
Prod.
EXAMPLES.
           Hlid. gal.  qt. pt.
   Multiply  12  42  3  1
   By_________8
   Prod.
        Hhd. gal.  qt. pt.
Multiply 24  16  2  1
By__________12
Prod.
L 
H8
ARITHMETIC.
          DIVISION OF  LIQUID MEASURE.

  i or directions  to  perform the operation, see Division  of
pounds, shillings, and pence.

                       EXAMPLES.
     T.  Hhd. gal. qt.                 Hhd. gal. qt. pt.
Div. 35  3  47  2,  by 8.        Div. 74  36  2   l,by9.
T. 8)35 Quot.	hhd. gal. qt. 3 47 2	11. 10.	T. 9)74	hhd. gal. 36 2	qt. 1
Hhd. Divide 64 Divide 121	gal. qt. 48 3, by 36 2, by		Divide Divide	Gal. qt. 146 3 97 3	pt. 0, by 12, 1, by 6.
                  PRACTICAL EXERCISES.
   1. A merchant has 4 sorts of wine in as many casks, which
measure  as follow: No. 1,  124 gals. 2 qts. 1 pt.—No. 2, 100
gals. 3 qts.—No. 3,  87 gals. 2 qts. 1 pt.—No. 4, 96 gal. 0 qt.
1 pt.; how many gallons has he in all ?  Ans. 409 gal. 0 qt. 1 pt.
  2. A farmer filled the following casks with cider :  No. 1, con-
taining 75 gals. 2 qts.—No. 2, 67 gals. 1 qt. 1 pt.—No.  3, 96
gals. 2 qts.—No. 4,  110 gals. 3 qts.;  how much cider do they
all contain ?                        Ans. 350 gals. 0 qts. 1 pt.
   3. A farmer made 627 gals'. 3 qts.  1 pt. of cider, and sold
249 gals. 1 qt., how much remains ?  Ans. 378 gals. 2 qts. 1 pt.
   4. A grocer purchased 5 hhds. 46 gals. 2  qts. of wine, and
sold  2 hhds. 54 gals, and 3 qts.; how much remains unsold ?
                                 Ans. 2 hhds. 54 gals. 3 qts.
  ■ 5. A farmer, at a cider-making, filled 6 casks, each contain-
ing 85 gals. 3 qts. 1 pt.; how much cider did he make in all ?
                                   Ans. 515 gals. 1 qt. 0 pt.
   6. A brewer filled 8 casks with beer, each containing 45 gals.
3 qts.; how many gallons were there in the whole ?  Ans. 366.
   7. A grocer purchased 25 bbls. 12 gals, and 3 qts. of cider,
which he would divide into 8  equal parts; how many barrels
allowing 32  gallons to the barrel, would there be in each part ?
                                  Ans. 3 bbls. 5 gals. 2f qts.
   8. A wine merchant imported 14 hhds. 47 gals, and 3  qts. of
 wine, which he would  put into 12 equal casks; how many gal-
 lons were there in each ?  Ans. 1 hhd. 14 gals. lqt. lpt. \% over. 
DRY MEASURE.
119
         ADDITION OF DRY MEASURE.
  For the table, and the denominations, see Introduction, under
Dry Measure.
  For directions  to perform the operation, see Addition of
pounds, shillings, and pence.
Bu.	V-	qt.	EX.A Bu.	MP. P-	LES. qt.	Bu.	7>-	qt.
75	3	1	125	2	A	226	0	6
84	2	0	264	3	1	420	3	5
46	3	7	136	2	0	731	2	4
26	2	6	45	0	7	129	0	7
16	3	5	84	3	2	314	5	2
10	0	6	275	0	6	114	2	1
Sum								
        SUBTRACTION OF DRY  MEASURE.

  For directions to perform the operation, see  Subtraction of
pounds, shillings, and pence.

                      EXAMPLES.
       Bu.  p.  qt.       Bu. p. qt.       Bu. p. qt.
From  219  2  4       74   1 6      274 1   5
Take  126  8  7       36   2  7       95  2  6
Rem.
      MULTIPLICATION OF DRY  MEASURE.

  For directions to perform  the operation, see Multiplication
of pounds, shillings, and pence.

                      EXAMPLES.
         Bu.  p.   qt.  pt.             Bu.  p.  qt.
   Mult. 117  3   6   1         Mult. 246   2   6
   by__________7_        by            9
   Prod.
      B.  p.  qt. pt.
Mult. 74   3  5  1
by___________11
Prod.
			
	B.	P-	qt.
Mult.	56	2	6
by			12
 
120
ARITHMETIC.
           DIVISION OF DRY MEASURE.
  For directions  to  perform the operation, see Division of
pounds, shillings, and  pence.

                       EXAMPLES.
Divide 112 bu. 3p. A qt. by 7-  Divide 2276m. 3p. 7qt. 1 pt. by 8.

      Bu.  p.  qt.                      Bu. p.  qt. pt.
    7)112   3  4                    8)227  3  7  1
Quot.
Div. 255 bu. 2p. 7 qt. 1 pt. by 9.   Div. 736 bu. 2 p. 5 qt. by 12.
Div. 274 bu. A p. 5 qt. 1 pt. by 10.  Div. 327 bu. 5 p. 6 qt.by 11.
                  PRACTICAL EXERCISES.
   1. A farmer raised of Indian corn 526 bu. 3 p. 7 qt.; of wheat
375 bu. 2 p. 6 qt.; of oats 216 bu. Op. 6 qt.; of barley 146 bu.
3p. 5 qt.; of rye 312 bu. 2 p. j how many bushels did he  raise
in all ?                             Ans. 1578 bu. 1 p.  0 qt.
   2. A miller purchased of one fanner 246 bu. 3 p. of wheat;
of another  327 bu. 2 p.; of a third 504  bu.  0 p. 6 qt.;  of a
fourth 462 bu. 3 p. 5 qt.; how many bushels did he purchase in
all ?                                Ans. 1541 6m. 1 p.  3 qt.
  3. A farmer raised  749 bu. 2 p.  7 qt. of wheat; and sold
247 bu. 3 p. 5 qt.; how  many bushels has he remaining?
                                    Ans. 501 bu. 3 p. 2 qt.
  4. A farmer raised 562 bu.  2 p. 6 qt. of the several kinds of
grain;  and estimating the consumption of his family and stock,
to  be 215 bu. 3 p. 7 qt-; how much will he have for sale.
                                   Ans. 346 6m. 2 p.  7 qt.
  5. A miller has in 5 several bins, 215 6m. 2 p. 6 qt. of wheat
each ; how many bushels has he in all ?  Ans. 1078 bu. 1 p. 6 qt.
  6. If a farmer raises 816 6w. 2 p. 6 qt. of wheat yearly ;  how
many bushels would he raise in 8  years ?   Ans. 6533 6m. 2 p.
  7. A miller has  2150 6m.  3 p. of wheat; and can  grind the
whole in 7 days; how many  bushels must he grind one day with
another, to finish the whole?               Ans. 307 bu. 1 p.
  8. If you divide 765  6m. 1 p. 6  qt. into 9 equal parts ;  what
will one part be ?                    Ans. 85 6m. 0 p. 1% qt.
  ADDITION OF MOTION, OR CIRCLE MEASURE.
  For the table, and the denominations, see Introduction under
Motion, &c.
* 
MOTION OR CIRCLE MEASURE.          121
  For directions to perform the operation,  see Addition  of
pounds, shillings, and pence.
				EXAMPLES,				
Sig. 2	0 14	i 46	n 25	0 ' 21 34	u 25	0 16	i 27	33
7	27	54	56	32 16	24	12	24	43
8	17	27	25	17 42	37	11	17	16
10	11	12	14	35 16	21	10	42	24
14	21	42	24	15 13	12	9	16	15
11	9	29	30	11 11	17	8	21	31
i.								
   SUBTRACTION OP MOTION, OR CIRCLE MEASURE.
  For directions to perform the operation, see Subtraction of
pounds, shillings, and pence.
					EXAMPLES.						
	Sig.	o	t	n	Sig.	o	i	n	0		
From	11	16	10	12	9	29	21	16	75	41	48
Take	5	24	35	4	6	22	29	46	36	50	55
Rem.								v-			
 MULTIPLICATION OF MOTION, OR CIRCLE MEASURE.
  For directions to perform the operation, see  Multiplication
of pounds, shillings, and pence.

                       EXAMPLES.
       Sig. °    '    "                 Sig. °    '    "
  Mult. 3  25  27   31           Mult. 5   12  16  29
  by____________6           by__________8
   Prod.                        Prod.
      O    I    II                    Oil'
Mult. 47  37  27               Mult. 54  26   56
by   ________11               by________12
Prod.                          Prod.
      DIVISION OF MOTION, OR CIRCLE MEASURE.
  For directions to perform the operation, see Division of
pounds, shillings, and pence.
            L 2 
122
ARITHMETIC.
                       EX AMPLFS
Divide8 sig. 25° 31' 46", by 6.  Divide 3 sig. 46° 56' 44", by 7.
     sig.  °    '    "                sig. o    '    »'  .
    6)3   25   31  46              7)3  46  56   44
Quot.
Divide 11 sig. 16° 45' 55" by 8.   Divide 154° 56'46" by 9.
Divide 9 sig. 17° 45' 55" by 5.     Divide 241° 35' 27" by 11.
                  PRACTICAL  EXERCISES.
  1. There are 3 places on the same parallel of latitude; the
first is 34°, 26', 35", in east longitude; the second is 45°, 27',
35", east of the first; and the third is 28°, 39', 46", east of the
second ; what is the longitude of the last place ?
                                    Ans. 108 deg. 34', 7".
  2. There are 2 places  in the  same latitude ;  one situated in
56°, 27', 45.", east longitude; and the other in 75P-, 46' 25'^ west
longitude; how far are they from  each other ?
                                    Ans. 132 deg. 14', 10"
  3. There are 2 places in the same latitude, one in 96°, 27',
32" east longitude, and the other in 37°, 46' 29" east longi-
tude ; what in the difference of their longitude?
                                     Ans. 58 deg. 40', 53".
  4. When the earth has moved  through 5 sig. 26 deg. 27 min.
 36  sec, how far  has she  to  go, to complete one revolution
 around the sun ?                 Ans. 6 sig. 3 deg. 32', 24"
                ADDITION OF TIME.
  For the table, and the denominations, see Introduction, under
Time.
  For directions to perform the operation, see Addition  of
pounds, shillings, and pence.
			EXAMPLES.						
Y.	771.	d.	D.	h.	TO.	sec.	M.	d.	h.
12	6	20	35	16	24	20	8	20	16
16	7	16	27	21	46	55	10	21	18
42	11	24	24	16	30	32	9	17	9
19	9	21	17	12	18	00	11	23	23
24	3	22	41	15	10	12	22	9	10
36	7	25	14	9	9	9	31	11	11
Sum 
TIME.
123
             SUBTRACTION OF  TIME.
  For directions to perform the operation, see Subtraction of
pounds, shillings, and pence.
                      EXAMPLES.
       Y.  to.   d.     Y.  to.  d.      D.    h.   to.   sec.
From  25   7  21     55  8   20     2l6   12   20  30
Take  16  8  24     31  9   22     174   16   31  45
Rem.
           MULTIPLICATION  OF  TIME.
  For directions to perform the operation, see Multiplication
of pounds, shillings, and pence.
                      EXAMPLES.
         Y.  to. d.   h.              D.   h.   to.  see.
Multiply 5   7  24  8     Multiply 162   12   17  31
By	6
Prod.	
Multiply By Prod.	Y. to. d. 24 5 10 9
By	8
Prod.	
D. h. Multiply 217 15 By Prod.	to. sec. 12 45 12
                DIVISION OF TIME.
  For  directions to perform the operation,  see Division of
pounds, shillings, and pence.
                      EXAMPLES.
      Y.   M.   d.  . h.            D.   h.  min.  sec.
Div. 116   10  26  6, by 7.  Div. 241  16  20   25, by 8.

   7)116   10  16   6          8)241   16  20  25
Quot.
      Y.  to.   d.   h.             d.   h.  min.  sec.
Div. 17   9  24   8, by 6.   Div. 324  17   34  44, by 8.
Div. 23  10  16  10, by 9.   Div. 235  19   36  55, by 11.

                  PRACTICAL EXERCISES.
  1. A lady has 3 daughters ; the age of the eldest is 15 years,
5 months, and 16 days—of the next, 12 years, 10 months, and
22 days—of the youngest, 10 years, 11 months, and 12 days;
what is the sum of their ages ?        Ans. 39 yrs. 3 m. 20 d. 
124
ARITHMETIC.
   2.  A person resided in Philadelphia from his birth  24 years,
9  months, and 22  days—afterward in New York, 6 years, 8
months,  and 10  days: from thence he  removed to Boston,
where lie resided 10 years, 3 months, and  12 days; what is his
age ?                                Ans. 41 yrs. 9 m. 14 d.
   3.  A gentleman has two sons; the elder is 17 years, 5 months,
and  19 days, and the difference of their ages  is 4 years,  7
months, and 24 days; what is the age of the younger ?
                                   Ans.  12 yrs. 9 m. 25 d.
   4.  A young lady  finished her education when  she was  16
years, 4 months, and 15 days old; she  had  been at school  7
years, 9 months, and 24 days; at what age did she begin ?
                                     Ans. 8 yrs. 6 m. 21  d.
   5.  A gentleman took a lease  of a farm for 7 years, and after
he had held it 4- years, 3 months, and 16 days, how much time
remained ?                          Ans. 2 yrs. 8 m. 14 ds.
   6. What will be the product of 5 years,  10 months, 16 days,
7 hours, and 20 minutes, multiplied by 6?
                         Ans. 35 yrs.  3 m. 7 d. 20 h. 0 min.
   7. A gentleman resided 7 years, 5 months, and 12 days in
Philadelphia, which was exactly one seventh part of his age;
how old was he?                    Ans. 51 yrs. 1 m. 24  d.
   8. A married lady is 36 years, 10 months, and 24 days old,
and her eldest daughter is 14 years, 11 months, and 19 days;
how old was she when her daughter was born ?
                                   Ans. 21 yrs. 11 m. 5  d.
  9. What will be the  quotient  of  64  years, 10 months, 17
days,  5 hours, 24 minutes, and 36 seconds, divided by 6?
               Ans. 10 yrs. 9 m. 22 d. 20  h. 54 min. 26 sec.
              VULGAR FRACTIONS.

  A vulgar fraction is a part or  parts of an  integer, or whole
number, and is expressed thus, §, J, which  mean two divided
by three, or two-thirds, and one divided by four, or one-fourth.
  The upper number is called the numerator, and expresses the
integer.
  The number below the line is  called the  denominator, and
expresses the parts into which the integer is divided.
  Vulgar fractions are proper, improper, compound, and mixed.
  A proper fraction has the numerator less than the denomina-
tor, thus, §, £, f. 
VULGAR FRACTIONS.
125
   An improper fraction has the numerator equal to, or greater,
 than the denominator, thus, f, f, |.
   A compound fraction is a fraction of a fraction, thus, f of f of J.
   A mixed number is a whole number and a fraction, thus, 7%, 5§.
   A whole number, or integer, is expressed in the form of a
 fraction thus, T, -f.
   Let the  pupil express by figures, in the form of a fraction, the
 following  numbers:  one  divided  by two, or  one-half;  two
 divided by three, or two-thirds; three-fourths; five-sixths; seven-
 eighths; nine-tenths; eleven-twelfths; thirteen-fifteenths; twenty
 divided by twenty-four; twenty-five divided by one hundred;
 forty divided by  twenty-two; eleven divided  by five;  eight
 divided by four; five divided by three; two divided by one, &c.
 What kind of fractions are they ?
                   PRACTICAL EXERCISES.
   1. How do you set down the J of 6 in the form of a vulgar
 fraction?—A. I set down J and \, and multiply the numerators
 of the fractions together, for a  new numerator, and the denomi-
 nators together for a new denominator; and I have this frac-
 tion |, or  6  divided by 2.  This gives 3,  the  half of 6; fqr
 6 contains  2, three times.
   2. How do you set down the J of 12 in the form of a vulgar
 fraction?—A. I set down £ and \f, and I multiply the nume-
 rators together  for a new numerator, and the  denominators
 together for  a  new  denominator; and then I have this frac-
 tion, lg.  This gives 4, the third of 12; for 12 contains 3, four
 times.
   3.  How do you set down the ^ of 20 in the form of a vulgar
 fraction ?—A. I set down A and 2T° and multiply the numerators
 together for a new numerator, and the denominators together
 for a new denominator, and I have 2?°. This gives 4, the fifth
 of 20; for  20 contains 5, four times.
   4.  How  do you set down £ of 7 in the form of a vulgar frac-
 tion ?—A.  I  set down £ and \, and multiply as in the preceding
 examples,  and I have £.  This being divided  gives 2, and £
 over; for 7 contains 3 twice, and  one-third, or one divided by
 three, over.
   5.  Will you set down  2 divided by 3 in the form of a vulgar
 fraction?—How much is it?—What kind of a fraction is it?
 i  6. Will you set down 11 divided by 12?—How much is it?
 —What kind of a fraction  is it ?
  7. Will you set down 9 divided by 16?—How much is it?—
What kind  of a fraction is it? 
126                 ARITHMETIC.
  8. Will you set down 7 divided by 5 ?—What kind of a frac-
tion is it?—How much is it?
  9. Will you set down 46  divided by 16?—What kind of a
fraction is it ?—How much is it ?
   10. What is the J of 63 ? Ans. 21.—How do you find it?
  11. What is the % of 33 ? Ans. 22.—How do you find it ?
  12. What is the | of 75 ? Ans. 50.—How do you find it ?
  13. What is the | of 30 ? Ans. 24.—How do you find it ?
  14. What is the £ of 120? Ans. 100.—How do you find it?
  15. What is the § of 16 ? Ans. 14.—How do you find it ?
  16. What is the || of 60 ? Ans. 55.—How do you find it ?
  17- What is the fv of 100? Ans. 5.—How do you find it?
  18. If you have f of a farm valued at  2500 dollars, what is
your part worth ? Ans. 2083 dolls. 33£.—How do you find it?
  19. If you own f of a ship, valued  at 5000 dollars, what is
your share worth ? Ans. 3750 dolls.—How do you find it ?
  20. A lot of ground is valued at 1000 dollars: what is £ of it
worth ?  What is § of it  worth ? What  is | of it worth ? What is
J of it worth ? What is ^ of it worth ? What is T*T of it worth :
What is J^ of it worth ?—What are these several answers ?—
How do you find them ?
  21. If you  own a house worth 1200 dollars :  for what sum
would you sell \ of it?  For what sum would you sell \ of it ?
For what sum would you sell |- of it?  For what sum would you
sell -j^j- of it ?   For what sum would you sell \ of it ?—What  are
these several answers ?—How do you find them?
  22. A cargo of flour is valued at 5500 dollars; how much
must you pay for £ of it ? How much for £ of it ?  How  much
for \\ of it ?  How much for T7^- of it ?—What are these  sev e-
ral answers ?—How do you find them  ?
  23. A gentleman'3 income arising from public stock is 2000
dollars  a year;  how much is it for \ a year?  IIow much for a
J of a  year?  How much for £ of a year ?  How much for |
of a year ?—What  are these several answers ?—How do you
find them ?

      REDUCTION  OF VULGAR FRACTIONS.
  A common measure of a fraction is a number, that will divide
the numerator and denominator without a remainder.
  The greatest common  measure is the  greatest number, that
will divide the numerator and denominator without a remainder;
as, 3 is the greatest common measure of the fraction T\, because
it is the greatest number that will divide them without a remain-
der.  It will then become §, which is of the same value of ■£%. 
VULGAR FRACTIONS.
127
  A fraction is said to be in its lowest terms, when no number
above a  unit,  will divide  both parts of it  without leaving a
remainder ;  thus, £ is in its lowest terms, because no number
above 1 will divide the parts without a remainder.

                         CASE 1.
  To reduce a fraction to its lowest terms.
  Rule.—Divide both parts of the fraction by the greatest num-
ber that  will leave no  remainder; and, if no number above a
unit will so divide them, the fraction is already in its  lowest
terms.
   1.  Reduce \\ to its lowest terms.                 Ans. £.
   Here by inspection, you see that 12 will divide both parts of
the fraction  without leaving a remainder, and it is the greatest
number  that will so divide it.  The fraction then becomes \;
which is of  the same value with if.
   2.  Reduce T8? to its lowest terms.                  Ans. §.
   Here  by  inspection you find that  4 is the greatest number,
 which will divide both parts of the fraction without a remainder.
 Let it be divided, and it will  become §, which is of the same
 value with T82-
   3.  Reduce if to its lowest terms.                   Ans. §.
   4.  Reduce \\ to its lowest terms.                   Ans. %.
   5. Reduce -^^ to its lowest terms.                  Ans. \.
   6. Reduce T5T to its lowest terms.                   Ans. \.
   7. Reduce f- to its lowest terms.                    Ans. §.
    8. Reduce f| to its lowest terms.                   Ans. \.

    If jhe greatest common measure cannot readily be discovered
 by inspection, the division may  be repeated, until you reduce
 the fraction so that no number above unit will divide it  without
 a  remainder.  The fraction then will be  in  its lowest terms.
 Thus :  let  T\27 be reduced to its lowest terms.  Divide, in the
 first  place, by 2, which  will  reduce the fraction to ||-  ; which
 will divide  by 4. It will then be reduced to T9¥.  Here I find by
 inspection  that 9 will divide both parts of the fraction without a
 remainder.   The division being made,  the fraction becomes £,
 which is the same value of T7^.  In like manner,
    9. Reduce ££ to its lowest terms.                 Ans. \.
    10. Reduce || to its lowest terms.                 Ans. \.
    11. Reduce £fc to its lowest terms.                Ans. ^.
    12. Reduce £f£ t0 'ts  lowest terms.                Ans. £.
    13. Reduce iff- to its lowest terms.                Ans. £.
    14. Reduce -iff- to its  lowest terms.               Ans. T£. 
128
ARITHMETIC.
   Or, divide the greater part of the fraction by the less, and this
 divisor by the number  remaining; and so on till  there is no
 remainder: the last divisor will be the greatest common measure;
 by which divide both parts of the fraction, and it will be reduced
 to its lowest terms.
   15. Reduce -£gr to its lowest terms.               Ans. j|.

          60)125(2           5)tVt—H the lowest terms.
             120
               5)60(12
                 60

   16. Reduce £$$■t0 'ts lowest terms.                Ans. 4.

          288)480(1
             288
             192)288(1      96)f||=| the lowest terms.
                  192
                  "96)192(2
                      192

  In the first of these examples, I divide 125 by 60, and I have
5 remaining ; by which I divide 60, the previous divisor,  and
nothing remains ; 5 therefore is the greatest common measure.
Divide both parts of the  fraction by 5, and you have ||, the an-
swer, which are the lowest terms of the fraction.  In the same
manner proceed with the following fractions.
  17. Reduce -fifo to its  lowest terms.                Ans. ||.
  18. Reduce |f| to its  lowest terms.                 Ans. £.
  19. Reduce 7Vtbt t0 'ts lowest terms.              Ans. £.
  20. Reduce £f |s to 'ts l°west terms.            Ans. £f§|.

                         case 2.
  To reduce several single fractions to others retaining the same
value, but having a common denominator.
  Rule.—Multiply each numerator into all the denominators
except its own, for its respective numerator; then multiply all
the  denominators together for a common  denominator.
  Note.—Each fraction, in the first place, should be reduced
to its lowest terras, if it is not so already.
                     EXAMPLES.
1.  Reduce f, T87, and £ to a common denominator.
                                             m it? 2«-
Ans. ££ i 
VULGAR FRACTIONS.
129
  Before you begin the operation, reduce the fractions to their
lowest terms; thus, £- divided by 3 becomes £ : T8T divided by 2
becomes |: and £ divided by 2 becomes £.  These are of  the
same value with the original fractions f, TV, and f.   The frac-
tions to be reduced to a common denominator are,   £,   |,  £.
         1X5X2=10 0
         4X2x2=16 > numerators.
         1X2X5 = 10)
       2x5x2=20 the common denominator.
  The fractions become,} g, £§,i&; which express the values
of |, TV» and f •
  2. Reduce \. £. W, to a common denominator.
            *' 8' 12'                      in,  24  24  44
                                          Ans. -5-5-, -£•£, 4g.
  Here you observe the two first fractions are not in their lowest
terms.  Let them be reduced, and they become £, and £;  the
last fraction is already in its lowest terms.   You then have these
fractions:  £,  £,  |L

              1X2X12=24 ~)
              1 X2 X 12=24 > the numerators.
              11X2x2=44 )
       2x2x12=48  the common denominator.

  The fractions become ££-, £|, f|.  These fractions are of the
same value with f, |, and }-i, the given fractions. In like man-
ner, let the following fractions be reduced to a common denomi-
nator.

  3. Reduce 4-, if, and &, to a common denominator.
                                          Ans 32   24  i2
                                          Ans. 3^, 47, ^j.
  4. Reduce §, /T, and -ff, to a common denominator.
                                          A r»e  1 8   9  18
                                          Ans. gr, -57, 2T'
  5. Reduce £, 4, and &, to a common denominator.
             4' 5'     8'                   Ans i*   i«  20
                                          ■rtlJS- 70 5 lO') TO"
  6. Reduce T<U |, 4, and 4, to a common denominator.
                                  Anc  378  315  70   540
                                  AHS. "6"?p'> STOJ TT¥5 6~-3"0*
  7. Reduce t4^, i, 4,1, to a common denominator.
             10  3     "'            AnB  120  100  120  225
                                  AnS' ?T)0) SUV "3~0"0"> "3"0"0"'
  8. Reduce £, T9¥, and I \, to a common denominator.
                                        Anc 840 860 880
                                        ■^"'•."960^ StVS 9"6"ZT"
  9. Reduce |, £, f, and |, to a common denominator.
                  ■r-               An«96603040
                                 AI1S.  yj^, -j^^, 720-, yj^.
  10. Reduce f, T7r, |, and \, to a common denominator.
                                  All(5  240  352  120   90
                                  AIIS. -g-j^, -j-ij-^, -y^, -jg-fl-.
            M 
130                  ARITHMETIC.
                        CASE 3.
  To reduce a mixed number to an improper fraction :
  Rule.—Multiply the whole number by the  denominator of
the fractional part;  to the product add the numerator of the
fractional part; and under the sum, place the denominator.

                       EXAMPLES.
  1. Reduce 19T7o- to an improper fraction.         Ans.  \9J.
  Here  you multiply 19 by 10, and add to thp product 7, the
numerator of the fractional part, which will make 197 ; under
this, place 10, the denominator of the fractional  p^rt, thus,  'y9/;
which is the answer.  In like manner reduce the following mixed
numbers.
  2. Reduce 27§ to an improper fraction.
  3. Reduce 5^T to an improper fraction.
  4. Reduce 9§ to an improper fraction.
  5. Reduce 467r to an improper fraction.
  6. Reduce 54^| to an improper fraction.
  7- Reduce 125T40-50-to an improper fraction.

                        CASE 4.
  To reduce an improper fraction to a mixed number :
  Rule.—Divide the numerator by the denominator.
  Note.—This case is the reverse of the preceding one.

                       EXAMPLES.
  1. Reduce  W9 to its proper terms.
  2. Reduce 3TST4 to its proper terms.
  3. Reduce || to its proper terms.
  4. Reduce 4|1 to its proper terms.
  5. Reduce 9Tsyl to 'ts ProPer terms.
  6. Reduce 1-|^8 to ',s ProPer  t«'rms.
  7. Reduce  y3 to its  proper terms.
  8. Reduce V«* to Ils proper terms.
  9. Reduce 2f6 to its proper terms.
 10. Reduce a^|4 to its proper  terms.

                        case 5.

  To reduce a- compound fraction to a single one :
  Rule.—Multiply all the numerators together for a new nume-
rator; and all the denominators together for a new denominator.
    Ans. y.
    Ans. %\.
    Ans y.
  Ans. *f».
 Ans. '»48.
Ans. »"$*.
Ans. 12jf
Ans. 19jf.
 Ans. 5-fe.
 Ans. 46|.
 Ans. 56^
 Ans. 5A^
   Ans. 1|
    Ans. 9.
  Ans. 64.
 Ans. 126. 
VULGAR FRACTIONS
131
But, in the first place, reduce the fractions to their lowest terras,
if they are not so already.

                        examples.
   1. Reduce § of *- of-^ to a single fraction.          Ans. ±f.
   Here you perceive £ and ■& are not in their lowest terms.
By dividing them  by 2, they become § and f, which  are their
lowest terms.  The compound fraction to be  reduced  then will
be,f off of*.
       2x2X4 = 16 product of the numerators.
       3 X 3  X 5 = 45 product of the denominators.   The
single fraction  therefore is ±f.
   2. Reduce \\ of £„ of f of % to a single fraction. Ans. Ts/T.
   Here the first, second, and fourth fractions are not in their lowest
terms.  Let them be reduced, and the fraction becomes f of f"
off of£.
      6x3x2  X 1 = 36 the product of the numerator.
      7x5x5x3= 525 the product of the denominators.
The single  fraction is, therefore, -g/j.
   3. Reduce £ of \ of f to a single fraction.     Ans. /? = £.
   4.  Reduce £ off off  to a single fraction.  Ans. T\8g- = 58T.
   5.  Reduce § of f of T£ to a single fraction. Ans. £j*s = £|.
   6.  Reduce f of £ off of-}f to a single fraction.  Ans. y/yy.
   When the same figure or figures occur both in the numerator
and denominator, they may be cancelled  without affecting the
result of the operation.   This depends upon the well known
axiom, that if equals  be multiplied or divided  by the same num-
ber, the product, or quotient will be equal  accordingly.
   7. Reduce § of f of ^  to a single fraction.           Ans. *.
   Here you find 3 in the denominator of the first, and in  the
numerator of the  second, fraction.   Also 4 in the denominator
of the second, and in  the numerator of the third: all  of which
may be cancelled, leaving f for the answer, which  are the only
remaining terms, or parts of the given fractions.
   8. Reduce | of T8g- of f to  a single fraction.        Ans. T3g-.
   9. Reduce f- of T9f  of T8T to a single fraction.  Ans. 746^ = |..
   10. Reduce  | of f of £ of ^ to a single fraction. Ans. 4|f.

                         case 6.-

   To reduce a fraction of a  less denomination, to the fraction
of a greater denomination, retaining the same  value : 
 132                   ARITHMETIC

   Rule.—Compound the given fraction with the several denomi-
 nations, expressed in the form of a fraction, between its own, and
 the one, to which you  would reduce it: this compound fraction
 reduce to a single one, as directed in the preceding case.
   Note.—Observe to place the several  denominations, with
 which the given fraction is to be compounded, as denominators,
 and a unit or 1, as the several numerators.

                        EXAMPLES.
   1. Reduce £ of a cent to the fraction of a dollar.  Ans. Tjff.
   Here between  |-  of a  cent, and a  dollar are two denomina-
 tions, cents and dimes.  You therefore compound  it  in the
following manner, | of ^ of Aj;  because 10 cents make a dime,
and 10 dimes make one dollar.
   Therefore f X TV * TV = T «T -*- 5 =  f^.
   The product of the  numerators is  5, and that of the denomi-
nators 600.  The fraction z^, divided by 5, the greatest com-
mon measure, gives y^- of a dollar,  for  the  value of £ of a
cent.
   2. Reduce-J of an oz. troy to the fraction of a pound.
                                              Ans. 77-5. lb.
   Here there is only one denomination between the ounce and
the pound, namely, 12.  Let £ be  compounded with  it, and
you have -j of -jL, or -g- x T\- =  -§V   I" 1'kc  manner proceed
in  the following examples.
   3. Reduce £ of a penny to the fraction of a £.
                                     Ans. 7-^77 = svv *■
   4. Reduce *- of a gallon to the fraction of a hhd.  Ans. 7^T.
   5. Reduce f- of a pound avoirdupois to the fraction of a cwt.
Reduce f of a cent to the fraction of a dollar.
Ans.
T7J0T-
Ans. t^tt. dol.
Reduce -fo of a minute to the fraction of a day.
i\rs'
                                       Ans. t^I^tt—tsVo-
  8. Reduce § of an hour to the fraction of a year of 365 days.
                                      Ans- *?Ity= rTT40-
  9. Reduce f of a pint to  the fraction of a barrel  of 32  gal-
lons.                                           Ans. y^*.
  10. Reduce  | of a perch to  the fraction of an acre.
                                               Ans.Ti^
  11. Reduce  f of a yard to the fraction of a mile. 
VULGAR FRACTIONS.
133
                         CASE  7.
   To reduce a fraction of a greater denomination to the fraction
 of a less denomination, retaining the same value :
   Rule.—Compound the given fraction with the several deno-
 minations  between it and the one, to which you would reduce
 it: this compound fraction reduce to a single one, by case 5th,
 and this again reduce to its lowest terms, by case 1.
   Note.—Observe  to  place  the several denominations, with
 which you compound the given fraction, as numerators, and a
 unit or 1, as the several denominators, contrary to the order of
 the preceding case.

                        EXAMPLES.
   1.  Reduce y^ of a cwt. to the fraction of a pound.
                                                 Ans. f lb.
   Here there is one  denomination, namely, 112 : which place
 as a numerator with  a unit for its denominator, thus,  A2. Mul-
 tiply  ihis by T1J557, in the following manner:
  . 7tro"F X' t2 = -iVifff 5 which reduced  to its  lowest terms, by
 dividing by 112, the  greatest common measure, will give f.
   2.  Reduce 1T of a pound troy to the fraction of an ounce.
                                                Ans. f oz.
   Here there is only one denomination, namely, 12.  Place this
 in the form of  a  fraction thus, »T2, and multiply it by ?2T, and
 you have -22Tx >■/* =£$=», by dividing by 3, the greatest com-
 mon measure.
   3.  Reduce y|7 of a dollar to the fraction of a cent.  Ans. |.
             ih*  'T°X xt°=t-SS=J of a cent.
   Here you divide by 20, the greatest common measure.
   4.  Reduce  T^5 of a ■£•to the fraction of a  penny.   Ans. |.
   Here by  compounding T-575 with the denominations between
a penny and a pound, you have nJ„x \2X  *f° ; which being
reduced, becomes T|£g.   This divided by the greatest common
measure, gives -£ of a  penny.
  5.  Reduce ^^ of a hhd. to the fraction of a pint.    Ans. |.
  6.  Reduce ^y- of a day to the fraction of a minute. Ans. 4-5.
  7.  Reduce yg7^ of an acre to the fraction of a perch.  Ans. I.
  8.  Reduce To|fl5 of a mile to the fraction of a yard.  Ans. -£.
  9.  Reduce gL Gl a £. to the fraction of a shilling.
                                              Ans. £=£».
  10. Reduce ,|5 of a cwt. to the fraction of a lb.  Ans. i lb.
       M 2 
134                   ARITHMETIC

                        CASE 8.
  To find the value of a fraction in the known parts of an inte-
ger, or whole number:
  Rule.—Multiply the numerator of the fraction by the parts
of the integer, and divide by the denominator.
  Note.—The questions under this case are similar to the prac-
tical exercises, page 125 et seq., and may be solved by the direc-
tions there given, which are in substance the same with the above
rule.
                        examples.
  1. Reduce f  of a dollar, or 100 cents, to its proper quantity.
                                            Ans. 75 cents.
                 axioo_300—7-5 cents-
  2. Reduce T|- of a pound troy to its proper quantity.
                                                Ans. 9 oz.
  3. Reduce T  of a mile to its proper quantity.
                       Ans. 4 fur. 125  yds. 2 ft. 1 in. 2}, b.c.
  4. Reduce -|  of a yard to its proper quantity.
                                          Ans. 3 qr. l^na.
  5. Reduce r35 of a day to its proper value.   Ans. 7 h. 12 m.
  6. Reduce ^  of a dollar, or 100 cents to its proper value.
                                              Ans. 12£ cts.
  7. Reduce -§$% of a hhd. to its proper quantity. Ans. 50 gal.
  8. Reduce Jf of a day to its proper quantity.
                                 Ans.  16 h. 36 m. 55TT. sec.
  9. Reduce |f of a hhd. to its proper quantity.
                                 Ans.  33 gal. 3 qt. 1 pt. ff.
  10. Reduce -li of a £. to its proper quantity.    Ans. 18«. 4a.

                         case 9.
  To  reduce any  given sum, or quantity to  a fraction of any
greater denomination of the same kind :
  Rule.—Reduce the given sum or quantity,  to the  lowest
terms  mentioned ;  and the integer to the same  denomination,
for a denominator to the fraction.  Lastly, reduce the fraction to
its lowest terms.
                        examples.
  ]. Reduce 9  oz. troy, to the fraction of a pound.  Ans. %lb.
  Here 79?  is  the fraction,  which divided by 3, the greatest
common measure,  will give %.
  2. Reduce 6s. and Sd. to the fraction of a £.      Ans. $£.
   Here you must reduce  the  shillings  and  pence to  pence,
 which make 80, for the numerator of the fraction.   Then re- 
VULGAR FRACTIONS.
135
duce 20s. to pence, making 240, for a denominator, and the frac-
tion is ,%, which divided by 80, the greatest common measure,
gives £.
  3. Reduce 1 JR. and 30 P. to the fraction Of an acre. Ans. TV
  Here reduce  1  R. and 30  P. to perches,  which make  70
perches: and 160 perches are an acre; therefore, y^V will be
the fraction, which reduced to its  lowest terms, gives y£.
  4. Reduce 13 hours and 30 minutes to the fraction of a day.
                                                Ans. yV
  5. Reduce 3  qrs. and 2 na. to  the fraction of a yard.
                                               Ans. £ yd.
  6. Reduce 1  month and 5 days to the fraction of a year, the
month being 30 days, and the year 365 days.         Ans. T77.
  7. Reduce 3  dolls, and  2 dimes to the fraction of an eagle.
                                                 Ans. ■£■%.
  8. Reduce 3 fur. and 25 rods  to the fraction of a mile.
                                                 Ans.  f|.
  9. Reduce 125 yards and 2 feet to the fraction of a mile.
                                               Ans  3 7 7_
                                               Ans. -s^-tjs.

       ADDITION OF VULGAR FRACTIONS.

  Rule.—Reduce the given fractions to a common denomina-
tor : add  the several numerators together, under which place
the common  denominator : then reduce the fraction to its low-
est  terms.                             *
  Note.—If there are compound fractions, reduce them to sin-
gle ones by case 5.
  If there are whole numbers given, they may be omitted in the
operation,  and  afterward added  to  the value of the fractional
part..
                        EXAMPLES.
  1. Add £ and £ together.               Ans. ||=1 ^=11

       1X8=8?,,
       _,    „ __ 74 > the numerators.

                 22=sum of the numerators.
  2x8=16  the  common denominator.  2| is  the fraction,
which, reduced to its lowest terms by dividing by 2, will be  \
This, divided by the denominator, gives if.  In like manner
add the following fractions.
  2, Add yTg-, jA, and | together.                 Ans. 2^\\.
  3. Add ^, |, and £ together.                   Ans. li||.
  4. Add f, -|, and y^ together.                    Ans. l|. 
136
ARITHMETIC.
   5.  Add \7\ and § together.                     Ans. 18^.
   Here you may omit ihe whole number, and  find the sum of
£ and §, to which al'tei ward add the 17.
       1X3=3)
       2 X 2 = 4 S numerators-
                7 the sum of the numerators.
   2x3=6, the denominator.  The fraction is -J, which divided
by 6,  the denominator, gives 1\. To which add  the integer, and
you have 184;.
  6.  Add 7, and £ of § together.                    Ans. 7\,
  Here you reduce the compound  fraction  to a single  one;
which gives |.. This,  in its lowest terms, is £.  To this add the
integer 7, nnd the whole is 7^.
  7.  Add \ of-J, and  \  of \\ together.           Ans. l,y„.
  Here you see ihat  2 and \ are lo  be reduced to  their lowest
terms, in the  first place.  The  fraction then becomes, \  of J,
and § of |-§.  Reduced to single fractions, we have T7ff and ££..
These reduced to a common denominator,  and to  their lowest
terms, give 1 £\.
  8.  Add 12J, 3§ and 4% together.               Ans. 20\\.
  Here omit the integers, and the fractions to be added will be
J, §, and f; which reduced as above directed, will  be 1\\ :  to
which add the several integers, and you have 20j.|
  9.  Add £ of § of 3 and  14 -f° together.         Ans. 1A\\\.
  30. Add"£ of 9b, ami $ of 14  together.          Ans.  43',^
  Here you  will make  the  integers  improper fractions, by
placing them  above a  unit, thus, 9S and 'j*.   The  fraction will
then be £ of 91s, and £ of 1T*.  Reduced, to single fractions you
have  9,5 and 9ff8.  These reduced, as in  the preceding examples,
you will have 43{^.
  11. Add 19,7, a»d £ of% together.               Ans  26J.

  To add fractions of money, weights,  measures, &c. together:
  Rule.— Find  the  value of each  fraction  separately, as in
case 8, and add these values together as in compound addition.
  12. Add \ an ounce troy, to § of a pound. Ans. 8 oz. 10 dwt.
       £ X  y° = \°= 10 dwt. the value of half an oz.
       | X y2 =  274 = 8 oz, the value  of two thirds of a pound.
  These added together make 8 oz. 10 dwt.
  13. Add £ of a  mile, to ^ of a furlong. Ans.  6 fur. 28 poles.
  Here multiply 4 by  ■«-, because 8 fuilongs make a mile; and
you have ^ = 6 furlongs, for the value.   Proceed  now  to
find the value of yj of  a furlong.  Multiply  it by  40,  because 
VULGAR FRACTIONS.
137
40 poles make a furlong, and you will have  y8^ =28 poles
for the value of J of a furlong.  These sums added together
make 6 fur. 28 poles.
  14. Add £ a yard to £ of a foot.           Ans. 2 ft. 2 in.
  15. Add f of a week, J of a day, and \ an hour together.
                                   Ans. 2 d. 14 h. 30 m.
  16. Add § of a dollar to \ a cent.        Ans. 67 c. 1,60 m.
  17- Add | of an eagle to \ of a dollar.        Ans. $3,37£.
  18. Add \ of a £. to f of a shilling.           Ans. 3s. 3rf.
  19. Add § of a league, 4s of a mile, and f of a furlong together.
                                     Ans. 2 m. 270 rods.
  20. Add | of a hhd. to % of a gal.         Ans. 53 gal. 1 qt.
     SUBTRACTION  OF  VULGAR FRACTIONS.

   Rule.—Reduce the fractions to a common denominator, and
 subtract the less numerator from the greater.  Place the differ-
 ence above the common denominator.   Which fraction reduce
 to its lowest terms, if it is necessary.

                       EXAMPLES.
   1. From \\\, take |.                          Ans. Ty5.
             111X  4=444}
               3X112=336 5 numerators-
                        108=difference of numerators.
             112x4=448 the common denominator.
   The fraction becomes igf.  Which reduced  to  its lowest
terms,  by  dividing by 4, the greatest  common  measure, you
have T2T72.
   2. From T%, take 3.                          Ans. $$£
   3. From *f, take |.                           Ans. ^.
   4. From |, take f.                            Ans. £1.
   5. From fl, take T*T.                           Ans. TVT.
   6. From 9fy, take 14T.                      Ans. 81^.8.
   Here you reduce the mixed  numbers to improper fractions as
in  case 3d.  The fraction then will be 2|9, and 1Qrx:  which re-
duced to a  common denominator, will be
           289X7=2023 )   .
           101X3= 3Q3 ^he numerators.

                    1720=difference of the numerators.
              3x7=21  the  common denominator.
   The fraction then becomes 1JH°=8lj^. 
138                 ARITHMETIC.
r the numerators.
  7. From 71 J, take Tf                        Ans. 70«f
  8. From 96, take j.                           Ans. 95}.
  Here you make 96 an improper fraction, thus, 9TS; and reduce
the fractions to a common denominator;
           96x5=480)
              IX1=__3S
                   477=difference of the numerators.
             1X5=5 the common denominator.
  The fraction then will be *J:7=95|.
  Orr subtract the numerator of the fraction from the denomi-
nator for a new numerator, und^r which write the denominator,
and add 1  or unit to the place of integers, thus:
      From 96
       Take   0 |
            95  f the Ans. as before.  This is a  very short
method, but is  of very limited use.  In like manner may any
other examples of the kind be solved.
  8. From 121, take \\.                      Ans. 120TV
  9. From 79, take £.                           Ans. 7«|-
  10. From y, take | of §.                        Ans. -ft.
  Here J of § is a compound fraction.  It must be reduced to a
single one, thus, \ of § = \ or £ by dividing by 2.
      3 y ^ = O )
       a   *    . >  the numerators.
       4 x 4=_4J
                5 = the diff. of numerators.
  4 X 3 = 12, the denominator.  The fraction then becomes -,'j.
  11. From % of ♦, take \.                        Ans. J9.
  12. From 14¥, take § of  19.                   Ans. I-ft.
  13. From 19% take § of 19.                     Ans. 7.

  To subtract fractions of money, weights, measures, &c.
  Rule.—Find the proper value of the fractions separately, and
subtract the less from the greater, as in compound subtiaction.
  14. From J of a ^.take \ of a shilling.       Ans. 5*. llrf.
  15. From T6ff of an oz. take §ofa dwt.   Ans. 11 dwt. 3 gr.
  16. From § of a league, take T7^ of a mile.
                                  Ans. 1 m.  2 fur. 16 rods.
   17. From | of a dollar, take £ of a dime.       Ans. 55 cts.
f   18. From § of an acre, take £ of a rod.
                                   Ans. 2 rods 6§ perches.
   19, From & of a day take § of an hour.
                                 Ans. 19 h.  54 m. 17} sec. 
VULGAR FRACTIONS.
139
 20. From | of a great circle, take J of a degree. Ans.l34°30'.
 21*. From | of an hhd. take £ of a gallon.        Ans. 52 gal.
 22. From J of a ton wt. take f of a lb.  Ans. 2239 lb. 10 oz.

   MULTIPLICATION OF VULGAR FRACTIONS.

  Rule.—Multiply the numerators of the given fractions for a
new numerator;  and the denominators together for a new deno-
minator.
  Note.—Mixed numbers are  to be reduced to improper frac-
tions,  and compound fractions to  single  ones, if there are  any
such given ; then proceed as the rule directs.

                      EXAMPLES.
  1. Multiply %, by f.                      Ans. /, = ,V
  2. Multiply 7, by t\.                           Ans. -ff.
  3. Multiply i, by ,•,.                         Ans. T|y.
  4. Multiply 7?, by 8£.                        Ans.  6lf.
   Here the fractions must be reduced to improper ones,  as  in
 case 3d.  They (hen become \9  and y, which, multiplied by
 the rule, are 4|3 = 6 if.
   5. Multiply 3 of 8, by % of 5.                    Ans. 21.
   Here make the integers 8 and  5 improper fractions, thus, f-,
 and \. The fractions will be compounded, thus, $ of \, and £ of
 4, or 2/ and \s ; which, multiplied by the rule, will be **» = 21 ;
 by dividing by 40,  the greatest common measure.
   6. Multiply $, by 13TV                       Ans. 12J$.
   Here you must reduce 13T9^ to an improper fraction, thus y^.
 Then multiply £, nnd  'TV9, vou will have 9f7o3 = 12||.
   7. Multiply | of T,  by f of Sf.                  Ans-. ||.
   Here you must  reduce 3f to an improper fraction, thus, y.
 The fractions reduced to single  ones are z%, and y/ ; which
 multiplied  become ^V   Thls divided by 30, the greatest
 common measure, trives |?.
   8. Muhiplv £ of 4, by ,V of {4.                 Ans. ^.
   9. Multiply $ of 91, by 71 £•                 Ans. 5205J
  10. Multiply | of y of £, by 7 of £.               Ans. T'TV
  11. Muhiplv | of j% of £ of §, by J of 7-           Ans. 3^.
  12. Multiply I of 112, by 5 of 48.              Ans. 1008.

        DIVISION  OF VULGAR FRACTIONS.

   Rule.—Multiply the numerator of the dividend  into the
 denominator of the divisor, for a new numerator, and the nume-
 rator of the divisor, into the denominator of the dividend, for a 
140                  ARITHMETIC.
new denominator. Then reduce the fraction to its lowest term*,
if necessary.
  Note.—Mixed numbers must be reduced to improper frac-
tions, and compound fractions to single ones.

                        EXAMPLES.
  1. Divide ]T, by $.                            Ans. ljf
                17x5=85 new numerator.
                21 x3=63 new denominator.
  The fraction is therefore f f=l££.
  2. Divide £, by 7.                          Ans. 41=-^.
  3. Divide T, by TV                         Ans. £f=2.
  4. Divide 7, by TV                   Ans. 7o = lff% = lf
  5. Divide l£, by A\.                       Ans. \\=-^t.
  Here you must reduce the mixed numbers to improper frac-
tions, thus, | and */, and proceed as in the preceding examples.
  6. Divide-J, by 4.                              Ans. ,\.
  Here reduce 4 to an improper fraction, and then proceed as
in the above.
  7.  Divide 4, by §.                         Ans. V =4J.
  Here again you make 4 an  improper fraction, and multiply
the fractions as the rule directs.
  8.  Divide * of 19, by § of \.               Ans.  y =7f
   Here reduce the compound  fraction to a single one, and the
19 to an improper fraction ; and you will have \ of L? and -ft.
Reduce now \ of \9 to a single fraction, and you will have >T9.
The  fractions are \9, and Tsf,  or £, dividing by 6 the common
 measure.
              19 X 2 = 38 the new numerator.
               1  x  5 = 5 the new denominator.
 which divided gives as above.
   9-  Divide £ of %, by § of g.                    Ans. <=f.
   These reduced to single fractions become f, and  -ft, or \.
              2  x 2 = A the new numerator.
               1 x 6 = 6 the new denominator,
 which reduced gives as above.
   10. Divide | of 4, by 4f                  Ans. \ 8|=£f
   Here | x | = Vj and 44== V* The fractions, thus reduced
 and multiplied, become \\%, which divided by 9,  the greatest
 common measure, reduces it to \\.
   11. Divide 71 by 9f.                           Ans. |4.
   12. Divide I of 6 by £ of 7 of }f              Ans. 8}°.
   13. Divide 5205} by 7 of 91.                   Ans. 71 £ 
VULGAR FRACTIONS.
141
  Question?.—1. What is a vulgar fraction ?—2.  How do
you represent it ?—3.  What  is the upper number called ?—4.
What does it express.—5.  What is the number below the line
called ?—6.  What  does it express ?—7-  How are vulgar frac-
tions commonly divided ?—8. What is meant by a proper frac-
tion ?—9. What is meant by an improper fraction ?—10.  What
is  meant by a  compound  fraction?—11. What by  a mixed
number ?—12.  How  would you represent, or express a whole
number in the form of a fraction ?
   13. What do you mean by a common  measure of a  fraction ?
   14. What do you mean by the greatest common measure of a
fraction ?—15.  Will you give an example?
   16. When is a fraction said to be in its lowest terms ?—17-
Will you give an example to show  this?
   18.  How do you reduce fractions to their lowest terms ?—19.
But, if you cannot readily discover their greatest common mea-
sure, what do you do?—20. There is another method; and what
is that ?
   21. How do you reduce fractions to a common denominator ?
   22.  What should always be done previously ?
   23.  How do you reduce mixed numbers to improper fractions ?
   24.  How do you reduce compound fractions to single ones?
   25.  What should always be done previously ?—26. How do
 you reduce a fraction to its  proper value?— 27- How do you
 add fractions, which have  the same denominator?—28. If the
 given fractions  have different denominators, what do you do ?—
 29. After you  have reduced them to a common denominator,
 what do you do ?—30.  If the fractions are mixed numbers or
 compound fractions, what  must you previously do with them?
   31. How do you add fractions of  money, weights, and mea-
 sures ?
   32. How do you subtract vulgar fractions, when the denomi-
 nators are the same ?—33. When they have different  denomina-
 tors, how do you proceed ?—34. How do you subtract fractions
 of money, weights, and measures?
   35.  How do vou multiply vulgar fractions ?—36.  When the
 fractions are mixed numbers, or compound fractions, how do you
 proceed ?
    37.  How do you divide vulgar fractions ?—38. If the frac-
 tions  are mixt numbers,  or compound fractions, how do you
 proceed ?
             N 
142                   ARITHMETIC.
             DECIMAL FRACTIONS.

  A decimal fraction is a part or parts of a unit, denoted by a
point or dot, prefixed to one or more figures; thus, .5, .45, .255.
The first figure, after the point or dot, denotes so many tenths
of a unit;  the second  figure denotes so  many hundred parts
of a unit; the third figure denotes so many thousand parts of a
unit, &c.
  The value of any decimal fraction may be expressed in the
form of a vulgar fraction, thus, fV> tVoj tWo"* &c-
  A mixed number consists of a whole number and a decimal,
thus, 125.7,-74.125,-65.25.
  The value of a decimal  fraction decreases from  the decimal
point in a tenfold  proportion, thus .5, .05, .005.  The .5 in the
first place is ten, times the value of it  in the second, and a hun-
dred times its value in the  third place; and  so on to any num-
ber of places.
  Cyphers added to decimals neither increase nor diminish their
value.  But, if they are prefixed to them, they lessen their value,
in a tenfold proportion.  Thus, by adding  a cypher to .5 we do
not affect its value.   It then becomes fifty hundredths of a unit,
which  is the same  with 5  tenths; and so if two,  three, or four
cyphers were added to it. But placed before the decimal, they di-
minish its  value  by removing it farther from the  decimal point,
thus, .005,  ,05, .5, all express different values.  The first is one
tenth part  of the second, and the second one tenth part of the
third; and so for any greater number of places.
	TABLE		OF DECIMAL PARTS.					
.1	2	3	4	5	6	7	8	9
H	S3	H	*	p	5="	*	p	H
CD	c							3-
3	a a.	o c	H	H	5'	3	3.	0 c
	-1	C/l	3-	3"				</]
	n	P		O	S			V
^	a.	s	C	Zl	3"	5	o	3
"!	B"	q.	P		T3	3	^.	O.
o —1	-3	3" "a	S a.	a a.	S3	3"	T3	s.
S3	£5.	p	S-	5"	Q	S3	&>	ETt
c	o	-*		•a	—>	r*	•-1	O 3
3	-•»	o	w	»	P	o	O	
3-"	p	"»»	-^	t	e	—»	—»	Er
	3	p	o	o	st	63	P	-a
	3_	c	►*»	•»>	3/	a	c	p
	J»	5.	p	p		gt	D	•*.
			c 3_	5.				—» B 3
 
DECIMAL FRACTIONS.               143
  The denominations of Federal currency are arranged in a
decimal or tenfold ratio.  An eagle is ten times the value of a
dollar; a dollar is ten times the value of a dime;  a dime is ten
times the value of a cent, and a cent ten times the value of a
mill, thus:
I00c=l    dot.
 75c.= .75  " =7d.5c.
 50c.= .5   " =5d.
 25c.= .25  "  =2d. 5 c.
 12£ = .125"  =ld.2c.5m.
                 or 12 c. 5wi.
1 Eagle = 10 dollars.
1 dollar = 10 dimes.
1 dime  = 10 cents.
1 cent  = 10 mills.
J of a dollar = 33,3,32 cents.
| of a dollar = 66,6,60 cents.
  Let the pupil be  required to express the following sums in
their proper decimals :—Three, tenths ; twenty-six, hundredths;
one hundred and  fifteen,  thousandths; five hundred and forty,
thousandths ; one hundred  and twenty-five, thousandths ; five,
hundredths; fifteen, thousandths; one hundred and twelve, ten
thousandths; twenty-four,  millionths; nine,  ten thousandths;
three,thousandths;  sixty-five, one hundred thousandths; eighty-
four millionths.
  Let him write, in figures, ten eagles, five dollars, six dimes,
four cents, and seven mills; sixty-four cents and  seven mills ;
five dollars, and twenty-five cents; three dollars, and two cents;
six dollars, and one cent; four cents; two mills; one dime;
one dollar, and twenty-five  hundredths ; two dollars, eight hun-
dred and  seventy five thousandths; ten  dollars, and fifty  five
hundredths ;  nine dollars,  one  hundred  and thirty-seven thou-
sandths;  six dollars, one  hundred and twenty-five thousandths;
one dollar, three hundred and seventy-five  thousandths: and any
other sums, the teacher may think proper.
              ADDITION OF  DECIMALS.
  Addition of Decimals is performed like addition of whole
numbers.   Tenths must be placed under tenths, hundredths un-
der hundredths, thousandths under thousandths,&c. In the sum,
observe to  make as many decimal places as are in any one of
the given fractions.
  If mixed numbers are given to  be  added together, place the
integers, units under units, tens under tens, &c. beginning at the
decimal point, from the right to the left.  The fractional parts
are to  be placed on  the right of the decimal point,  as above
directed, tenths under tenths, &c. 
144                   ARITHMETIC.
Fractions.	Mixed Numbers.	Mixed Numbers
.16457	146.02145	345.62171
.84321	364.14215	38.21054
.92456	562.52068	6.7020-1
.00256	142.35421	35.00164
.74021	813.21543	376.543
.54216	321.54216	75.04754
3.21727	2349-79608	877.12647
  Let the pupil be required to add the following fractions :
  1. .42105, .002541, .016421, .7042, .5 1,  .25241.
  2. .9874602, .745210, .000014, .34210, .2467, .789-
  3. Let him add the  following mixed  numbers:  74.62741,
5.00421, 4261.3407,  174.32164, 9876.2542.
  4. 10.6215, 107.35421,  84.542106,  42.81,  1.7542.
  5. Also the following mixed  numbers:  54.004, 12.54764,
1.2554, 746.2104, 25.46201, 246.00012,  4621.3456.
  6. 333.3333, 64.6666, 246J407, 246.22102.
  7. William purchased the following articles,  viz.; a copy of
Virgil for $2.12,5, an exercise for .37,5 cts., a quire of paper for
.18,7,5 cts., quills for .12,5 cts., ink for ,5 mills ; what was the
amount ?                                   Ans. $2.81,7,5.
  8. Jane purchased an  English grammar for .50 cts., a  geo-
graphy for $1.37,5 cts., a slate for .37,5 cts., paper and quills
for .56,2,5 cts., ink for ,6 mills; how much money did she lay
out?                                       Ans. $2.81,8,5.
  9. A farmer carried to market the following articles,  which
he sold for these several sums:  viz. turkeys for $5.37,5, fowls
for  $3.75,4,5,  butter for $1.25, potatoes for $3.87,5, turnips
.93,7,5, Indian meal for $2.12,5 ; how much did he  receive in
all?                                        Ans. $17.31,7.
  10. A gentleman bought a chaise for $210.56,2,5, a horse
for $116.75,  a saddle for $18.75, a bridle for $4.87,5,  a har-
ness for $31.37,5; how much was the amount paid ?
                                        Ans. $382.31,2,5.
  11. A lady laid out the following sums of money in a tour of
shopping; and, on her return home, desired her daughter to tell
her the amount: $1.37,5 for a pair of gloves, $2.87,5 for mus-
lin, $5.12,5  for crape, $7-50  for silk, $3.93,7,5  for linen,
.56,2,5 cts. for silk, and cotton thread ; what was the amount ?
   ' '                                      Ans. $21.37,5. 
DECIMAL FRACTIONS.               145
  12, A servant purchased  in market  a  turkey for .87,5 cts.,
a pair of chickens for .56,2,5 cts., potatoes for .37,5 cts., other
vegetables for $1.25, beef for $1.87,5; how much did he lay
out for the whole ?                          Ans. $4.93,7,5.
  13. A gentleman purchased a suit of  clothes : a coat  for
$18.75, a vest  for $3.25,  pantaloons  for $7-87,5,  linen  for
$5.56,2,5, a pair of boots for $5.75, stockings for $2.75, and a
hat for $6.37,5 ; what was the amount ?     Ans. $50.31,2,5.
  14. A merchant built a house, and when  finished, the bills
were as follow :  The  carpenter's bill $475.56,2,5, the brick-
layer's bill $384.50, the lumber merchant's bill $450.37,5,  the
brickmaker's bill $275.12,5, the plasterer's  bill  $185.93,7,5,
the painter's bill $94.17,9, the ironmonger's bill $49-85 ; how
much did the house cost, including $1165.75, which he gave for
the lot of grourd ?                        Ans. $3081.27,9-

            SUBTRACTION OF DECIMALS.

   Subtraction of decimals is  performed like subtraction of
whole numbers; but observe to make as many decimal places in
the difference, as there are in either of the  given numbers.

                        EXAMPLES.
       Fractions.       Mixed Numbers.    Mixed Numbers.
      .874698          4765.008476       47-62845
      ■23987426        2104.7628         31.046289
      .63482374        2660.245676       l6.582l6l
  1. Take 47.504,.from 1746.80456, what will be left?
                                        Ans. 1699,30056.
  2. Take 1.755, from 5.8755, what will remain? Ans. 4.1205.
  3. Take 74.8064 from 975.006, what will remain ?
                                          Ans. 900.1996.
  4. Take 25.0074 from 97-8462, what will be left ?
                                           Ans. 72.8388.
  5. A  man  borrowed $674.87,5, and paid $346.50; how
much remains due?                        Ans. $328.37,5.
  6. A servant went to market with $5.37,5, and purchased to
the amount of $2.43,7,5;  how much must he have remaining?
                                          Ans. $2.93,7,5.
  7. Jane took with her, for the purchase of books and sta-
tionary, $4.87,5, and her bill was $3.62,5 ; how much change
must she receive ?                              Ans. $1.25.
              N2 
146                  ARITHMETIC.
  8. A gentleman's income is $1145.87,5; and his family's
expenses  amount to $824.25 ;  how much does he save yearly ?
                                         Ans. $321.62,5.
  9. What is the difference between one dollar, and one mill?
                                          Ans. 99 c 9 m.
  10. What is the difference between $99.99, and 4 dollars and
2 mills ?                                   Ans. $95.98,8.
  11. A man commenced a journey with $174.85,5, and soon
after he set out, he found that he had spent $47.95,5 ; what sum
had he left?                                Ans. $126.90.
  12. A  merchant purchased  a quantity of goods at a public
sale  for $1784.12,5, and sold $847.62,5  before he removed
them;  what amount had he left ?               Ans. $936.50.
         MULTIPLICATION OF DECIMALS.

  Multiplication of decimals is performed like multiplication
of whole numbers; but observe to make as many decimal places
in the product, as there are in the multiplicand and multiplier
taken together.
                       EXAMPLES.
Fractions. .4675 .27	Mixed Numbers. 24.6021 24.02 592042 > 984084 492042 591.042442	Mixed Numbers 742.0245 1.54
32725 9350 ( .126225		29680980 37101225 7420245 1142.717730
		
  1.  Multiply 743.56815, by 52.647.
  2.  Multiply 79-0342, by 102.51.
  3.  Multiply 125.74, by 56.84.
  4.  Multiply 274.174, by 125.57.
  5.  Multiply 125.0478,  by 12.042.
  6.  Multiply 70.3046, by 54.321.
  7.  Multiply 217-1462,  by 740.025.
  8.  Multiply .04621, by .00241.
  9.  Multiply .00027, by .00742.
 10.  Multiply .074, by .00025.
  1.  A farmer sold 47.5 bushels of wheat at 75 cents a bushel;
required the amount.                        Ans. $35.62,5.
  2.  A gentleman sold a farm containing 127-875 acres of land, at
$11.37,5 an acre; required the amount. Ans. $1454.57,8,125.
Ans. 39146.63239305.
   Ans. 8101.795842.
      Ans. 7147.0616.
   Ans. 34428.02918.
   Ans. 1505.8256076.
   Ans. 38190161766.
 Ans. 160693.616655.
    Ans. .0001117661.
   Ans. .0000019234.
      Ans. .00001850. 
DECIMAL FRACTIONS.               147
  3. A gentleman bought 4.25 yards of broad cloth, at $5.56,25
a yard; required the amount.                   Ans. $23.64.
  4. A merchant bought a piece of cloth containing 31.6 yards,
at $3.12,5 a yard ; required the amount.        Ans. $98,75.
  5. A lady  bought a  piece of silk containing 7.55 yards, at
$1.12.5 a yard ;  required the amount.       Ans. $8.49,3,75.
  6. Required the amount of 37.5 yards of muslin, at 16,5 cents
a yard.                                     Ans. $6.18,7,5.
  7. Required the amount of 40.25 bushels of wheat, at 93,7,5
cents, a bushel.                            Ans. $37-73,44.
  8. Required the amount of 5.625 cords of wood, at $3.75 a
cord.                                 ^  Ans. $21.09,3,75.
  9. A housekeeper purchased a bag of coffee containing 112.25
pounds, at 12.5 cts.a lb; what was the amount ?Ans. $14.03,1,25.
  10.  A grocer bought a box  of tea containing 41.657 lb. at
$1.12,5 a lb.; what was the amount ?      Ans. $46.86,4,125.
  11.  A lady purchased  the following articles;  4.25 yards of
muslin at 18,7,5 cents  a yard ; 6.75  yards of linen at  56,2,5
cents a yard;  12,5 yards of tape at 5  mills ayard ; and 5 skeins
of silk at 3,5 cents  each; what was the amount of her bill?
                                          Ans. $4.83,1,25.
  12. The expenses of a family is $1,75,5 a day; how much
 will the amount of expenses be in a year, estimating 365.25 days ?
                                        Ans. $641.01,3,75.

               DIVISION OF DECIMALS.

   Division of decimals  is performed  like division of whole
 numbers ; but, observe that there must be as many decimal places
 in the divisor and quotient, taken together, as there are in the
 dividend.
   Note.—If there are not a  sufficient number of decimal places
 in the dividend, cyphers must be added.

                        EXAMPLES.
   Divide 234.70525, by  64.25.           Ans. 3.653.
                 64.25)234.70525(3.653
                        19275 • • •
                        41955
                        38550
                         34052
                         32125
19275
19275 
148
ARITHMETIC.
Divide .746894756, by .0216.            Ans. 34.57846+
           .0216).746894756(34.57846+
                  648-.....
                   988
                   864
                   1249
                   1080
995
864
1316
1296
   20
I
  1. Divide 2508.92806, by 92.41035.     Ans. 27.14984-.
  Note..—Here  there is not a sufficient number  of  decimal
places in the  dividend; cyphers must be added to it, so that
the quotient may have three or more decimals, as in the  answer.
2.	Divide 1836.88305, by 23.15.		Ans. 79.347.
3.	Divide 3673.7661, by 158.694.		Ans. 23.15.
4.	Divide .00178600398, by .00463.		Ans. .385746.
5.	Divide 3632.46, by 78.00.		Ans 46.57.
6.	Divide 9, by .9.		Ans. 10.
7.	Divide .9, by 9.		Ans. .1.
8.	Divide .3, by 3.		Ans. .1.
9.	Divide 47.0546, by .0164.	Ans. 2869,1829.4-	
10.	Divide 174.62154, by 1.674.	Ans.	104.31334.+
11.	Divide 984.20462, by 4.026.	Ans	244.46214.4-
                  PRACTICAL EXERCISES.

  1. A miller bought 210.5 bushels of wheat for $195.75, how
much was it a bushel ?                   Ans. 92 cts. 9,9.+
  2. A merchant bought 547-5 lbs..of tea for $497.87.5, how
much was it a pound ?                  Ans. 90 cts. 9,36.4-
  3. A piece  of broad-cloth containing  25.335 yards, cost
$116.76.5,  how much was it a yard ?      Ans. $4.60,8,8.+
  4. A gentleman boards at 3 dolls. 87,5 cts. a week; how many
weeks can he board for 54 dolls. 75 cts. ?  Ans. 14.129 weeks. 
DECIMAL FRACTIONS.
149
  5. A farmer raised upon one acre of land, 50.25 bushels of
corn; how many  acres will yield, at the same rate, 526.875
bushels?                               Ans. 10.485+acres.
  6. A gentleman left 4567 dolls. 93,75 cts. to be  divided
equally among his 5 sons; what was the share of each ?
                                      Ans. $913.58,7,5.+
  7. A farmer bought a plantation containing 425.5  acres for
2116 dolls. 87,5 cts.; how much was the land an acre ?
                                         Ans. $4.97,5.+
  8. A  merchant bought 575.25 bushels of salt for 174 dolls.
62,5 cts.;  how much  was it a bushel?    Ans, 30 cts. 3,56.+
  9. If 21.5 yards of cloth cost 46 dolls. 75 cts., how much was
the  cloth a yard ?                          Ans. $2.17,67.+
  10. A  farmer sold a load of hay, weighing 23.25  cwt. for
15 dolls. 93,75 cts; how  much was it a cwt ?
                                    Ans. 68 cts. 5,48 mills.
  11. The sum of 846  dolls.  47,6 cts. was divided equally
among 7 persons, how much was each one's share?
                                          Ans. $120.92,5.
  12. A labourer has 87,5 cts. a day ;  in how many days will
he earn 48 dolls. 50 cts.?                 Ans. 55.428+ days.
  13. The tax of a certain town amounts to 647 dolls.  75,5 cts.
The population is 1000 taxable persons. How much must each
one pay ?                          Ans. 64 cts. 7,75+milIs.

            REDUCTION OF DECIMALS.
  To reduce a vulgar fraction to a decimal fraction:
  Rule.—Add a sufficient number of cyphers, or decimal places
to the numerator, and divide by the denominator.
  Note.—There  must be as  many places  of decimals in  the
quotient, as in the dividend.

                        EXAMPLES.
  1. Reduce | to a decimal  fraction.               Ans, .25
                        4)1.00
                            .25

  2. Reduce J to a decimal fraction.                  Ans. .5
                          1)1.0
                          ~5

  3. Reduce § to a decimal fraction.                Ans. .75
                        4)3 00
                           •75 
150
ARITHMETIC.
  3. Reduce £ to a decimal fraction.                Am. .334
                         3)1.000
                            .33*

  When the same figure recurs in the quotient, as in this exam-
ple, it is called a  repeater; and  the true value of  the fraction
cannot be ascertained.  It will continue  to repeat to any inde-
finite number of places.  It is usual to extend it to three or four
places, after which it becomes so small in value, that, in ordinary '
cases, it may be rejected.  The  same is the case in the follow-
ing example.  A line is commonly drawn across the last figure,
which implies  that the figure is  a repeater, and may be added
indefinitely without continuing the operation.
  When the same figure recurs at intervals of one, two, three, or
more places, as it sometimes  happens, it is called a circulating
decimal.
4.	Reduce y to a decimal fraction. 3)2.0000 .6660	Ans. .6660
5.	Reduce £ to a decimal fraction.	Ans. .875
6.	Reduce £ to a decimal fraction.	Ans. .834
7.	Reduce -J to a decimal fraction.	Ans. .8
8.	Reduce -| to a decimal fraction.	Ans. .88$
9.	Reduce -^ to a decimal fraction.	Ans. .9
10.	Reduce +| to a decimal fraction.	Ans. .91660
11.	Reduce 2^- to a decimal fraction.	Ans. .4375
12.	Reduce M to a decimal fraction. Reduce § to a decimal fraction.	Ans. .45614+
13.		Ans. .375
14.	Reduce yf- to a decimal fraction.	Ans. .7058 +
15.	Reduce T^*g- to a decimal fraction.	Ans. .01111 +
16.	Reduce f-g-g- to a decimal fraction.	Ans. .3624+
17.	Reduce TT2- to a decimal fraction.	Ans. .01785 +
18.	Reduce -gV to a decimal fraction.	Ans. .011236+
19.	Reduce *y^ to a mixed number.	Ans. 11.3683 +
                         Case. 2.
  To reduce any sum or quantity to the^decimal of a given
denomination of the same  kind :
  Rule.—Reduce the given sum or quantity to its lowest terms
for a numerator, and the proposed integer to the same denomina-
tion  for a denominator.   To the  numerator affix a sufficient
number of cyphers or decimal places, and divide by the deno-
minator, as in the preceding case. 
DECIMAL FRACTIONS.              151
                    EXAMPLES.

1.  Reduce 12«. and 6d. to the decimal of a £.    Ans. .625.
          s.  d.
          12  6
          12
         150^ the numerator = the pence in 12*. 6d.
20 X 12 = 240, the denominator = the pence in 20s., or 1 £.
                  240)150.000(.625
                      1440 • •
                        600
                        480
                        1200
                        1200

 2. Reduce 7s. and 6d. to the decimal of a £.     Ans. .375.
   s.   d.
   7   6        20  x  12 = 240 the den. = the d. in 1 £.
   12
   90. the num. = the pence in 7s. 6d.
                  240).90.000(.375
                       720 •♦
                       1800
                       1680
                       ~1200
                         1200

 3. Reduce l6s. to the decimal of a £.              Ans. .8.
 4. Reduce 76 yards, to the decimal of a mile. Ans. .04318 +
 5. Reduce 3 furlongs to the decimal of a mile.    Ans. .375.
 6. Reduce 112 perches to the decimal of an acre.    Ans. .7-
 7. Reduce 15 dwts. to the decimal of a pound trov.
                                           Ans. .0625.
          dwt. in an oz. = 240)15.0000(.0625
                              1440--
                              ~~600
                                480
                                1200
                                1200

 8. Reduce 7 minutes to the decimal of a day.  Ans. .0048+
 9. Reduce 14 gallons to the decimal of a hhd.   Ans. .222;*.
 10. Reduce 3 cwt. 3 qrs. to the decimal of a ton. Ans. .1875. 
152
ARITHMETIC.
  11. Reduce 2 qrs. and 1 nail to the decimal of a yard.
                                              Ans. .5625.
  12. Reduce 23 days to a decimal of a year. Ans. .006301 +
  13. Reduce 3 qrs. 16 lb. to the decimal of a cwt. Ans. .8928 +
  14. Reduce 12 drams to the decimal of a pound avoirdupois.
                                           Ans. .046875.
  15. Reduce 1 pint to the decimal of a gallon.    Ans. .125.
  16. Reduce 52 days to the  decimal of a year of 365 days 6
hours.                                       Ans. .142368.
  17. Reduce 1 rood, 16 perches to the decimal of an acre.
                                                Ans. .35.
                        case  3.
  To reduce a decimal fraction to its proper value:
  Rule.—Multiply it by the several denominations of the inte-
ger, and make as many decimal places in the product, as there
are in the giysn decimal.

                       EXAMPLES.
   1. Reduce .785416 of a £ to its value.      Ans. 15*. fyd.
                 .785416
                 "_____20=1  £.
              s. 15.708820
                        12=1 shilling.
              d.  8.499840
                         4=1 penny.
              gr. 1.999360
   2. Reduce .76 of a £ to its value.           Ans. 15*. 2\d.
   3. Reduce .861 of a cwt. to its value.
                               Ans. 3 qr. 12 lb. 6 oz. 14 dr.
   4. Reduce .7 of a lb. troy to its value.    Ans. 8 oz.  8 dwt.
   5. Reduce .761 of a day to its value. Ans. 18 h. 15m. 50 sec.+
   6. Reduce .67 of a league to its value.
                              Ans. 2 m. 3 rods, 1 yd. 1 in.+
   7. Reduce .128 of an acre to its value.
                                 Ans. 20 per. 14 yd. 4 ft.+
   8. Reduce .216 of a year of365 days to its value.
                               Ans. 78 d. 20 h. 9 m. 36 sec.
   9. Reduce .5625 of a yard to its value.    Ans. 2 qrs. 1 na.
  10. Reduce .375 of a mile to its value.           Ans. 3  fur.
  11. Reduce .4875 o/a dollar to its value. Ans. 48 cts. 7,5. m.
  12. Reduce .6875 of a yard to its value.     Ans. 2 qr. 3 na.
  13. Reduce .07 of a hhd. to its value. Ans. 4 gal. 1 qt. 1.28 pt. 
DECIMAL FRACTIONS.
153
  14.  Reduce .078 of an acre to its value.
                                Ans. 12 per. 14 yd. 4.68 ft.
  15.  Reduce .46 of a great circle of 360 degrees to its value.
                                       Ans. 165 deg. 36 m.
  16.  Reduce .146 of a sign of the zodiac to its value.
                         \        Ans. 4 deg. 22 m. 48 sec.
  17.  Reduce .875 of a quadrant or 90 degrees to its value.
                                       Ans. 78  deg. 45 m.
  18.  Reduce .0425 of a century, or 100 years to its value.
                                          Ans. 4 yrs. 3 mo.
  19.  Reduce .027 of a yard to its value.       Ans. 2,916 b.c.
  20.  Reduce .00897 of a  mile to its value.
                                  Ans. 15 yds. 28 in. 1 b.c.
  21.  Reduce .79 lb. troy  to its value.
                                 Ans. 9 oz. 9 dwt. 14+ gr.
  22.  Reduce .56 of a yard to its value.          Ans. 2.24 qr.
  23.  Reduce .04 of a year to its value.
                              Ans. 14 d. 14 h. 38 m. 24 sec.
  24.  Reduce .65 of a gallon to its value.   Ans. 2 qt. 1.04 pt.
  25.  Reduce .7546 of a cwt. to its value.
                            Ans. 3 qr. 0 lb. 8-oz. 3.89+ dr.
  26.  Reduce .3467 of an  acre to its value.
                                 Ans. 1 rood, 15.47+  per.
  27.  Reduce .42891 of a  mile to its value.
                        Ans. 3 fur. 17 roods, 1 yd. 1.1448 ft.
  28.  Reduce .186405 of a £. to its value,
                                    Ans. 3*. 8d.  2.9488 qr.

   Questions.  1. What is a decimal fraction?—2. How do
you distinguish a decimal from  a whole number or integer ?__
3. Will you  give an example?—4. How much does the first
figure to the right hand of  the decimal point, signify ?—5. How
much does the second figure signify?—How much does the
third  signify ?—How  much does the  fourth  signify ?—6. In
what  proportion does  the value of a decimal decrease, from
the decimal point?—7- Will you give an example?—8.  Do
cyphers, added to decimals in any way, affect their value?__
0. If  they are placed  before them, what  effect have they?__
10. Will you  explain this  by an example, and assign the reason
of this ?
   11. Will you repeat the table of decimal parts?—12. Are
the denominations of Federal currency arranged in decimal pro-
portion ?—13. Will you explain this more fully ? 
154
ARITHMETIC.
  14. What do you mean by a mixed number ?—15. How are
decimal fractions added together?—16. What do you observe
in placing  them for addition?—17. When they  are  added
together, what rule do you observe in  placing the separatrix,
or decimal  point?
  18. When mixed numbers are given  to  be added together,
how do you  place the integers ?—How do you place the deci-
mals ?—At what place do you begin ?
  19- How are decimals  subtracted?—20. What rule do you
observe in placing the decimal point?—21. How do you mul-
tiply decimals ?—22. What rule do you observe in pointing oil
the product ?—23. How do you  divide decimals?—24. What
rule do you observe in placing the decimal point?—25.  When
there are not sufficient decimal places in the dividend, what do
you do ?
  26.  How do you reduce a vulgar  fraction, to a decimal ?—
27. What  do you observe in placing the decimal point ?—
28. When  is a decimal said to repeat?—When is it  said to
circulate ?—29. Will you explain this by an example ?—30. How
do you reduce a decimal fraction to its proper value ?—31. Will
you explain this by an example ?
          REDUCTION OF INTEGERS.
  Reduction is the operation of changing any sum or quantity
of a given denomination to another denomination, retaining the
same value.
  Reduction is  two-fold ; descending and ascending.
  Reduction descending  brings  higher  denominations  to
lower, and is performed by multiplication.
  When any sum or quantity is to be reduced to a lower deno-
mination ; multiply  the  highest denomination by the number,
required of the next lower denomination  to make 1  or unit of
the higher; multiply this product by the  number, required of
the next lower denomination to make 1 or unit of the higher;
and  so continue to do, till you reduce it  to the denomination
required.  But, if the  given sum consists of several denomina-
tions, reduce the highest denomination  to  the next lower one,
and to the product add the parts of the same denomination in
the given sum; and so continue to do, till you come totthe last
or lowest denomination in the given sum.
   Reduction descending is proved, by Reduction ascending. 
REDUCTION OF INTEGERS.
155
                      EXAMPLES.
  1. Reduce 26 Eagles to cents.           Ans. 26000 cents.
  2. Reduce 126 dollars to mills.         Ans. 126000 mills.
  3. Reduce 187 dimes to mills.           Ans. 18700 mills.
  4. Reduce 746 dollars to dimes, cents and mills.
                 Ans. 7460 dimes, 74600 cts., 746000 mills.
  5. Reduce £25,12"shillings, to shillings and pence.
                            Ans. 512 shillings, 6144 pence.

  Here you multiply the pounds by 20, because 20 of the  low-
er denomination or shillings, make 1 or unit of the higher; and
to the  product add  the 12  shillings in the given sum; which
make 512  shillings.   This  multiply by 12,  because 12 pence
make one shilling, and the product will be 6144.   If there had
been pence in the given sum, these must have been added ; and
the  sum would have been  the pence  required.  In like  manner
proceed to reduce any other sum or quantity to a lower deno-
mination, whatever the denomination may be.

  6. Reduce 15 lbs. troy  weight to ounces.       Ans. 180 oz.
  7. Reduce 7 lbs. troy to dwt. and grains.
                                 Ans. 1680 dwt. 40320 gr.
  8. Reduce 12 lbs. 2 oz. 16 dwt. to grains.   Ans. 70464 gr.
  9. Reduce 25 tons to qrs. and pounds.
                                 Ans. 2000 qrs. 56000 lbs.
  10. Reduce 7 cwt. 3  qrs. 12 lbs. to pounds.    Ans. 870 lbs.
  11. Reduce 45 lbs. 11 oz. to drams.        Ans. 11696 dr.
  12. Reduce 5 lbs. 6 oz. to drams and scruples.
                                   Ans. 528  dr. 1584 scru.
  13. Reduce 6 lbs. 3 oz. to grains.             Ans. 36000 gr.
  14. Reduce 2 lbs. 5 oz. 4 drams 1 scruple to grains.
                                           Ans. 14180 gr.
  15. Reduce 6 miles to furlongs and rods.
                                    Ans. 48 fur. 1920 rods.
  16. Reduce 5 furlongs to rods and yards.
                                  Ans. 200 rods, 1100 yds.
  17- Reduce 24 degrees to geographical miles.
                                          Ans.  1440 miles.
  18. Reduce 15 degrees to statute miles of 69£ to a degree.
                                        Ans. 1042.5 miles.

  When  it is required to multiply by \, \, or \, the operation
may frequently be rendered easier, by taking the decimal values
of the fraction, and annexing them to the integer;  and multiply 
156
ARITHMETIC.
by the sum.  In the preceding example, reduce the A mile to
its decimal value, or .5;  which annexed to 69 becomes  69.5
for the statute miles in a degree.   By this  multiply the given
degrees, and the product will be found 1042.5, or  1042 miles,
and 5 tenths, or half a mile.

  19. Reduce 120 yards to quarters and nails.
                                   Ans. 480 qrs.. 1920 na.
  20. Reduce 12 yards to nails.                Ans. 192 na.
  21. Reduce 127 ells English to quarters.     Ans. 635 qrs.
  22. Reduce 175 ells Flemish to quarters and nails.
                                    Ans. 525 qrs. 2100 na.
  23. Reduce 15 acres to roods and perches.
                                       Ans. 60 R. 2400 P.
  24. Reduce 24 acres, 3 roods to perches.     Ans. 3960 P.
  25. Reduce 46 acres, 2 roods, 19 perches, to perches.
                                             Ans. 7459 P.
  26. Reduce 4 acres, 1 rood, 39 per. to perches.  Ans. 719 P.
  27. Reduce 15 square feet to square inches.
                                     Ans. 2160 sq. inches.
  28. Reduce 4 hhds. 16 gals, to gallons and quarts.
                                   Ans.  268 gals. 1072 qts.
  29. Reduce 15 gals, to quarts and gills. Ans. 60 qt. 480  gills.
  30. Reduce 54 gallons, 3 quarts, to quarts, pints, and  gills.
                            Ans. 219 qt. 438 pt. 1752  gills.
  31. Reduce 25 bushels, 2 pecks, to pecks and quarts.
                                       Ans. 102 p. 816 qt.
  32. Reduce 24  pecks, 1 pint, to pints.         Ans. 385 pt.
  33. Reduce 3 years, 7 months, to months.      Ans. 43 mo.
  34. Reduce 7 years, 4 mo. 22 days, to days. Ans. 2662 days.
  35. Reduce 16 days, 22 hours, to hours and minutes.
                                   An3. 406 h. 24360 min.
  36. Reduce 27 days, 10 hours, 16 seconds, to seconds.
                                        Ans. 2368816 sec.
  37. Reduce 16 degrees, 48 minutes, to minutes  and seconds.
                                 Ans. 1008 min. 60480 sec.
  38. Reduce 4 signs, 13 degrees, to minutes.    Ans. 7980'.
  39. Reduce 15 lb. 7 oz. 10 dwt. 4 gr. troy weight, to grains.
                                           Ans. 90004 gr.
  40. In an ingot of silver, weighing 6 lb. 3 oz. 10 gr. how
many grains?                               Ans. 36010gr.
  41. In 5 cwt. 2 qr. 16 lb. how many pounds ?  Ans. 632 lb.
  42. In 4 miles, 6 furlongs, 22 rods, how many rods ?
                                          Ans.  1542  rods. 
REDUCTION OF INTERGERS.            157
  43.  In 360 deg. how many geographic miles ?    Ans. 21600.
  44.  In 360 deg. how many statute miles ?       Ans. 25020.
  45.  In 456 yds. 3 qrs. of cloth, how manv quarters and nails ?
                                  Ans. 1827 qr. 7308 na.
  46.  In 64 acres, 1 rood, 39 perches, how many perches ?
                                         Ans. 10297 per.
  47.  In 12 acres, 3 roods, 39 perches, how many perches ?
                                           Ans. 2079 per.
  48.  In 5 hhd. 12 gallons, 3 quarts, how many quarts ?
                                            Ans. 1311 qt.
  49.  In 46 bush. 3 pecks, how many quarts ?  Ans. 1496 qt.
  50.  In 5 years, 174 days, how many hours and minutes?
                              Ans. 48006 h. 2880360 min.
  51.  In 49 weeks, 4 days,  12  hours, how many hours  and
minutes ?                         Ans. 7904 h. 474240 min.
  52.  In 9 signs, 12 degrees, 50 minutes, how many minutes ?
                                             Ans. 16970'.
  53. A grocer purchased 16 cwt. 3 qrs.  12 lb. of sugar; how
many  pounds are there in the whole ?           Ans. 1888 lb.
  54. A  tobacconist bought ll6cwt. 2  qr. of tobacco; how
many  pounds were in the whole ?              Ans. 13048 lb.
  55. A silversmith has 7 lb. 10  oz. 11 dwt.  of silver; how
many  grains are in the whole ?                Ans. 45384 gr.
  56. An apothecary has several sorts  of drugs,  weighing
together, 47 lb. 6 oz. 4 dr.; how many scruples and grains are
in the whole ?                   Ans. 13692 scru. 273840 gr.
  57- The sum of the distances of 4 places from Philadelphia
is 156 miles, 5 fur. 6 poles; how many poles and yards are in
the whole ?                  Ans. 50126 poles, 275693 yds.
  58. There are two places, one situated 40 degrees east of the
meridian of Washington, the other 25 degrees west of it; what
is the  distance in geographic  miles between them, they being
situated on the equator ?                       Ans. 3900 m.
  59. A merchant bought 174 yards, 3 qr. 2 na. of broad-cloth;
how many quarters and nails are in the whole ?
                                    Ans. 699 qr. 2798 na.
  60. A farmer has several fields, containing in the whole 54
acres, 2 roods, 24 perches; how many perches in the whole ?
                                           Ans. 8744  per.
  6l. A gentleman has 4 lots of ground, containing as follow:
No. 1. 2 acres, 1 rood, 30 perches—No. 2.  1 acre, 3 roods, 36
perches—No. 3. 2  roods, 39 perches—and No. 4. 5 acres, 0
roods, 10 perches;  how many perches are in the whole ?
                                           Ans. 1635 per.
          o 2 
158                   ARITHMETIC.
  62. A merchant has several casks of wine, containing in the
whole 156 gallons, 3 quarts; how many pints are in the whole?
                                            Ans. 1254 pt.
  63. There are 4 sisters; the age  of the eldest is 13 years, 2
months, 16 days—the age of  the next is 11 years, 1 month, 5
days—the age  of the third is 8 years, 6 months, 11 days—and
that of the youngest is 6 years, and 5 months ;  what is the sum
of all their ages in days, hours, and minutes ?
                   Ans. 14132 d.  339168 h.  203500S0 min.
  64. A gentleman  has a field, in which is a  large pond: the
content of the  field,  including  the pond is 47 acres, 2 roods, 36
perches; and  surveying the ground, exclusive  of the pond, he
found it to be  26 acres, 3 roods, 39 perches;  what is the con-
tent of the pond in perches ?                   Ans. 3477 per.
  65. A lady  purchased for the use of her family the following
articles: 18 yds. 3 qr. 2 na. of muslin—26 yds. 2 qr. 1  na. of
linen—14 yds. 3 qr. 2 na. of crape—12 yds. 1  qr. of  silk; and
having returned home, desired her daughter to  inform her, how
many quarters, and  nails were in the whole ?
                                     Ans. 290 qr. Il6l na.
  66. A miller has  in several bins the following quantities of
wheat:  in No. 1, he has  175  bu. 3  p. 6 qt.—in No. 2, 254 bu.
1 p. 7 qt.—in No. 3, 58 bu. 3  p. 5 qt.—and in No. 4, 546 bu. 2
p. 4 qt.; how many  pecks, and quarts are in the whole ?
                                    Ans. 4143 p. 33150 qt.

              REDUCTION  ASCENDING.

   Reduction  ascending   brings lower denominations  to
higher, and is  performed by division.   When any given sum or
quantity of a lower denomination is to be reduced or brought to a
higher denomination ; you must divide by the number, required
of the lower denomination to make  1 or unit  of the higher;
and so you must continue to do, till you bring the given sum, to
the denomination required.
   Reduction ascending is proved, by Reduction descending.

                        EXAMPLES.
   1. In 467890 mills, how many dollars ?
                              Ans. 467 dol.  and 89 cts. over.
   2. In 74654 dimes, how many dollars?
                            Ans.  7465 dol. and 4 dimes over.
   3.  In 246700 mills, how many dollars ?
                              Ans. 246 dol.  and 70 cts. over. 
              REDUCTION OF INTERGERS.            159

  4. In 17460 pence, how many pounds ?
                                 Ans. .£72, and 15*. over.
  Here you divide, in the first place, by 12; because 12 pence
make one shilling, or 1 of the next higher denomination, and
the quotient will be  the  number of shillings : these you must
divide by 20, because 20 shillings make £l, or 1 of the next
higher denomination; the quotient thence arising will be pounds.
  On the same principle, you  proceed to bring any given sum
or quantity of a lower denomination, to  one that  is higher,
retaining the same value.

   5. In 74616 farthings,  how many shillings and pounds ?
                           Ans. 1554s. £77, and 14s. over.
   6. In 167400 grains troy, how many pounds ?
                              Ans. 29 lb. and 15 dwt. over.
   7- In 4765 dwt., how many ounces and  pounds ?
                        Ans. 238 oz. 19 lb. and 10 oz. over.
   8. In 114 qrs. how many cwt. ?    Ans. 28 cwt. 2 qr. over.
   9. In 17640 pounds, how many quarters and cwts.?
                     Ans. 630 qrs.  157 cwt. and  2 qrs. over.
  10. In 467456 drams, how many pounds and quarters ?
                         Ans. 1826 lb. 65 qr. and 6 lb. over.
  11. In 1746880 ounces, how  many tons?
                 Ans. 48 tons, and 14 cwt. 3 qrs. 14 lbs. over.
  12. In 17645 grains apothecaries'weight, how many pounds ?
                       Ans. 3 lbs. and 6 dr. 0 scru. 5 gr. over.
  13. In 168474 feet, how many miles ?
                      Ans. 31 miles, and 7 fur. 58 yds. over.
  14. In 274755 feet, how many miles ?
                            Ans. 52 miles, and 65 yds. over.
  15. In 7468 rods, how many miles?
                          Ans. 23 miles, and 108 rods over.
  16. In 11462 rods, how many furlongs?
                            Ans. 286 fur. and 22 rods over.
  17« In 316 geographic miles, how many degrees ?
                                    Ans.  5° and 16' over.
  18. In 11456 geographic miles, how many degrees ?
                                  Ans. 190° and 56' over.
  19. In 164736 inches, how many miles ?
                     Ans.  2 miles, and 4 fur. 176 yds. over.
  20. In 14764 nails of cloth, how many yards ?
                             Ans. 922 yds. and 3 qrs. over.
  21 In 2746 quarters, how many ells English ?
                              Ans. 549 ells, and 1 qr. over. 
160
ARITHMETIC.
22. In 16217 quarters, how many yards ?
                           Ans. 4054 yds. and 1 qr. over.
23. In 74678 perches of land, how many acres ?
                        Ans. 466 acres, and 118 per. over.
24. In 174 roods, how many acres ?
                          Ans. 43  acres, and 2 roods over.
25. In 4761 square feet, how many square yards ?
                                       Ans. 529 sq. yds.
26. In 17647 square yards, how many square perches ?
                                Ans. 583,371+ sq. per.
27. In 1476 pints, how many gallons  ?
                            Ans. 184 gals, and 2 qts. over.
28. In 167840 gills, how many barrels of 32 gallons each ?
                          Ans. 163 bbl. and 29 gals. over.
29. In 1789500 minutes, how many years of 365 days, 6 hours ?
                             Ans.  3 yrs. and 3527 h. over.
30. In 7854 hours, how many days ?
                               Ans. 327 d.  and 6 h. over.
31. In 47820 seconds of a degree, how many degrees ?
                                   Ans. 13° and 17' over.
32. In 7647 pounds, how many quarters ?
                              Ans. 273 qr. and 3 lb. over.
33. In 10752 pounds, how many tons ?
                            Ans. 4 tons, and 16 cwt. over.
 34. In 16840 pints of wine, how many hogsheads ?
                            Ans. 33 hhds. and 26 gal. over.
 35. In 4746 grains of silver, how many ounces r
                        Ans. 9 oz. and 17 dwt. 18 gr. over.

   Questions.  1.  What is  Reduction ?—2. How is it divid-
ed?— 3- How do you perform Reduction descending?—4. When
any sum of a given denomination is  to be reduced to a lower
one, how do vou  proceed ?—5. By what number do you multi-
ply ?_6  When the given sum consists of several denomina-
tions, with which do you begin?-7- What do you  do with
the other  denominations ?—8. How  is Reduction descending

P  9  How do  you perform Reduction ascending ?—10.  By
what  numbers  do  you divide?—11.  How is it proved?—
12 When you have to multiply or divide by the fractions £ or
I, how would you manage them ?-13. Will you explain  this
by an example? 
REDUCTION OF CURRENCIES.            161
The following table shows the Currencies of the principal countries
  with which the United States have intercourse, reduced to Federal
  Currency, or Dollars, Cents, and Mills.
    Great Britain and Ireland.
 20s. sterling=$4.44,4,44=£l.
  Is.     "   «   .22,2,22 « 12d.
  Id.     "   '     1,8,51 «4 qrs.
  1 Guinea   «  4.66,6,66 ' 21s.
  4s.6<£or54rf.=1.00
 20s. Irish=$4.10,25=£l.
  Is.   '   '   .20,5,2=12^.

             France.
 1 livre   =$0.18,5,04=  20 sols.
 1 sol      <   .00,9,25+=12 den.
 1 denier  «   .00,0,77+
 1 franc   '   .18,3,33=100 cent.
 1 centime*   .00,1,83.
 1 crown=$1.10.

            Holland.
 1 florin or gilder=$0.40=20sto.
 1 stiver    =$0.02.
 1 shilling1   '   12=6 stivers.
 2£ florins   '   1.00 * 1 rix-dollar.
 20s.Flemish«   2.40 ' l£.
 1 ducat     «   2.00 « 5 florins.
 1 groat     «    .01 4  8 pennings.

             Spain.
 1 dollar plate=$1.00 or 100 cents.
 1 real plate  '   .10.
 1 real vellon '   .05.
 1 marvedies  «   .00,1,47+

             Russia,
 1 ruble  =$0.75*  =100 copecs.
 1 copec   «   .00,7,5 « 2 dcnuscas.
 ldenusca'   .00,3,75.

         Leghorn—Italy.
1 piaster or pezzo=$0.86,5,55.
1 livre  =$0.14,44=3 J soldis.
1 soldi   '   .04,3,33=12 deniers.
1 denier  «   .00,3,61+
 1 ducat  «  1.08,3=7£ tores.

            Turkey.
1 piaster=$0.38,8  = 4 solotas.
1 asper   «   .04,8,5 « 4 mangars.
1 xeriff  '  1.43,7   « 15 solotas.
1 rupee=5
1 ana   '
1 tale
1 rupee
1 mace,
            Calcutta.
          0.50   =16 anas.
            .03,12 ' 12 pices.

             China.
            =$1.48    =10 maces
            «   .50
            «   .14,8   «10 cawc?.
1 candareen '   .01,4,8 * 10 caxas

     Jamaica— West Indies,
20 shillings=$3.00    =1£.
12 pence    •   .15     * 1 shilling.
 1 penny   «   .01,2,5 « 4/ar.
    6s. 8d.  or 80d., 1 dollar,

            Portugal.
1 millrea=$1.24=1000 rais.
1 rea    «   .00,1,24.
8 reas   '   .01.
800reas '   1.00.

        Trieste, Austria.
1 florin  =$0.48,4,85= 5J livres.
1 livre   «   .09,2,35  « 20 so/c£.
1 soldi   «   .00,4,617.
1 cruitzer«   .00,0,808.

          .Easf Indies.
1 rupee  =$0.55,5,5=10/anama
1 fanam   *    .05,5,5 ' 8 pices
1 pice    «    .00,6,9+
1 pagoda  '   1.94.
1 rial
8 rials
     Cuba.
50.12,5.
 1.00.
           Hamburg.
1 mark-banco=$0.33,3,33=16s&\
3 marks    =$1.00=1 rix-dollar
2\ rix-dollars «  2.50 «1 £. F/em.
1 stiver      '   .02,0,84.

    British Possessions, N. A.
20 shillings=!=$4.00=-l£.
 1 shilling '   .20 * 12rf.
 1 penny   «   .01,6,66=4/ar.
    5s. or 60d.=$1.00.
* This is the value of a ruble, by an essay, lately made in the mint of the U. States. 
162                    ARITHMETIC.
  Note.—By the help  of the preceding table, any sum of money,
in the currency of any one of the above-mentioned places, may be
easily reduced to dollars and cents.
  If the given sum consists of one denomination only, multiply it by the
value of a unit or 1 of that denomination in dollars, cents and mills, as
given in the table, and the product will be the sum required.   Hut, if
there are several denominations in the given sum, then multiply each
one by its respective value in dollars, cents, and mills; and the sum of
these products will be the amount required.

The following table shows  the  currency of the several states,
               reduced to Dollars, Cents, and Mills.
N. E. States, Virginia &f Kentucky.    New  York and N. Carolina.
20 shillings=$3.33,3,33=l£.
 1 shilling '   .16,6,66 « 12rf.
 1 penny   «   .01,3,88 ' 4 far.
 1 farthing '   .00,3,47.
 6s. or72rf. «  1.00.
20shiHings=S2.50   =1£.
 1 shilling  «   .12,5 • 12d.
 1 penny   «   .01,04 ' 4 far.
 1 farthing '   .00,2,6.
 8s. or96rf. '  1.00.
   South Carolina and Georgia.
20 shillings=$4.28,5,7 =»1£.
 1 shilling  '   .21,4,28 « 12d.
 1 penny   «   .01,7,85 ' 4 far.
 1 farthing '   .00,4,44.
 4s.8tf. or 566?.=$ 1.00.
 New Jersey, Pennsylvania, Dela-
      ware, and Maryland.
20 shillings=$2.66,6,66«=l£.
 1 shilling <   .13,3,33 « 12e?.
 1 penny  «   .01,l,ll+=»4/ar.
 1 farthing '   .00,2,78.
 7s. 6d. or 90c?.=$1.00.
  Note.—By the help of this table, any sum of money in the currency
of any of the above mentioned states, may easily be reduced to dollars,
cents, and mills.
  If there are pounds, shillings, and pence in the given sum, multiply
each denomination by the value of a pound,  shilling, and penny, of
that  particular currency, given in the above  table; and the sum of
the products will be the sum required in dollars and cents.  If pounds
only are given; then multiply by the value of a pound in that particu-
lar currency as given in the table, and the product will be the answer
in dollars, cents and mills.

       TABLE OF FOREIGN AND DOMESTIC  COINS.
This table exhibits their names, standard weight, and value in Dollars,
                        Cents, and Mills.
 Gold Coin.
Johannes
Doubloon
Moidore
E. Guinea
F. Guinea
Sp .Pistole
Fr.Pistole
St. Wt.	Value.
dwt gr.	D. C.
18 0	16.00,0
16 21	14.93,0
6 18	6.
5 6	4.66,6
5 5	4.60
4 6	3.77,3
4 4	3.66,7
 Silver Coin.
Fr. Crown
Spa. Dollar
A rix-dol. of
 Denmark,
 Sweden,
 and Ham-
 burg.
St. wt
dwt gr
19  6
17  0
17  6
Val
O. C.
1.10
LOO
1.00
U. S. Coin
Eagle
Half Ea-
 gle
Dollar
Half dol
 lar
Dime
dwU gr.
11  4$

 5 14|
17  i|
 8 12|

 1 16.9
D.C.
10.00

 5.00
 1.00

  .50
  .10
  Other gold coin passes in  the  United States  by its weight, at 89
cents per dwt., if it has sufficient fineness.  Other silver coin is valued
at 111 cents per ounce, by an act of congress. 
              REDUCTION OF CURRENCIES.           163

  To reduce sterling to dollars and cents,  the  dollar being
valued at 54 pence.
  If the given sum is pounds only : multiply by the value of a
pound sterling, in dollars, cents, and mills, and the product will
be the answer in  hundredths of a mill; which  reduce to dollars
and cents, for the sum required.  But, if there  are also shillings
and  pence  given, multiply the shillings by 22,2,22, and  the
pence by 1,8,51.   The sum  of these products will  be  the dol-
lars, cents and mills required.  Or,
  Reduce the given sum to pence, which multiply  by  100, and
divide the product by 54; the  quotient  will be the answer in
cents.   See the table of foreign currency.

                        EXAMPLES.
   1. Reduce £l65, 12s. and 6d, sterling, to dollars and cents.
                                         Ans. |736.10,3,7.
 1st method. l65x4.44,44=733.32,60=dol.and cts. ini£l65.
              12x.22,22=   2.66,66=dol. and cts. in 12s.
                6x1,851=    .ll,ll=centsin6J.
                          736.10,37=the sum required.
                     £    s.   d.
      2d method.     165  12   6
                      20
                     3312
                        12
                    39750=the pence of the given sum.
                         100
                     3975000-^54=736,11,1 +

   This  gives a few mills  more  than the former  method, and is
the most correct.   The  difference is  owing to the omission of
the very small fractions of the pound,  shilling and penny, in the
table.
   2. Reduce .£455, 16s. and 9d. to dollars and cents.
                                        Ans. $2025.94,4+
   3. Reduce 154 guineas, to dollars and cents. Ans. $718.66,0.
   Here again the latter method gives a few mills more than the
former ;  for $4.66,6,60 are not exactly the value of a guinea.
   Gold is the only legal tender in England.  The dollar is merely
an article of commerce, and consequently varies in value with
the quantity in market.   Its  true value, as  determined by  the
the British mint, is something less than 50 pence sterling.  The
Congress of the United States fixed its value at 54 pence sterling j
which is too high. 
164                   ARITHMETIC.
  4. A lady, having a legacy of ,£476, 10s. lrft to her by a
relative in London, what sum in dollars and cents should she
receive i« New York for the same, the dollar being in London
51 pence sterling?                        Ans.$2242. 35 +
            £    s.
          A76   10
                 20
                9~530
                  12
               114360
              ______100
               11486000-r 51 =2242,35 +

  When the value of the dollar is not 54 pence, you must always
use the second  method, and observe to divide  by the current
value of the dollar, for the true sum in dollars and cents.
  5. A merchant in London draws upon his  agent in Philadel-
phia, for the sum of £716, 8s. and 9d.; how much in dollars and
cents will be the  sum, the value of the dollar  being 53 pence
sterling?                                  Ans. $324415,1.

  To reduce dollars and cent3 to sterling, the dollar  being
valued at 54 pence.
  Reduce the given sum to cents; which multiply by 54, and
divide by 100, the quotient will  be pence:  these reduce to shil-
lings, and pounds for the true sum.   Gr,
  Reduce the given sum to hundredths of a mill, and divide by
4.44,4,44 for pounds;  the remainder (if more than 22,2,22 the
value of a shilling) divide  by 22,2,22 for shillings ;  and  the re-
mainder (if  more than  1,8,51 the value of a penny) divide by
18 51 for pence :  but, if the remainder after  the first division is
less'than  2,22,22, then divide  by  1,8,51  for  pence, and  the
remainder by 0,4,6+ the \alue of a farthing, for farthings.
                        EXAMPLES.
  6. Reduce 17465.45 cents, to pounJs sterling.
                                    Ans. £3929 14s. 6^d.
       1st method.         17465.45
                                54
                     1.00)943134.30
                       12)943134.
                       2.0)  7859.4  6.
                     Ans. £3929,14«. 6c/. 
REDUCTION OF CURRENCIES.           l65
   2d method. 4.44,4,44) 17465.45,0,00(3929£.
                       1333332  •  ••
                         4132130
                         3999996
                          1321340
                           888888
                           4324520
                           3999996
                    22.2,22)3.24,524(14*.
                           2 2222
                             102304
                              88888
                        1.851)13.416(7^.
                              12957
                          ,4.60)4.59(lgr.
                                 460

  Here after the first division $3.24,524 remain, which divided
by the value of 1 shilling, or 22.2,22 cents, gives 14 shillings.
The  remainder is 13.416 cents, which divided by ,1.851, the
value of a penny, gives 7 pence;  the remaider 4.59 mills, divided
by ,4,60 the value of a farthing, gives 1 farthing nearly.
  The first method gives the true sum. The latter gives a few
mills too much, because we do not divide by the exact value of a
pound, shilling, &c.  When  the  value of the dollar is not 54
pence, you  must use the first method, but observe to multiply
by the current value, and divide by 100.
  7. A gentleman with  his wife being about to make  the tour
of Europe, would invest his funds amounting to 2764 dolls. 37%
cts.  in English funds ; how much sterling will be the sum, the
dollar being worth only 51  pence ?        Ans. £587-  8s. 7^d.
   8.  How much sterling will 3740 dolls.  25 cts. amount to, the
dollar being 52 pence?                    Ans. £810, 7s.  9d.
  9.  How much sterling  will  1574 dolls. 62.5 cts. amount to,
the current value of the dollar being 53 pence ?
                                      Ans. £347. 14s. 7%d.
   10. How many guineas will  846 dolls. 12.5 cts. purchase,
the current value of the dollar being 52 pence?
                       Ans. 174 guineas, and 12s. 6%d. over.
   Having  reduced the given sum to shillings, you must divide
by 21, the number of shillings in a guinea, and the quotient will
be the answer.
             P 
166                  ARITHMETIC.
  11. How many guineas may be purchased for ll67dol|j.
87.5 cts. the dollar being valued at 51  pence?
                        Ans.  240 guineas, and 6s. 9§d. over.

  To reduce the currency of France  to dollars and cents :
  If the given sum is livres, or livres, sols, and deniers ; multiply
the livres by  the value of a livre as given in the table of foreign
currency, and the sols and deniers by their respective  values,
as given in the said table. These seveial products, added toge-
ther, will be the amount required.  But,
  If the given sum is francs and centimes, then multiply by the
value of a franc, as given in the table, and the centimes by the
value of a centime; the sum of these products will be the amount
required.   In all cases reduce the sum to dollars and cents.

                        EXAMPLES.

  12. What will 4764 livres, 12 sols, 6  deniers, amount to in
dollars and cents?                          Ans. $881.64,2.
  13. A lady in Philadelphia has the  sum of 14764 francs, 16
centimes, left to her by her father in Paris:  what is the value in
dollars and cents ?                      Ans. $2706.75,9,28.
  14. A gentleman in France,  about  to remove to the United
States, invested his  funds amounting to 11764 livres in public
stock at 18.5 cents per livre; how much did he purchase ?
                                            Ans. $2176.34.
  15.  A merchant imported from France a case of silk, invoiced
at 1087 francs, 75 centimes . what is  the amount in dollars and
cents?                                   Ans,  $199-42,0,5.
  16. What is the value of 1476 livres, 10 sols, in dollars and
cents, the current value  of the livre being 17 cents 8 mills r
                                            Ans. $262.81,7.
  17. In 12784  francs, how many dolls, and cts.; the current
value of the franc being 17 cts. 9 mills ?    Ans. $2288.33,6.

  When the current value of the livre and  franc, as in these
last  examples, is different from  that given  in the table, yu must
multiply by  the current value, for the  true  sum.  The same
thing is to be observed in the reduction of any other currency.

  To reduce dollars and cents to French  currency :
  Reduce the given sum to hundredths of a  mill, so that there
 may be as many decimal places in the divideml, as there are in
 the  divisor.   Then divide by 18.5,04  cents for  livres, and by
 18.3,33 cents for francs, as given in  the  table of foreign cur- 
              REDUCTION OF CURRENCIES.           167

rency; and the remainder, if there is any, by the value of a sol,
denier, or centime as the case may be.
   But, if the value of a livre  and franc  is different from that
given in the table, observe always to divide  by that value, for
the true sum.

                       examples.
   18. Reduce 1074.27 cents, to livres.
                         Ans. 5805 livres, 12 sols, 2 deniers.
          18.5,04)1074.27,000(5805 livres.
                  925 20
                  149070
                  143032
                    103800
                     92520
                925)11,280(12 sols.
                      9 25
                      2030
                      1850
                   .77)1,80(2 deniers.
                       154
                        26

   Here you add three cyphers or decimals to  the cents, which
reduce the  sum to hundredths of a mill; these divide by 18.5,04,
the quotient will be livres ; the remainder divide by 9 mills 25
hundredths, the value of a  sol, the quotient will be sols; the
remainder, if any, divided by ,77 hundredths of a mill,  the
value of a denier, will give the  deniers.  But,
   If the given sum is to be reduced to francs, you must divide
by 18.3,33, and the quotient will be the francs ; the remainder
divided  by 1,83, the value of a centime, will give the centimes.
   19. Reduce 1764 dollars to  francs.
                             Ans. 9621 francs, 99 centimes.
   20. General Lafayette received of the congress of the United
States the sum of 200,000 dollars; what is the amount in francs ?
                        Ans. 1,090,928 francs. 93 centimes.
   21. A lady has the sum of  1576 dollars, 15 cents, and desires
to know how many livres and sols they would amount to : what
is the number ?                  Ans. 8528 livres, 13 sols + .
   22. The expense of a young lady at a  boarding school in
France is $285.50 cents; what is the amount in francs and cen-
l,raes ?                       Ans. 1557 francs, 30 centimes. 
168                   ARITHMETIC.
  23. Reduce $4789.17, to francs.     Ans. 26,123 fr. 22 cen.
  24. Reduce $13546.37,5, to livres, and also to francs.
          Ans. 73207 Hv. 16 sols, 4+den. 73890 fr. 66+cen.

  To reduce the currency of Holland to dollars and cents:
  Multiply the given florins by 40, the value of  a florin in the
table  of foreign  currency, the product will be the amount in
cents: If shillings and stivers are also given, or either of them,
multiply them  by their respective values: the  sum of all the
products will be the amount required.  If ducats, rix dollars, or
pounds are given, multiply them  by their resptctive values, as
given in the above table, for their value in dollars and cents.

                        examples.
   25. A  merchant in  Amsterdam shipped to Philadelphia, a
quantity of goods,  valued at 17645 florins, 16 stivers ; what is
the value  in dollars and cents ?               Ans. $7058.32.
   26. A lady leaving  Amsterdam  sold her  effects,  which
amounted to 1246 florins, 12 shillings: on her arrival at New
York, she desired to know the amount of her  funds in dollars
and cents: what was the amount ?              Ans. $498.52.
   27. A  gentleman has in Dutch funds the sum of 1225 pounds,
 16 shillings Flemish, which he is desirous to transfer to the cur-
 rencv of  the United States; what is the sum in dollars and
 cents^?                                     An..*294U>2.
   23. A gentleman, with his family, arrived from Holland to
 make a tour in the United States, and  brought with him 5165
 florins, 15 stivers ; how much  is the sum in dollars and  cents?
       '                                     Ans. $2066.30.
   29. A young lady has a fortune of 37491 guilders, 3 stivers,
 in the funds of the Dutch government, which she wishes to trans-
 fer  to the Bank of the  United States, and  receive  stock in
 exchange ; how much stock must she receive, if it be transferred
 to her at  par?                             Ans. $14996.46.

   To reduce dollars and cents to the currency of Holland :
   Reduce the given sum to cents or mills, and divide  by  the
 value of that  particular denomination, mentioned in the ques-
 tion, as shown in the table of foreign  currency.
                         EXAMPLES.
    30 Reduce $2450.25, to florins.  Ans. 6125 flor. 12+ stiv.
    31* A merchant of New York remitted to Holland the sum
  „f  2468 dollars, 50 cents ; what will be the amount in  pounds
  F^ush?                          Ans. £1028, 10s. 5 stiv. 
REDUCTION OF CURRENCIES.
169
   32. A farmer of Virginia sent as an adventure to Amsterdam,
 50 hhds. of tobacco,  valued here at 4785.47 cents, which was
 sold so as to cover cost  and  charges;  what was the amount in
 florins and stivers, when the freight and other charges amounted
 to 1264 dols. 12£ cts. ?             Ans. 15123 flor. 19| stiv.
   33. Reduce $5467-15 cents, to florins and stivers.
                                  Ans. 13667 flor. 17% stiv.

                  PROMISCUOUS EXAMPLES.
   See the note at the  foot of the table of foreign currency.
   34. Reduce 1624 piasters, 25 aspers of Turkey, to dollars
 and cents.                               Ans. $630.23,3,25.
   35. Reduce 625 rubles, 26 copecs of Russia, to dollars and
 cents.                                    Ans. $468.94,5.
   36. Reduce 542 piasters, 5 livres of Leghorn, to dollars and
 cents.                                      Ans. $469-85.
   37. Reduce  1156  marks-banco, 12 stivers of Hamburg, to
 dollars and cents.                           Ans. $385.58^.
   38. Reduce 256 tales, 5 mace, 4 candereens ofChina, to dol-
 lars and cents.                            Ans. $379-67,92.
   39. Reduce 175 millreas, 645 reas of Portugal, to dollars and
 cents.                                   Ans. $217-77,9,8.
   40. Reduce 564 florins, 5 cruitzers of Austria, to dollars and
 cents.                                    Ans, $273.49,6.
   41. Reduce 316 pagodas, 25 fanams of India, to dollars and
 cents.                                   Ans. $614.42,7,5.
   42. A merchant of Jamaica, (West Indies,) shipped to Phila-
 delphia goods amounting  to 567 pounds, 10 shillings; how
 much is the sum in dollars and cents ?        Ans.  $1702.50.
   43. A  merchant in Montreal,  (Canada,) purchased a draft
 upon  the Mechanics' Bank of the city of New York, for 476
 pounds, 16 shillings; what is the sum in dollars and cents ?
                                           Ans. $1907.20.
   To reduce New England and Virginia currency to dollars
 and cents:
            6 shillings or 72 pence = 100 cents : or
                        18 pence =  25 cents, by dividing by
4, the greatest common measure: therefore
  Multiply the Virginia or New England currency in pence, by
 25, and divide by 18, for dollars and cents :  Or,
  Multiply the given  sum, if in pounds only, or in  pounds,
shillings and pence, by the respective values of these denomina-
tions, as given in the table of domestic  currency.  The sum of
            p  2 
170                   ARITHMETIC.
the products will  be the amount required.  See the table of
domestic currency.
   44. Reduce 146 pounds, 16 shillings, and 4 pence, to dollars
an" cen's.                              Ans. $489-38,8,8.

     First method. £146,16s. Ad.= 35236 pence.
                                    25
                                880900-4-18=489.38,88.
     Second method.
         146x3,33,3,33=486.66,63
           l6x ,16,6,60=  2.66,66
           Ax   1,3,9 =='     5,56
                       $489.38,85  nearly as before.

   45. A lady  in  Virginia has an income of 450 pounds, 12
shillings a year; how much is it in dollars and cents ?
                                     Ans. $1501.99,9,9+
   46. A gentleman  in Richmond received an order for the col-
lection of 1764 pounds, 5  shillings; what was  the amount in
dollars and  cents ?                     Ans. $5880.83,2,96.
   47- A gentleman has an income of 746 pounds New England
currency, and is desirous of knowing the amount in dollars and
cents;  what is it ?                       Ans. $2486.66,6+
   48. A gentleman  placed his daughter at a boarding school in
New Haven, at an expense of 117 pounds, 15 shillings per an-
num ; what is the  sum in dollars and cents ?    Ans. $392.50,
   49. A lady of  Boston purchased goods to the amount of 46
pounds,  17 shillings,  and  desired her daughter, who had just
returned from school, to inform her of the amount in dollars and
cents;  what was the amount ?             Ans. $156.49,9+

   To reduce dollars and  cents to New England, or Virginia
currency:
   Reduce the given sum to  cents, which  multiply by 18 and
divide the  product  by 25, the  quotient will  be pence; which
reduce to pounds,
   50. Reduce $1746.16, to Virginia currency.
                                    Ans. £523, 16s.  ll£d.
   51. A merchant shipped an invoice  of goods  to Boston,
amounting to $2416.75; what was the amount in New England
currency ?                              Ans. £725, 0«. 6d.
   52. A gentleman has an income of 1746 dollars a year, but
residing in  New England, desires to know the amount in that
currency; what is it ?                    Ans. £524, As. Ad. 
REDUCTION OF CURRENCIES.           171
  53. A gentleman  has a  son  at a boarding school in  New
England, at an expense of $275,25 cts. per annum ; what is the
sum in that currency ?                    Ans. £86, Os. 3fa.
  54. A lady has in public stocks the sum  of $10475,75 cts.
but residing in Virginia, is  desirous to know the amount in the
currency of that state : what is the sum ? Ans. £3273,13s. 5%d.

  To reduce sterling to New England currency; and the con-
trary :
     54 pence sterling=72 pence New England: or
      3 pence  do.  = 4 pence     do.     by dividing both
by 18, the greatest common measure :  therefore,
  Multiply sterling  by 4, and divide by 3, the quotient will be
New England, or Virginia  currency:  And
   Multiply Virginia currency by 3,  and divide by 4, the quo-
tient will be sterling.
   55. Reduce £476, 16s.  sterling to Virginia currency.
                                       Ans. £635,14s. Bd.
                       £    s
                      476  16
                    _______4
                   3)1907    4
                      635  14  8

   56.  A merchant in New England received from London  an
 invoice of goods, amounting to £784, 12s. sterling: what is the
 sum in that currency?                    Ans. £1046, 2s. 8rf.
   57.  A farmer in  Virginia shipped to Liverpool a quantity of
 tobacco, amounting to £1564, 10s.  Virginia currency; what is
 the amount in sterling?                  Ans. £l 173, 7s. 6d.
   58. A lady has a fortune of £1416, 5s.  in the hands of a bank-
 er in London, which  she is desirous to transfer to Boston; how
 much in New England currency will it amount to?
                                       Ans. £1888, 6s. Sd.
   59.  A merchant of Richmond shipped to Liverpool a quan-
 tity of tobacco, amounting to £1746, 15s. Virginia currency;
 how much would it amount to in  England, provided he neither
 gained nor lost  by the shipment ?         Ans. £1310,  Is. 3d.

    To reduce New Yoik  and  North Carolina currency to dol-
 lars and cents ;  and the contrary :
      96 pence =  100 cents; or
      24 pence =   25 cents, by dividing by 4, the greatest com-
 mon measure:  therefore, 
172
ARITHMETIC.
  Reduce the  given sum  to pence, which multiply by 25, and
divide the product by 24, for cents.  And
  Multiply the given  sum in  cents by 24, and divide  the pro-
duct by 25, the quotient will be pence, which reduce to pounds.
  60, Reduce 125 pounds, 6 shillings and 8 pence, to dollars and
cents.                                     Ans. $313.33J.
  71. A lady has in cash 2174 pounds, 10 shillings, which she
desires to invest in stock of the bank of the United States; how
much can she purchase, the stock being at  par ? Ans. $f)436.25.
  62. A lady has a school worth to her the sum of 1756 dollars
45 cents  per annum; how much  is the  income  in New York
currency ?                            Ans. £702, lis. 7\d.
  63. A lady made a purchase of  goods in New  York, and on
her return home found her bill amounted to the sum of 56 pounds,
16 shillings,  and 4 pence.  Not  understanding the currency
very well, she  gave it  to  her  daughter, desiring her to tell the
amount in dollars and cents: what  was that amount ?
                                           Ans. $142.04.
  64. A gentleman finding the expenses of  his family, in  the
city of New York, to be annually $2162.68 cents, desired his son
to tell him the amount in the currency of the state;  what was the
amount?                             Ans. £865, Is.  5rf.+
  65. What will 4769 dollars, 14 cents, 6 mills,  amount to  in
New York currency ?                  Ans. £1907, 13«. 2d.

  To reduce sterling  to  the currency of New York or North
Carolina ; and the contrary:
     54 pence sterling =  96 pence New York: or
      9 pence  do.    =  16 pence    do.    by dividing by
6, the greatest common measure :  therefore,
  Multiply the sterling  by 16, and divide the product by 9, for
New York currency.  And
  Multiply the New York currency by 9, and divide by 16, for
sterling.
  66. Reduce 4l6 pounds, 10  shillings sterling,  to New York
currency.                           Ans. £740,8s. 10£+rf.
  67. A merchant in  Liverpool shipped  to New York goods
to the amount of 1764 pounds sterling; how much is the sum in
New York currency ?                          A ns. £3136.
  68. A merchant wishes to purchase a  draft on London for
785 pounds, 16 shillings.  How much must he give in New York
currency, the exchange being at par? Ans. £1396, 19*. 6J+rf.
  69. Reduce 2457 pounds, 16 shillings to sterling.
                                     Ans. £1382, l6«. 3d. 
REDUCTION OF CURRENCIES.           173
  70. A gentleman remits to London 8740 pounds, 10 shillings:
how much sterling is the sum, the exchange being 51 pence, per
dollar ?                               Ans. £4643, 7s. 9f <*•
  When the dollar is not worth 54 pence, as in the above ex-
ample, you must multiply by the current  value, and divide by
96 for sterling;  and the contrary, to reduce sterling to  New
York currency.

  To reduce  New Jersey, Pennsylvania and Maryland cur-
rency to dollars and cents ; and the contrary:
     90 pence = 100 cents ; or,
      9 pence = 10 cents, by dividing by 10, the greatest com-
mon measure: therefore,
  Multiply the given sum, reduced to pence, by 10, and divide
by 9 for cents.  And
  Multiply the sum  in cents by 9, and divide by 10 for pence :
which reduce to shillings and pounds.
  71. Reduce 564 pounds, 10 shillings, Pennsylvania  currency,
to dollars  and cents.                     Ans. $1505.33,33.
  72. Reduce $10746.75 cents, to Pennsylvania currency.
                                      Ans. £4030, Os. 7%d.
  73. A merchant received an invoice of goods amounting  to
6470 dollars : how  much is the sum in Pennsylvania currency ?
                                          Ans. £2426, 5s.
  74. A  gentleman in Maryland has an estate, which he values
at 3650 pounds j how much is it worth in dollars and cents ?
                                        Ans. $9733.33,33.
  75. A  gentleman in  Philadelphia has in ground  rents and
real estate, the sum  of 12640 pounds, 15 shillings ; how much is
the sum in dollars and cents ?                 $33706.66,60.
  76. A lady has, left by her father, an annual income  of 1246
dollars, 49 cents ; what  is the sum in New Jersey currency ?
                                        Ans. £467, 8s. 8rf.
  To reduce sterling to Pennsylvania currency;  and the con-
trary:
     54 pence sterling = 90 pence Pennsylvania ; or,
      3 pence   do.   =  5 pence     do.        by dividing
by 18, the greatest common measure: therefore,
  Multiply the sterling  by  5, and divide the product by  3, for
Pennsylvania currency.   And,
  Multiply Pennsylvania currency by 3, and divide by  5, for
sterling.
  77. Reduce 178 pounds, 18 shillings, and 6 pence sterling, to
Pennsylvania currency.                Ans. £319, lis. 2+rf. 
174                   ARITHMETIC.
  78. Reduce 416 pounds, 10 shillings, Pennsylvania currency,
to sterling.                                Ans. £249,18«.
  79. A  merchant of Baltimore  received from Liveipool an
invoice of goods, amounting to 789 pounds, 10 shillings sterling;    l
how much is the amount in Maryland currency ?
                                      Ans. £1315, l6s. 8</.
  80. A gentleman having 7641 pounds, 15  shillings Pennsyl-
vania currency, removes to London ; how much sterling is the
sum ?                                     Ans. £4585, If.
  81. A lady placed her daughter at a boarding school in Lon-
don, at an annual expense of  110 pounds,  5  shillings sterling ;
what is her bill in Pennsylvania currency ?   Ans. £183, 15«.
  82. A  gentleman commenced travelling, and took with him
the sum of 1260 pounds, 15 shillings Maryland currency, which
he  placed in the hands of a banker in London, subject to his
draft; what was the amount in sterling ?       Ans. £756, 9».

  To reduce South Carolina and Georgia currency to dollars
and cents; and the contrary:
            56 pence=100 cents: or
            14 pence= 23 cents, by dividing by 4, the greatest
common  measure: therefore

  Multiply the sum in South Carolina currency,  reduced to
pence, by 25, and divide the product by  14, for cents.  And
  Multiply the sum in cents by 14, and  divide  by 25, the quo-
tient will be pence ; which reduce to shillings and pounds.

  83. Reduce 746 pounds, 10 shillings, Georgia  currency to
dollars and cents.                       Ans. $3199.28,5+.
  84. Reduce 1074.25 cents, to South Carolina currency.
                                       Ans. £250, 13s. 2d.
  85. A merchant in  Charleston sends to New York a quan-
tity of cotton, amounting to 579 pounds, 10 shillings, (Georgia
currency ; how much will it amount to in dollars and cents ?
                                      Ans. $2483.57,14 + .
  86. A gentleman has a plantation in South Carolina,  valued
at 7896  pounds,  which he exchanges for property in Philadel-
phia ; what must be the value of the properly received, in dol-
lars and cents ?                               Ans. $33840.
  87. A  lady has a cotton  plantation in Georgia, and is desir-
ous of disposing of it,  to remove to New York ; what must be
its value  in the currency of  South Carolina,  when she receives
for it 10140 dollars ?                           Ans. £2266. 
REDUCTION OF CURRENCIES.            175
  88. A father left his daughter the sum  of 1265 pounds, 10
shillings, Georgia currency, to be  paid to her without interest
upon her marriage day; what is its value  in dollars and cents?
                                     Ans. $5423.57,14+.

  To reduce South Carolina and Georgia currency to sterling 3
and the contrary:
     54 pence sterling=56 pence South Carolina ; or,
     27 pence   do.   =28 pence  do.   by dividing by 2, the
greatest common measure: therefore
  Multiply the sterling by 28, and divide  by 27 for South Caro-
lina or Georgia currency.  And
  Multiply South Carolina currency  by 27, and divide by 28
for sterling.

  89.  Reduce 312 pounds, South Carolina, to sterling.
                                   Ans. £300, 17s. 1%+d.
  90.  Reduce 516 pounds, 10 shillings sterling, to South  Caro-
lina.                                Ans. £535,  12s.  7+d.
  91.  A merchant in Liverpool  ships to his correspondent in
Charleston,  goods amounting to 1764 pounds, 10s. sterling;
what is the sum in Georgia currency ?  Ans. £1829, 17s. 0%d.
  92.  A merchant of Savannah remitted  to London, the sum of
6174/. 5s. Georgia currency; for what sum must he have credit,
the exchange being at  par ?            Ans. £5953, 14s. 9$d.
   93.  A  planter of Georgia shipped  to Liverpool, a quantity of
cotton, amounting to the sum of 2160 pounds, 15 shillings, South
Carolina currency;  what sum in sterling  must he receive for it,
if he neither gains nor loses by the adventure ?
                                   Ans. £2083, lis. 7+d.

   To reduce New England  and Virginia currency to  New
 York currency ; and the contrary:
     72 pence Virginia = 96 pence  New York ; or,
       3 pence  do.   = 4 pence    do.    by dividing by
 24, the greatest common  measure :  therefore,
   Multiply the New England currency by 4, and divide  by 3,
 for New York currency.  And
   Muhiplv the New York currency  by 3, and divide by 4, for
 New England.
   94.  Reduce  475  pounds 10 shillings, Virginia currency, to
 New York currency.                           Ans. £634.
   95. Reduce 4791 pounds, New York, to New England cur-
rency.                                    Ans. £3593, 5s. 
176
ARITHMETIC.
  96.  A merchant of Virginia sent to New York a quantity of
tobacco, amounting to 1140 pounds, 12 shillings; what will be
the amount in the currency of New York ?   Ans. £1520, 16*.
  97.  A gentleman from Virginia, visiting with his  family at
Balston Springs, on his way purchased for his wife and daugh-
ters sundry articles in the city of New  York, amounting to 55
pounds, 16  shillings, and 6 pence,  North Carolina currency:
how much is the sum in that of Virginia ?  Ans. £41,  17*. 4%d.
  98.  A young lady, at a boarding school in New Haven, finds
her annual expense to be 95 pounds, 10 shillings, New England
currency : what is it in that of New York ? Ans. £127, 6s. 8d.
  99. A gentleman from Virginia placed his eldest son at school
in New York; and, for  boarding, tuition, clothing, &c.  his bill
amounted to 135 pounds, 10 shillings :  what is the sum  in Vir-
ginia currency ?                         Ans. £101, 12«. 6d.

  To reduce Virginia currency to that of South Carolina and
Georgia ; and the contrary :
     72 pence Virginia = 56 pence Georgia;  or,
      9 pence   do.    =  7 pence   do.   dividing by 8, the
greatest common measure :  therefore,
  Multiply Virginia currency by 7, and divide by 9, for Georgia
currency.  And
  Multiply Georgia currency by 9, and divide by 7, for Virginia.
  100.  Reduce 475 pounds, 10 shillings, Virginia currency, to
that of South Carolina.                  Ans. £369, 16s. 8t/.
  101.  Reduce 764  pounds, 15 shillings, Georgia currency, to
that of New  England  or Virginia.              Ans. £'983, 5s.
  102.  A merchant of Charleston remitted  his  correspondent
in Richmond the sum of 1216 pounds, 12 shillings, South Caro-
lina currency : what is the sum in that of Virginia ?
                                          Ans. £1564, As.
  103.  The income  of a cotton plantation  was found to be
1024 pounds, 14 shillings, Georgia currency; what is the amount
in that of New England ?              Ans. £1317, 9s. 5+rf.
  104.  A cargo of provisions and  lumber was shipped  from
Boston, invoiced at 3465 pounds, 5  shillings, and consigned to
a merchant in Charleston :  what is the amount in South Caro-
lina currency ?                     Ans. £2695, 3s. 10^+rf.
  105. A lady had an estate in Georgia, valued at 4768  pounds.
But, for the purpose of educating her daughters, removed to Con-
necticut, and exchanged her property for  lands and  tenements
there:  what was  the  value of the  property  received in i\ew
England currency ; the exchange being made even ?
                                   Ans. £6130, 5s. 8£+cf. 
REDUCTION OF CURRENCIES.           177
  To reduce New England and Virginia currency, to Pennsyl-
vania, New Jersey and Maryland currency; and the contrary:
     72 pence Virginia = 90 pence Maryland ; or,
     36 pence   do.   = 45 pence   do.    by dividing by 2,
the greatest common measure: therefore
  Multiply the pence New England currency  by 45, and divide
by 36, for pence Pennsylvania. And
  Multiply the pence Pennsylvania by 36, and divide by 45, for
Virginia currency.
  106. Reduce 546 pounds, 10 shillings, New England currency,
to Pennsylvania.                         Ans. £683, 2s. 6d.
   107. Reduce 746 pounds, 15 shillings, Maryland currency, to
Virginia.                                   Ans. £597, 8s.
   108.  A merchant of Boston remitted to his correspondent in
Philadelphia the sum of 1016 pounds, 5 shillings, New England
currency ; what is the amount in Pennsylvania currency ?
                                      Ans. £1270, 6s. 3d.
   109.  A commission merchant of Baltimore sold goods for a
manufacturer in Massachusetts, amounting  to 1260 pounds, 16
shillings, Maryland currency; what is  the sum in New England
currency ?                         Ans. £1008, 12s. 9%+d.
   110. A gentleman in  New  England  possessed property which
he valued at 4176 pounds, 17 shillings, but removing to Philadel-
phia, he exchanged the  same  for houses and lots there ; what is
their value in Pennsylvania currency,  the exchange being made
even ?                                Ans. £5221, Is. 3d.
   111. A lady of Maryland  has a fortune of 7467 pounds, 10
shillings, but marrying  a gentleman of Richmond, is desirous of
knowing the value  of her estate in Virginia  currency;  what is
the amount ?                                  Ans. £5974.
   112. A  young lady  from Virginia, at a boarding school  in
Philadelphia, finds her  expenses to be 145 pounds, 12 shillings
per annum, in Pennsylvania  currency; what is  the amount  in
the currency of her native state ?        Ans. £ll6, 9s. 7-\-d.
   113. If a  young gentleman at Princeton college costs his
father 175 pounds,  12  shillings, per annum, New Jersey cur-
rency ;  what will it be  in Virginia or New England currency ?
                                     Ans. £140, 9s. 7+d.

   To reduce New  York and North Carolina currency to  New
Jersey, Pennsylvania and Maryland currency; and the contrary:
   96 pence New York  = 90 pence Pennsylvania ; or,
   48 pence   do.     = 45 pence   do.   by dividing by 2.
the greatest common measure: therefore
            Q                         3 
178
ARITHMETIC.
  Multiply New York currency in pence by 45, and divide by
48, for Pennsylvania currency. And
  Multiply Pennsylvania currency in pence, by 48, and divide
by 45, for New York currency.
  114. Reduce  125 pounds,  10  shillings, New York, to New
Jersey currency.                      Ans. £117, 13s. l£tf.
  115. Reduce £l67,12s. 6rf. Maryland currency,to New York
or North Carolina currency.                Ans. £178, l6s.
  116. A merchant of Philadelphia purchased goods  in New
York, to  the amount of 1746 pounds,  17 shillings, New York
currency ; but desired his bill to  be made out  in Pennsylvania
currency ; what was the amount ?      Ans. £1637, 13s. 5\d.
  117. A gentleman of New  York purchased an estate in New
Jersey, valued at 1576 pounds, 10 shillings, in the currency of
the state  : what sum must he pay in New York currency ?
                                        Ans. £1681, 12«.
  118. A manufacturer of Maryland consigned to a merchant
in New York, goods amounting to 1846 pounds, 11 shillings;
what is the sum in New York currency ?  Ans. £1969, 13+s.
  119. A lady of Maryland  has an income of 516 pounds, 10
shillings, but removing to New York, receives her money in the
currency of that state; how much is the amount ?
                                      Ans. £550, 18s. 8<f.
  120. A young lady of New York had a  fortune lefi to her by
a rich relation,  amounting to 5640 pounds, 17 shillings, which
was to be paid on her marriage day, but removing with her hus-
band  to Pennsylvania, would know the amount in the currency
of that state; what was it r            Ans. £5288, 5s. lljrf.

   To reduce New York or North Carolina currency, to South
Carolina or Georgia currency ; and the contrary:
     96 pence New York=56 pence South Carolina ; or,
     12 pence    do.   = 7 pence      do.      by dividing
by  8 the greatest common measure:  therefore
   Multiply  New  York currency by 7, and divide  by 12, for
 South Carolina.   And                                 t
   Multiply Georgia currency by 12, and divide  by 7, for New
 York currency.
   121. Reduce 546 pounds, 10 shillings New  York, to South
 Carolina currency.                    Ans. £318, 15s. lOd.
                    546   10
                   _____7
                12)3825   10
                  £318   15  10 
REDUCTION OF CURRENCIES.           179
   122. Reduce 847 pounds, 12 shillings, and 6 pence, Georgia,
 to North Carolina currency.           Ans. £1453, Is. 5+d.
   123. A merchant of Savannah shipped to New York a quan-
 tity of cotton amounting to 1042 pounds, 10 shillings; what is
 the amount in New York currency?    Ans. £1787, 2s. 10^d.
   124. A merchant of Charleston purchased goods of  a com-
 mission merchant in New York, amounting to the sum of 1574
 pounds, 16 shillings of that currency;  how much was the sum
 in the currency of his own state ?         Ans. £918, 12s. Sd.
   125. A gentleman of  South Carolina left home on an excur-
 sion of pleasure, with the sum of 416 pounds, 11 shillings ;on his
 arrival in the city of New York  he found that he had spent 27
 pounds, 5 shillings; what sum had he remaining, in New York
 currency ?                              Ans. £667, 7s. 5\d.
   126. What sum in the currency of Georgia is equal to 2546
 pounds, 5 shillings, and 9 pence, New York currency ?
                                       Ans. £1485,  6s: 8d.
   127. What sum of money in North Carolina  currency is
 equal to 1850 pounds, 17 shillings, and 6 pence, in that of South
 Carolina ?                             Ans. £1079,13s. 6d.
   128. A young lady is  to receive the  sum  of 21 pounds, 14
 shillings, and 6 pence, in money and presents, reckoning in Geor-
 gia currency, as soon as she shall be able to tell how  much that
 sum will be in the currency of New York or North  Carolina ;
 what will be the sum ?                   Ans. £37, 4s. 10£tf.

   To reduce Pennsylvania currency to that of South Carolina,
 and the contrary :

     90 pence Pennsylvania=56 pence South Carolina ; or,
     45 pence     do.     =28 pence         do.   by divid-
 ing by 2, the greatest common measure: therefore

   Multiply Pennsylvania currency in pence by 28, and  divide
 by 45, for South Carolina.  And
  Multiply South Carolina in pence by 45, and divide by 28,
 for Pennsylvania currency.

  129. Reduce 749 pounds, 10 shillings Pennsylvania, to Geor-
 gia currency.                           Ans. £466, 7s. Id.
  130. Reduce  742 pounds,  12 shillings, and 6 pence, South
Carolina to Maryland currency.      Ans. £l 193, 10s. 1+d.
  131. A young lady at school in Philadelphia, is informed by
her mother in Charleston  that she shall send her in presents and 
180
ARITHMETIC.
in money, the amount of 36 pounds, 7 shillings, and 3 pence,
reckoning in Pennsylvania currency, as soon as she shall inform
her, what that sum would be in the currency of Georgia;  what
is the sum ?                            Ans. £22, 12s. 7+d.
   132. A gentleman has a cotton plantation in Georgia, valued
at 7416 pounds, which he will dispose of to a gentleman in New
Jersey; what sum, reckoning in the latter  currency, must he
receive for it ?                         Ans. £11918, 11*. 5d.
   133. A merchant shipped to  Charleston  property, valued at
5476  pounds, 10 shillings,  Pennsylvania currency; what sum
will he receive, reckoning South Carolina currency  ?
                                    Ans.  £3407, 7*. 6%+d.
              SINGLE PROPORTION.

  This is commonly called the Rule op Three, because three
terms are always given to find a fourth.  The operation is found-
ed upon the principle that, if four numbers are proportional, the
product of the means (or middle terms) is equal to the product
of the extremes, (or the first and last terms.)  And the product
of the means, divided by either extreme, will give the other ex-
treme.*

  * Four numbers are proportional, when the first has the same ratio
to the second, as the third has to the fourth, as; 2 :6 :: 4 : 12; that
js, 2 has the same ratio (or bears the same relation) to 6,  that 4 has
to 12.
  The ratio,  which one number has to another, is found by dividing
the latter by the former.  Thus, the ratio of 2 to  6 is 3, because 6 con-
tains 2 three times . the ratio of 8 to 12 is 1.5, because 12 contains 8,
one and a half times: and the ratio of 12 to 6 is represented by <pj.
or 6 divided by 12: and so of other numbers.   When there are four
numbers such, that the ratio of the first to the second is the same with
the ratio of the third to the fourth ; these  four numbers are  propor-
tional, as 4 : 8 : : 3 : 6, or  as 12 : 4 :: 9 : 3.   This is called direct
proportion.   When four numbers are proportional directly, they will
be proportional in the following1 ways; for, in each case the  product
of the middle terms is equal to that of the extremes, as will appear by
inspection :
            1st. directly,              4 :8  : :3  : 6
            2d. by transposition,       4:3  : : 8 : 6
            3d. by inversion,          3 : 4  : : 6 : 8
  This property of proportional numbers has been made the founda- 
SINGLE PROPORTION.
181
  In all questions of the Rule op Three, the  answer is to be
considered the last term of the proportion.   Of the three terms
that are given, two are always of the same kind ;  and the third,
or remaining term, must be always of the  same kind  with the
answer, thus: If 2 bushels of wheat cost $2.50, what  will 4
bushels cost ?  Here there are two terms alike, namely, 2 bushels
and 4 bushels, which must  stand in the first and  second  places
in the statement;  and from the nature of the question, the an-
swer must be of the  same kind with the remaining, or  third
term, which is money.
  In  stating the  question, consider whether the answer must
be greater than the third  or remaining term ;  for then the first
term in the proportion must be less than the second ; but, if the
answer must be less than the third term, then the first term must
be greater than the second, for otherwise  the product  of the
extremes  could not be equal  to that of the means, or middle
terms.
  Having stated the question, as above directed, reduce the first
and second  terms to the  same denomination, if they are  not so
already.  Reduce  the third term  to the lowest denomination
mentioned, if it is a compound quantity.  Then multiply  the se-
cond  and third terms  together, and divide  the product  by the
first term  : the quotient will be the answer in the same kind, and
of the same denomination, with the third term.
  If  any  term has the  fraction %,  %, or  f, added to  it, the
operation commonly  may be  rendered easier  by reducing the
fraction to a decimal,  and adding it to the whole number or in-
teger.
  If the first and second terms will divide  by a common mea-
sure,  the operation may be shortened by dividing those terms by
that number, and using the quotient in the operation.
  Proportion  is expressed by  placing between the first and se-
cond terms, two dots or points, thus,: —between the second and
third,  four, thus,  •■ : —and  between the third and  fourth,  two,
thus,  : —as in this example, 4 : 8  :: 3 : 6, which means, that 4
is to 8, as  3 is to 6; or that 4 has the same ratio to 8, that 3 has
to 6.

tion of the  Rule of Three Inverse, as it is usually called.  But it is a
useless and unnecessary distinction.  For any question in inverse pro-
portion may, by changing the terms,  be made direct, and vice versa.
  Three numbers are proportional, when the first has the same ratio
to the second as the second has to the third, as 4 : 8 : : 8 : 16, and the
product of the extremes is equal to the square of the mean, or  middle
terms, as will appear from inspection.
          Q  2 
182
ARITHMETIC.
                       EXAMPLES.
  1. If 1 yard of broad-cloth costs #4.50 cents; what will 4 yards
cost?                                            Ans. $18.
             yd. yds.  cts.
              1:4:: 4.50 : £ proportion.
                        4
                    $18.00 Answer $18.00.
  Here, from the nature of the question, the answer, or fourth
term in the proportion, must  be greater than the  third term;
therefore the first term must be less  than the second ; 1  must
then be the  first term, 4 the second ; and 4.50, the third, or re-
maining  term must be in the  last place.   The two first terms
being of the same denomination, they require no reduction. You
do not divide by  the first term, because it is a unit, and would
not reduce the  dividend. Therefore the product of the means is
the answer to the question, or the fourth  term of the proportion;
for multiplied by 1, it will equal the product of the middle terms.
   2. If  6 bushels of wheat cost $7-50; what will 24 bushels
cost ?                                             Ans. $30.
   bush.  bush.    cts.
     6 : 24 : : 7-50  : £ proportion;  or,  by dividing  by 6,  the
common measure of 6 and 24,  you  have
               bush. bush.  cts.
                 1:4: =  7-50 : % proportion.
                             a'
                        $30.00
   Here again,  the answer must be greater  than the third term;
therefore the first term must be less than the second.   The first
term is 6, and the second is 24 ; both of which will divide by 6
without a remainder; 6 is therefore their common measure.  The
quotients are 1 and 6, which you use in the  operation, as will be
seen above.  In like manner, other questions may be reduced,
whose first and second terms will divide by a common measure.
   This operation is founded  upon the  principle, that if equal
or proportional numbers are divided by the same  number, the
quotients will be equal, or proportional  accordingly.
   3. If 3 pounds of butter cost 50 cents ;  what will 25 pounds
cost ?                                        Ans. $4.16,60.
                 lb.   lb.   cts.
                 3 : 25 : : 50 : | prop.
                      50
3)12.50
   4.16,60 
SINGLE PROPORTION.
183
  Here the answer terminates in a repeating decimal of a cent,
which is equal to § of a unit.  The answer, or fourth term in the
proportion, is 4 dollars, l6§ cents.
  4. If 1 pair of stockings cost 75 cents, what will 14 pair cost ?
                                             Ans. $10.50.
               pr.   pr.   cts.
                1 :  14 : : 75 : | prop.
                         14
                         300
                         75
                       10.50 the answer.
  5. If 10 pounds of beef cost 62% cents; what will 116 pounds
cost ?                                         Ans. $7.25.
                lb.    lb.    cts.
                10 : 116 : : 62.5 :  | prop.
                            116
                           3750
                           625
                          625
                       10)7250.r3
                           7.25 cents.

  In the third term, the half cent, I make 5 tenths of a cent, or
5 mills, and annex it to the integer; the price  becomes then 62
cents, 5 mills, for the third term.  The first and  second terms
may be reduced by dividing by 2, their common measure, as
directed, and shown in a preceding example.
  6. If 12 oranges cost 75 cents ;  what will 125 cost ?
                                           Ans. $7-81,2,5.
                or.   or.    els.
                12 :  125 : : 75 : ^ prop.
  7. If 24 yards of linen cost $14.50; what was the price of 1
yard?                                Ans. 60 cts. 4,1+min.
                yd. yd.     D.
                24 :  1 : : 14.50 s £ prop.
  Here  it is plain that  the fourth term  must  be  less than  the
third term ;  therefore the first term must be greater than the se-
cond ; otherwise the product of the extremes  would  not equal
that of the means.
  8. If 1 bushel of corn costs 42 cents; what is the value of 52
bushels?                                      Ans. $21.84.
               bu.  bu.    cts.
                1 : 52 : : 42 : | prop. 
184
ARITHMETIC.
  9. If 1 ton of hay sells for $8.50; for how much will 8 J tons
sell ?                                          Ans. $72.25.
               ton.  tons.   D. cts.
                1 : 8.5 : : 8.50 :  £ prop.
  Here the % in the 2d term I change  into  5 tenths of a ton,
and annex it to the 8. It becomes then, 8.5 tons.   The answer,
or fourth term, must be greater than the third term, therefore,
the  first term must be greater than the second ; for otherwise the
product of the means would not equal that of the extremes.
  10. A grocer bought a  chest of tea  containing  14 lbs. for
#12.75 cents; how much was it a pound ?  Ans. #91.07+cts.
                lb.  lb.   D. cts.
                14 : 1 : : 12.75 : ^ prop.
  11. If you give 5 cents a loaf for bread ; what will 48 loaves
come to, at the same rate ?                       Ans. $2.40.
                lo.  lo.  cts.
                1  : 48 : : 5 : J prop.
  12. If a man earns 75 cents a day; how much will he earn in
25fdays?                                Ans. $19.31,2,5.
                d.    d.     cts.
                1 : 25.75 -■•75 -. J prop.
  Here the f, in the second term, I reduce to a decimal, and
annex it to the integer; and it becomes 25,75.   The fourth term
must be greater than the third, therefore, the first term must be
less than the second;  for otherwise the  product of  the means
would not equal that of the extremes.
  13.  A gentleman bought a bag of coffee for the use of his
family, weighing 127 lbs., for which he gave #15.25  cts.;  what
was it a pound?                             Ans. 12+cts.
                 lb.  lb.     ct.
                127 ■ 1 ••• 15.25 = i prop.
  14.  A lady has an income of #675.75 cts.; how much may
she spend every day, and have nothing left at the  end  of the
year?                                       Ans. $1.85+
                 d.   d.     ct.
                365  : 1 :: 675-75 : J prop.
  15.  A gentleman has an  income of 1246 dol. 12 ct. 5 in.;
how much may his family spend daily, and  have  250 dollars
left at the end of the year ?                 Ans. $2.72,9,1 +
  Here you must subtract 250 dollars from the given sum; the
difference will  be what he spends in the support of his family.
The question is performed exactly, after the deduction is made,
as the preceding sum. 
SINGLE PROPORTION.
185
  16. If a gentleman spends 4 dols. 62% cts. every day, how
much will that amount to in a year ?         Ans. $l688.1?,J.
                d.  day.      ct.
                1 : 365 : : 4.62,5 : £ prop.
  17. A lady bought for the use of her family,  a piece of cloth,
containing 16yds. 3 qrs. 2 na., at 1.25 cents a  yard ; what was
the amount of the whole ?                    Ans. $21.09+
               yd. yd.  qr. na.   ct.
                1 : 16   3   2 := 1.25 : £ prop.
  Here the second term consists of several denominations:  it
must be reduced to the lowest one mentioned, which is nails.
The first  term  must also be  reduced to  nails.   When this is
done, then multiply the second and third  terms  together, and
divide  the product by the first term thus reduced; the quotient
will  be the fourth term of the proportion, or the answer to the
question.
  18.  If 48 men can perform a certain piece of work  in 24 days,
how many men can do the same in 192 days ?     Ans. 6 men.
     d.    d.    m.
     192 : 24 = : 48 : £ prop,  or, by  dividing by 24 the greatest
                        d.  d.   m.
common measure, we have 8 :1 :: 48 : £ prop.

  Here it  is evident from the nature of the question, that the
last term  of the proportion must be less than  the third ; there-
fore, the first term must be greater than the second ; for other-
wise, the  product  of the means could not equal that of the
extremes.
  19. A footman can perform a journey in 7 days, when the
days are 12 hours long;  how many days will he be, in  perform-
ing the same, when the days are 10 hours long ? Ans. 8f days.
                 h.  h.   d.
                10 : 12 : : 7  : i prop.
  Here the answer or fourth  term in the proportion must be
greater than the third  term ;  therefore,  the first term must be
less than the second.
  20. If 6 men  can perform  a piece of work in  12 days, how
many men can do it in 24 days ?                 Ans. 3 men.
                24 : 12 : : 6 : £ prop.
  21. A piece of silk containing 14 yds. 3 qrs., cost  $17.62,5 j
what is that a yard ?                         Ans.  $1.19,4,9.
               yd.   qr. yd.    D. ct. m.
                14  3 J Z ::  17.62,5 : £ prop. 
186
ARITHMETIC.
  Here the first term  must be reduced to quarters, and the
second term must be brought to the same denomination; after
which, multiply the means, and divide by the first term, for the
answer, or fourth term of the proportion.

  22. The quarter bill of a young lady at school is $18.62$;
what will her bill be for 2 quarters and a half, at the same rate?
                                            Ans. $4(i.56J.
                qr.  qr.   D. ct. m.
                1 :2.5 :: 18.62,5 ■ \ prop.

  Here the half quarter, I make a decimal, and annex it to the
whole  number.  The same I do with the % cent, making it 5
mills.   The answer must be greater than the third term ; there-
fore, the first term must be less than the second.

  23.  If 1 cwt. of sugar costs #12.50 cents, what will 6 cwt. 2
qr. 14 lb. come to, at the same rate ?         Ans. $82.81,2,5.
               cwt. cwt. qr.  lb.     ct.
                1:6  2   14 : : 12.50 : J prop.

   Here the second term consists of several denominations.   It
must be reduced to the lowest denomination mentioned.  The
first term  must be reduced to the same denomination.  After
which, multiply the means, and divide the product by the first
term ; the quotient will be the answer, or fourth term in the pro-

P°Observe always that the first and second terms must be of the
 same denomination; for,  as often as either term is taken, so
 often must the other be taken also;  otherwise, the proportion,
 and consequently the equality, of the terms will be destroyed.

   24   A farmer of Virginia sold 22 hhds. of tobacco, weighing
 nett, in the whole, 156 cwt. 8 qrs. 16 lb. at f 5.68 cents a cwt ;
 what is the amount of the whole ?             Ans. $891.15+
              cwt.  cwt.  qr.   lb.
               1 : 156  3  16 : = 5.68 : \ prop.
    25  A flour merchant sold 1574 bbls. of flour, at #5.12$ cents
 a barrel:  what was the amount ?              Ans. #8066.75.
    26.  A  gentleman bought a piece of ground, and agreed to
 eive #26.37* cts. an acre, and being surveyed, it was  found to
 contain 5 acfes, 2 roods, and 32 per.; what was the cost: of_the
 whole?                                  Ans. $150.33,7,5.
            acre. acre. r.  per.     ct.
               1:5   2  32  : : 26.37,5  : \ prop. 
SINGLE PROPORTION.
187
   27.  A farmer made from an orchard of apples, 146 barrels of
cider, which he afterwards sold at #3.12^ cents a barrel;  what
was the amount of the whole ?                 Ans. $456.25.
   28.  A farmer raised from a field 1267 bushels of corn, which
he sold at A7% cents a bushel; what did the whole amount to ?
                                           Ans. $601.82,5.
   29.  A merchant  failing, was  able to pay his creditors only
62% cents in a dollar ;  how much will a person receive, whose
claim is 746.25 cents ?                    Ans. $466.40,6,25.
                 ct.    ct.    dol. cts.
                 100 : 62,5 : : 746.25 : £ prop.
   30.  A person purchased a piece  of  cloth for #54.75 cents,
valuing it at #2.l6;§ cents a yard; required the number of yards ?
                                   Ans. 25 yds. l¥f ff,. qr.
               D. ct. m.   D. ct.   yd.
                2.16,7,5 : 54,75  : : 1 : £ prop.

   Here you  must reduce the second term to tenths  of a  mill,
that the first and second term may be of the same denomination.
Then proceed as the rule directs.

   31.  A farmer rents a  plantation, containing 316 acres, for
1.03 cts. an acre; what  is the rent of the whole ? Ans. $325.48.
   32.  A merchant sold 2 pieces  of  muslin, containing 74 yds.
2  qrs.  1 na. at ll£ cents a yard ; what was the amount of the
whole ?                                    Ans. $8.57,4+
             yd. yds.  qr. na.    cts.
              1-74   2  1 : : 11.5 = | prop.
   33.  A gentleman pays 3.50 cents a week for board; what
will 47^ weeks boarding cost him ?             Ans. #166.25.
   34.  If a  pound of tea cost 93f cents, what will a  box  contain-
ing 14£ lbs. amount to  ?                  Ans. $13.59,3,75.
                lb.  lb.      ct.
                 1 : 14.5 : : 93.7,5 : £ prop.
   35.  If 100 dollars in  12 months, gain 5 dollars;  what sum
will produce it in 8 months ?                     Ans. $150.
                  m. m.    D.
                  8 :  12 : : 100 : £ prop.

   It is plain here,  that the fourth term  of the proportion, or
answer to the question, must be greater than the  third term •
therefore, the first term must be less than the second.  In  this
case, and in all others of the same kind, the interest or produce of
the capital  has  no place in the  statement.  In the following 
188                   ARITHMETIC.
question, the sum at interest has no place in the statement.  So
in all other questions of the same kind.
  36. If 600 dollars, in 18 months, produce 54 dollars interest,
what will it produce in 10 months ?           Ans. 30 dollars.
                   m.   im.    D.
                   18 : 10 : :54: J prop.
  37. If a sum  of money at interest yields in 12 months 126
dollars, what will the amount of interest be in 17 months ?
                                            Ans. $178.50.
                m. m.   D.
                12:17 •■■ 126: J prop.
  38. A gentleman lent his friend 1000 dollars for 10 months;
how long must his friend in turn lend him 600 dollars to repay
the favour ?                             Ans. 16.660 months.
  39. A lady has an income of #965.75 cents, and is desiious of
saving at the end of the year, the sum of $342.87,5 ; how much
may she  spend weekly in her family ?      Ans. $11.97,8,3 +
                $965.75  = whole income.
                  342.87,5  ' sum saved.
                $622.87,5 ' sum to be spent in 52 weeks.
  40. If a staff 4 feet long casts a shadow 3 feet upon the level
ground;  and the shadow of a tree, standing perpendicularly to
the ground, was  measured, and found to be 154 feet; what was
the height of the tree ?                       Ans. 205 J feet.
                   ft.   ft.   ft.
                   3 :  154 ■■■ A-% prop.
   41.  If 4  cords of wood sell for #21.50 cents, what will 10$
cords come to at the same rate ?              Ans. $.36.43,75.
   32.  A gentleman purchased his  fuel for the winter for 72
dols. 25  cts.; how  many cords of wood were there, reckoning it
at #4.87£ cents  a cord ?     Ans. 14.825+cords, or 14j cords.
   43.  The stage-fare being 5 cents a mile, how far can a per-
son travel for #22.15 cents ?                  Ans. 443 miles.
   44.  A lady purchased a set of silver, weighing in the whole
5 lb. 6 oz. 5 dwt. at 1.50 cents an ounce ; what was the cost of
the whole ?                                   Ans. $99.37,5.
              oz. lb.  oz. dwt.  cts.
              1:5  6   5 :: 1.50 : J  prop.
   45.  The  floor of a  parlour contains 32  square yards, how
 much carpet # of a yard wide must be purchased to cover it ?
                                          Ans. 42.6^ yards.
                    yd. yd.  yds.
                   .75 '• 1 ■■•32 : J prop. 
SINGLE PROPORTION.
189
  It is manifest,  that it will require more yards of carpet than
there are  in the floor, because it is less than a yard in width ;
therefore, the fourth term, or answer  must be greater than the
third term; the  first term  will necessarily  be  less than the
second.  The £ I reduce to a decimal, which is  .75 and use it
in the operation.
  46. A lady,  intending to  make  a  bed quilt,  containing S%
square yards, desired herdaughter to inform her how much mus-
lin 1J yard wide would be required to line the same ; how many
did it take ?                        Ans. 6.8 yds. or 6|. yds.
                   yd.   yd.   yd.
                   1.25 : 1 : : 8.5 : £ prop.
  Here reduce the fractions to decimals,  and annex them to
their respective integers, and multiply as the rule directs.
  47. A miller purchased 576 bushels of wheat at $1.12% cents
a bushel;  what was the amount ?                Ans. $648.
  48. The rent  of a house is 526 dollars a year;  what will it
amount to in 4 years and a half?                Ans. $2367.
  49. The rent of a certain house is 647 dollars a year ;  what
will  it be a quarter ?                          Ans. $161.75.
  50. If 12 yards of muslin cost $2.25  cents; what is it a yard ?
                                        Ans. 18 cts. 7,5 m.
  51. If 10 persons in a family consume 4 bushels of corn in a
month ; how much will 25 persons use in the same time ?
                                          Ans. 10 bushels.
                  per. per.  bu.
                   10 : 25 :: 4 : J prop.
  52. The expense of a young  lady at a boarding school is
$275.50 a year; how much is it a week?    Ans. $5.29,8+.
  53. If 75 men consume 40 bushels of corn in 5 months; how
much will 216 men consume in the same time ?
                                      Ans. 114 |f. bushels.
                 771.  771.    bu.
                75 : 216 : : 40 : J prop.
  54. A certain  field will afford  pasture for 10 oxen, for 60
days ; how long will the same pasture suffice for 24 oxen ?
                                            Ans. 25 days.
                ox.  ox.   d.
                24 : 10 : : 60 : | prop.
  55. A pipe will drain off a cistern of water in 12 hours; how
many pipes of the same size will empty it in 30 minutes ?
                                           Ans. 24 pipes.
                   m.   h. pipe.
           R     30 : 12 :: 1 : J prop. 
190                   ARITHMETIC.

  56. The walls  of a  room measure 125 square yards; how
many yards of paper, § of a yard wide, will cover them ?
                                        Ans. 187-5+ yds.
                .666^ : 1  : : 125 : J prop.

  Here § of a yard,  reduced to a decimal, is .6660 of a yard,
which will be the first term of the proportion.

  57. A merchant purchased 476 barrels of flour, at  $4.87% cts.
a barrel; what did they amount to ?          Ans. $2320.50.
  58. If 4 thousand Spanish cigars cost $45-50 cents, what will
16 thousand amount to ?                        Ans. $182.
                4 : 16: : 45.50 : £ prop.
  59. A piece of work  can be done by 16 men in 12 days ; how
many men can perforin the same in 4 days ?     Ans. 48 men.
  60. If a pair of shoes cost  87% cents, what will 10 dozen
pair amount to?                                 Ans. $105.
  6l. A lady purchased a yard of silk for  $1.21 cents;  how
much money  must she have had to purchase 5% yards ?
                                            Ans. $6.65,5.
  62. A lady purchased 6% yards of merino stuff at $3.50 cents
a yard ; what did the whole amount to ?         Ans. $22.75.
  63. A lady purchased 6J  yards of silk, and for  the whole,
gave $10.25 ; what did it cost a yard ?           Ans. $1.64.
  64. A lady purchased 2£ yards of lace, at $12.37£ cts. a yard;
what did the  whole come to ?                 Ans. $27.84+.
  65.  A servant  purchased  24 lbs. of beef, at 5% cts. a pound ;
what did the  whole cost ?                       Ans. $1.32.
  66. A gentleman purchased a dinner set of silver, weighing in
the whole, 12 lbs. 6 oz. 10 dwt., at $1.50 cents an ounce ;  what
did the whole cost ?                          Ans. $225.75.
              oz.  lb.  oz. dwt.
               1  : 12   6   10 :: 1.50 : £ prop.
   67.  A  merchant bought a bale of cloth, containing 375 yd8,
 at $3.12£ a yard; what did the whole amount to ?
                                         Ans. $1171.87,5.
   68.  How long must a board be, that is 9 inches wide, to make
 12  square feet?                                Ans. 16 feet.
                in.  in.   ft.
                 9 : 12 : : 12 : J prop.
    69. The earth revolves on its axis in 24 hours ;  how far will
  an inhabitant on  the equator be carried by its motion in 1  hour?
                         Ans. 15 deg. or 1042£ statute miles. 
SINGLE PROPORTION.
191
   Every great circle of the earth contains 360 degrees, which,
 multiplied by 69%, will give the circumference of the equator:
 and 15 degrees of the equator equal 1042^ statute  miles, or
 900 geographic miles.
   70.  Two persons travel on  the same road, and in  the same
 direction.   One sets out 5 days before the other, and travels 20
 miles a day ; the other travels 25 miles a day ; how long before
 the latter will  overtake the former ?            Ans.  20  days.
   Here the latter gains 5  miles a  day ; and, as the former was
 100 miles a-head of him, it is  plain that in 20 days, he would
 overtake him.  The distance travelled will be 500 miles.
   71.  A gentleman, furnishing his  house, purchased 24 chairs,
 at $3.50 each ; what did they amount to ?          Ans. $84.
   72.  What will 5 dozen knives and forks amount to, at  $1.75
 a dozen?                                       Ans. $8.75.
   73.  If 1 bushel of apples costs 37% cents ; what will 25 barrels
 come to, each 2% bushels ?                 Ans. $23.43,7,5.
   74.  A widowed lady hired a house for 350 dollars a  year;
 but, by reason of sickness in her family, was enabled to pay no
 more  than  150 dollars, which her landlord accepted for  the
 whole  amount; what was that in the dollar? Ans. 42 cts. 8,57+
   70.  A mother allows her daughter, at a boarding  school, 3
 cents a day for spending money ; how much will that amount to
 in a year?                                   Ans. $10.95.
   76.  If you buy 3 oranges for 10^ cents ; how many can you
 buy for 55 cents ?                             Ans. 15.71 +
   77.  If a horse will trot in a gig 8 miles in  an hour; how far
 will he trot, at the same rate, in 3%  hours.       Ans.  28  miles.
   78. If 25 quills cost 12£ cents; what will 475 come  to,  at the
 same rate?                                  Ans.  $2,375.
   79.  If you buy 3 lb. of tea for, $4.16 cents what can you buy
 10£ for, at the same rate ?                     Ans. $14.56.
   80.  If you  pay $1.56 cents for 18 lb. of sugar;  how  much
 must you pay for 115 lb., at the same rate ?     Ans. $9.96,6$.
   81.  A merchant bought 5 pieces of muslin, each  containing
 26 yards, at 11 cents a yard; what  did they amount to ?
                                             Ans. $14.30.
   82.  If a gallon of wine cost $1.16%;  what will 28 gallons
 cost, at the same  value ?                        Ans. $32.62.
  83. What wiil 2  hhds. of wine come to, at 1 doll. 87* cents a
&[lon?    .                                 Ans. $236.25.
  84.  1 he interest of 100 dollars for a year, is 6 dollars;  what
 will it be for 3£ years ?                        Ans.  $19.50. 
192                   ARITHMETIC.
  85. If the interest of a certain sum for 1 year,  is $ 16.50;
what will it be for 4f years ?                  Ans. $78.37J.
  86. The income of a young lady is $765.50 cenls a year ;  her
expenses are $5.25 a week ; what sum does she save at the end
of the year?                                 Ans. #492.50.
  87. If 1 pair of gloves cost 87% cents ; what will 15 pair come
to, at the same rate ?                          Ans. #13.124.
  88. If a person earns $10.75 cents a month ; how much will he
earn in 6% months?                          Ans. >'(>!.).37£.
  89. If you purchase 55 yards of cloth, and pay at the rate of
$1.75 cents a yard; what will the whole come to ? Ans. #96.25.
  90. A farmer  raised  544 bushels of wheat, which he  sold at
$1.12 J cents a bushel; 674 bushels of corn, which he sold at 45
cents a bushel; and 7 fat oxen, at $25.50 each ; what was  the
profit of his farm ?                           Ans. S 1093.80.
  91. If 1 lb. of raisins costs 20 cents ; what will 3 boxes, each
weighing 24 lb.,  come to ?                     Ans. ,>' 14.40.
  92. If 1 ream  of paper costs $2.40 cents; what will 54 reams
come to?                                    Ans. $129.60.
  93. What weight, at the end of  a steelyard 39  inches  from
the  center of motion, will equipoise 208 lbs., suspended $ of an
inch from the center of motion, at the other end ?     Ans.  4 lb.
                39 : .75 • : 208  = \ prop.

  94. A garrison  of 800 men have provisions for 8 months;
how many must depart, that the provisions may last the remain-
der for 10 months ?                          Ans. 160  men.
                  m.  to.  men.
                 10:8:: 800 : £ prop.
  The  provisions will  be found sufficient for 640 men ; conse-
quently 160 must depart.

  95. How long must a  board  be, that is 9 inches wide, to
make 16 square  feet ?                       Ans. 21 ft. 4 in.
  96. The orbit of the earth is  about 596,900,000  miles, in
circumference; how many miles  does the earth move in 1 day,
to perform the whole in 365 days, and 6 hours  ? Ans. 1634223.
  97. Sound moves at the rate  of 1142 feet in a second of time;
how far distant was a cannon fired, whose flash was observed
56 seconds before the report was heard ?
                          Ans. 63,952 ft. or 12.112+ miles.
  98. How far distant  was a flash  of lightning, which  was ob-
served 71  seconds before the thunder was heard ?
                        Ans. 81,082 feet, or 15.356+ miles. 
SINGLE PROPORTION.
193
  99. If 160 pounds be placed on the end of a lever 10 feet long,
what weight will it raise on the other  end, supposing the prop,
or fulcrum, to be placed 1 foot from the weight? Ans. 1440 lbs.
               _____  ft. ft.    lbs.
               10—1=9 : 1 :: 160:J prop.
  In questions of this kind, the proportion will be, as the length
of the lever from the fulcrum, to the weight applied,  is to its
length from the fulcrum to the weight to be raised, so the weight
applied  is to the weight to be raised.
   100.  Venus performs her revolution around the sun in 224 of
our days ; how many degrees will she  pass through in 75 days ?
                                Ans. 120 deg. 32 m. 8 sec.
                224 : 75 : .-  360 :  £ prop.
   Every circle, whether great or small, is supposed to be divid-
ed into 360 equal parts, called degrees ;  and each of these de-
grees to be divided into  60 equal parts, called  minutes,  and
these again into 60 equal  parts, called seconds.

   Questions.   1. When are four numbers  said  to be propor-
tional r—2. Will you explain this by an example ?—3.  How
do you find the  ratio which one number has to another?—4.
Will you  explain this by an example  ?—5.  How will you find
the ratio which 12 has to  8 ?—Which  5 has to 15 ?—Which 16
has to 4?—Which 12 has to 8 ?
   6. When are four  numbers said to be in  direct proportion ?
7. If four numbers are proportional directly, will they be pro-
portional in any other way ?—8. Will you take the  proportion
4:8 ■•:  3 : 6, and show in what  other  way, the numbers will be
proportional ?—9.  How can you prove that they are  propor-
tional ?—10.  In  all cases, will  the product of the means  be
equal to that of the extremes?—11. What other property have
proportional numbers?—12.  Is this  the case, when  there are
only three numbers in the  proportion, as well as, when there are
four?—13. Will you show this by an  example?
  14. Is this properly of proportional numbers, the principle
on which the rule of three, is founded?—15. Why is it called
the rule  of three?—h  it  correctly  so  called?—What  then
should  it be called ?—16. What is the fourth term in  the pro-
portion ?—17. How do you proceed to find  it?—18.  On what
principle is this process founded ?
  19. What do you observe of the three terms, that are  given?
—20. Which term in the  statement of the question, must be of
the same kind with the answer?—21.  The two terms,that  are
       R2                                     ' 
194                   ARITHMETIC
alike, must stand in the first and second places in the proportion;
but how do you know the one to take theirs* place ?—22. Will
you explain this part, by an example ?—23.  Having stated your
question, if the first  and second terms are of different denomi-
nations, what do you do with them r—24. If the third term con-
sists of different denominations, how do you proceed ?—25. Of
what denomination, and  kind, will the answer or quotient be ?
—26. How do you find it ?
  27. If any term has the fraction %, £, f, %, &c, added to it,
what would you do with it ?—20. If the first and second terms
are divisible  by a common measure, may the operation be
shortened by dividing  them by it, and using their quotients ?—
29. How do you express  proportional  numbers?—30. Will
you show, by an example ?
           COMPOUND PROPORTION.
  This is commonly called the Double Rule of Three. There
are always  five terms given to find a sixth, which is the answer
to the question, and may be the fourth, fifth, or sixth term, in the
proportion.  The three first terms are a supposition, the other
two are a demand.
  In stating the question :  put the leading  term or the one, on
which the proposition depends, in the first  place;  the term im-
plying length of  time, distance of place, profit, or increase, in
the second  place; and the remaining term of the supposition, in
the third  place.
  Then place the terms of the demand under their correspond-
ent terms, in the supposition.  The answer will be of the same
kind with the term in the supposition, under which the blank is.
  If the  blank is  under the third term :  multiply the first and
second terms for a divisor, and  the other three for a dividend;
the quotient will be the answer.   But, if the blank is under the
first or second term": multiply the first, second, and sixth terms
for a dividend, and the other two for a divisor; the quotient will
be the answer.
                       EXAMPLES.
  1. If 6 men eat 10 pounds of bread in 8 days; how much
bread will 12 men eat in 24 days ?                Ans. 60 lb.
         to.    d.    lb.
          6 ..   8 .. 10,  terms of the supposition.
         12 ..  24        terms of the demand. 
COMPOUND PROPORTION.
195
  Here the blank is under the third term.   The leading term
in the question is men; therefore, 6 must be  put in the first
place ;  length of time must be in the second place; therefore 8
is the second term; and 10 must  consequently, be in the third
place.  The product of the  first and second terras will  be  the
divisor; and the product of the remaining terms will be the di-
vidend  ; the quotient  will be the  answer, and the sixth term in
the proportion.
   2. If 6 men  eat 10 lbs. of bread in  8 days;  in how many
days will 12 men eat 30 lbs. ?                  Ans. 12 days,
          m.   d.    lb.
           6 .. 8  ..  10, terms of the supposition.
          12         30, terms of the demand.
   Here the blank is under the second term of the supposition.
The product, therefore, of the third and fourth  terms will be the
divisor, and the product of the other three will  be the dividend;
the quotient will be the answer, and the fifth terra in the propor-
tion.
   3. If 7 men  can reap 84 acres of wheat in 12 days ; how
many men can reap 100 acres in 5 days ?        Ans. 20 men.
          to.    d.     a.
           7 . . 12 ..  84,  the terms of the supposition.
                  5  . . 100, the terms of the demand.
   Here 7 men  is  the foundation of the supposition;  it must
 therefore stand in the first place; length of time is the second
 term;  therefore, 12  days must take the second place; and 84
 acres must  be in the third place.   The blank  is under  the first
 term, because the answer is men.   The product of the third and
 fifth terms, that is 5 X84 will be the divisor;  and the product of
 the remaining terms will be the dividend ;  the quotient will  be
 the answer, and the fourth term in the proportion.
   4. If 12 oxen eat 20 acres of grass in 16 days;  how many
 acres will serve 24 oxen for 48 days ?               Ans. 120.
          ox.   d.    a.
          12 .. 16  .. 20,  terms of the supposition.
          24 .. 48        terms of the demand.
   The blank is under the third term ;  therefore the question is
 solved as the first example.
   5. If 8 men  in 4 days receive for their  wages 7 dollars, 68
 cts.; how much should 48 men receive for 16 days' labour ?
                                         Ans. $184.52 cts.
           to.     d.     cts.
           8 ..   4  .. 7-68, terms of the supposition.
          48 .. 16  . .      terms of the demand. 
196                   ARITHMETIC.
  6. If 500 dollars in 6 months, produce 12 dollars, 50 cts. in-
terest; how much will 600 dollars produce in 18 months?
                                              Ans. $15.
        dol.     to.     cts.
        500 . .  6 .. 12.50, terms of the sup.
        600 .. 18          terms of the demand.
  7- If 112 acres of meadow be mowed over by 16 men, in 7
days; how many acres can 24 men mow over,  in 19 days ?
                                          Ans. 456 acres.
       to.     d.     a.
       16  , .  7 •« 112, the terms of the sup.
       24  . . 19 • •      terms of the demand.
  8. If 180 dollars in 9  months gain 8 dollars, 10 cents inter-
est ; how much will 100 dollars gain in 12  months ?  Ans. $6.
      dol.      to.    cts.
      180  ..  9 . . 810, terms of the sup.
      100  .. 12         terms of the demand.
  9. If 4 men earn 3 dollars,20 cents, in 3 days; how many
will earn 12 dollars, 60 cents, in  16 days ?        Ans. 3 men.
  10.  If a  man travels 240 miles in 12 days, when the days are
12 hours long; how many days will he be in  travelling 720 miles,
when the days are 16 hours long ?             Ans. 27 days.
              d.    h.    to.
              12 .. 12 .. 240, terms of the sup.
                   16 .. 720, terms of the demand.
  11.  If 6  men can build a wall 20 feet long, 6 feet high, and -J
feet thick, in 16 days ; how long will it require 24 men, to build
one 200 feet long, 8 feet high, and 6 feet thick  ?  Ans. 80 men.
       to.    d.      ft.
       6 .. 16 ..  20x6x4, the terms of the sup.
      24         200x8x6, the terms of the demand.
  Here the blank falls under the second term.  Solved as the
second example.   The continued multiplication  of the length,
height  and  thickness of each wall,  will give the cubic feet in it,
or its contents.
  12.  If 8  cwt. of  iron  can be carried 128 miles for $12.80.;
what will be the expense of carrying 4 cwt. 32 miles ?
                                              Ans. $1.60.
         cwt.   TO.      D.
          8 .. 128 ..  12.80, terms of the sup.
         4 ..  32          terms of the demand.
  13.  If 5 men make 150 pair of shoes in 20 days ; how many
men can make 1350 pair in 60 days ?           Ans. 15 men. 
COMPOUND PROPORTION.
197
  14. If baggage weighing 200 pounds may be carried 40 miles
for 40 cents; how far may 2020 pounds be carried for $6.06
cts., at the same rate ?                       Ans. 60 miles.
             lb.      to.     ct.
            200 ..  40  ..   .40, terms of the sup.
           2020         6.06,  terms of the demand.
  15. A certain edifice was erected by 20 men in 12 months;
but, being demolished, it is required to build a similar one in 5
months; what number of men must be employed in the work ?
                                            Ans. 48 men.
            to.    mo.
            20 ..  12 .. 1  the terms of the sup.
                   5 . . 1  the terms of the demand.
  Here the blank is under the first term. The question is solved,
as the third example.
  The effect produced must be the  third term ; therefore, that
must be represented by a unit, or 1.

   16. If the freight of  a quantity  of goods, weighing in  the
whole 12 cwt.,  is $38.25.  for 60  miles; what will   be  the
freight of 42 cwt. for 100 miles ?           Ans. $176.45,8+.
            cwt.      m.      D.
            12  ..    60  .. 30.25, terms of the sup.
            42 ..  100          terms of the demand.
   17. If  the freight of 22  cwt. cost $25.50, for 116 miles';
 how far may 65 cwt. be carried for 74 dollars ?
                                        Ans. 113.9+miles.
           cwt.    TO.      D.
           22 ..  116 .. 25.50, terms of the sup.
           65              74, terms of the demand.
   18. If  240 dollars will pay the expenses of 6 scholars  at a
 boarding school for  6 months; how long will 500 dollars pay
 the expenses of 10 scholars, at the same school ?
                                          Ans. 7% months.
   19. If 100 dollars in 12 months gain 5 dollars interest; how
 much will gain 25 dollars in 20 months ?      Ans. 250 dolls.
   20. A  wall to be built 27 feet high has been raised 9  feet
 by 12 workmen  in 6 days ; how many men must be employed
 to finish the remaining 18 feet in 4 days ?        Ans.  36 men.
              to.     d.    ft.
              12  .. 6  ..   9, terms of the sup.
                     4  .. 18, terms of the demand.
   21. If  15 men eat 36 cents worth of bread in 6 days, when 
198                   ARITHMETIC.
wheat is worth 72 cents a bushel; how many days will 30 men
require to eat 1.60 cents worth, when wheat is 1.08 cts. a bushel r
                                            Ans. 20 days.
         to.    d.        cts.
         15 .. 6 ..  .36x .72, terms of the sup.
         30          1.60x1.08, terms of the demand.
  22. A cellar 22.5  feet long, 17-3 feet wide, and 10.25 feet
deep, may be dug by 6 men  in 2.5 days of  12.3  hours long;
in how many days of 8.2 hours long, can 9 men dig one, which
measures 45 feet long, 34.6 feet wide, and 12.3 feet deep ?
                                            Ans.  12 days.
  to.       d._______ft_______
  6 ..  2.5X12.3  ..  22.5 X17-3X10.25, terms of the sup.
  9         8.2  ..   45X34.6X  12.3, terms of the demand.
  Here the blank falls under the second term.  This  question
is solved,as the second example; but observe that the 8.2, stand-
ing under the second term, is to be multiplied into the product of
the third and fourth terms, for a divisor.   See question 11.
  Questions.   1.  What is compound proportion  commonly
called?—2. How  many  terms are  always given?—3. Is the
answer to the question always the sixth term, or the last, in the
proportion ?—4. What are the first three terms In  the propor-
tion, called ?—5. What  are the  remaining terms, called ?—
6. In  stating the question,  how do you proceed?—7-  What
term must stand in the first place ?—What in the second placer
8.  How do you place the terms of the demand ?
  9. When your question is stated, which terms do you multi-
ply for a divisor ?—Which for a dividend ?—10. Do you always
multiply the same terms for a dividend, as you do in single pro-
portion ?—11.  What rule then do you observe ?—12.  Will you
explain this matter by an example ?
PRACTICE.
  Many questions in single proportion may be solved in a short
way or method, which has been denominated Practice.  But
this is properly no rule in arithmetic.                    #
  When the  price of one article or thing is given, by making
use of the  parts of the integers, as given in the table below, the
value of any number of articles of the same  kind, and at the
same rate, may be found. 
PRACTICE.
199
TABLE OF PARTS OF THE INTEGER.
Pts. Cts.

 1=75
 f=62£


tV= 6i I  »
 4=33&
 §=66§J
Cts.    Pts.
50 = £-)
334= %
25 = |
20 = t v.
124= * f &
10 =TV

  5 — ?o J
Qrs.   lbs.   pts.
2, or 56= % )
1, or 28= |
     14= I
     16= |
      7=TV
P3
n






K3
  2
yard ?
       4 = 2V
       2=tVJ

                      examples.
    What will 326 oranges come to, at 61 cents ? Ans. $20.37*.
         6J=TV)326
                 20T6^=20|=20.37^.
     What will 126 yards of muslin come to, at 124 cents  a
                                          Ans. $15.75.
         12£=4)126
                 15f=15f=15.75.
  3. What will 75 bushels of apples come to,  at 25 cents  a
bushel ?                                    Ans. $18.75.
        25=^)75
               18f=18.75.
  4. What will 174 gallons of molasses come  to, at  334 cents
a gallon ?                                     Ans.
         334=4)174
                 58
  5. What will 56  yards of muslin  come to, at 50 cents  a
yard ?                                        Ans. $28.
        50=4)56
              28 
200
ARITHMETIC.
  6. What will 345 bushels of wheat amount to, at 75 cents a
bushel ?                                   Ans. $258.75.
         50=4)345
         25=i)l724
         75     86^
                258|=258.75.

  7. What will 648 quills come to, at 5 mills, or 4 a cent each ?
                                             Ans. $3.24.
  8. What will 58 yards of tape come to, at J of a cent, or 2.5
mills a yard ?                              Ans. 144 cents<
  9. What will 29 peaches come to, at  5  mills, or % a  cent
each?                                    Ans. 144 cents.
  10. What will 174 apples come to, at 2.5 mills, or £  of a
cent each ?                                Ans. 43| cents.
  11. What will 17  lbs.  of beef amount to, at  104 cents a
pound?                                     Ans. $1,784.
         4)17
            104
           170
              84
          $1,784

  12. What will 146 yards of muslin come to, at 14j cents a
Vard?                                     Ans. $20.80*.

          _14j
           584
          146
          2044
            364
        $20,804

  13  What will 12 cords of wood come to, at 4.75  cents a
cord?                                         Ans. $57.
          50=4)12
          25  i)_4|
         75     48=at 4   dollars.
                 6=at  50 cents.
                 3=at  25 do.
                57=at4T5. 
PRACTICE.
201
  14. What will 14 lbs. of tea come to, at $2,334 cents ?
                                            Ans. #32.66§.
         334=4)14
                _2.334
                 28

                 32§=32.66§
  15. What will 5 cwt. 2 qrs. of sugar, come to, at  12 dollars,
50 cents a cwt. ?                               Ans. #68.75.
         2 qrs.=4)12.50
                   ___5
                   62.50=amount for 5 cwt.
                    6.25=  do.   for 2 qrs.
                   68.75=  do.   for 5 cwt- 2 qrs.
  In this, and in the  following examples, multiply the price by
the highest denomination  in the given quantity, and take the
parts for  the other denominations; the sum will be the amount
required.  For  the parts, see the preceding table.
  16. What will 12 cwt. 2 qrs. 14 lb. of tobacco, come to, at 4
dollars, 75 cents, a cwt. ?                      Ans. $60.56£.
         2 qrs.=4)  4.75
        14  lb.=l)___12
                  57.00  =the amount of 12 cwt.
                    2.374=    do.       2 qrs.
                    1.18f=    do.      14 lbs.
                  $60.56i
   Here I multiply $4.75  by 12, and take the parts for the 2
quarters, the half of a cwt. and  place it under the product.  I
then take the parts for 14 lbs. which are a  fourth  part oftwo
quarters, or 56  lbs. and place the result under the product also;
the sum of the whole is the amount  required.
   17. A farmer bought a piece of land,  containing  12 acres, 1
rood, 20  perches, at the rate of  16  dollars, 50 cents, an acre;
what was the amount of the whole ?           Ans. $204.18s
  18. What is  the amount of 46 gals. 2 qts. of wine, at 1  doh
40 cents a gallon ?                             Ans. #65.10.
  19. What is the value of a tract of land, containing 25 acres,
2 roods, 20 perches, at the rate of 14 dols. 45 cents an acre?
                                          Ans. $370.28 +
  20. A  lady purchased for the use of her  family, 42 yds. 2
qrs. 2 nails, of linen, at the rate of 44 cents, 4 mills a yard; what
did the whole come to ?                    Ans. $18.89,7+ 
                   V
 202                  ARITHMETIC

    21.  A barrel of rice, weighing 2 cwt. 1 qr. 16 lbs. was sold
 at 4 dols. 16 cents a cwt.; what did it come to ?   Ans. $9.95i'
          1  qr.=4)4.l6                                 T
         l61b.= i)___2
                   8.32=the price  of 2 cwt.
                   1.04=   do.    of 1 qr.
                    .59^=  do.    of 16 lb.
                  $9.95^
    22. A piece of broad cloth, containing 2§ yards, cost 5 dols.
 25 cents a yard ; what did the whole amount to ? Ans. xi-j. l;>tf.
          2 qrs.=4) 5.25
           1 do.=|)____2
                    10,50=price of 2 yds.
                     2.62j=do. of 2 qrs.
                     1.3l|=do. of 1 qr.
                  $14.43^
    23. A farmer sold a load of hay, weighing 16 cwt. 2 qrs. at 75
 cents a cwt.; what did it come to ?             Ans. > 12.37%.
    24. A gentleman bought a set of silver spoons, weighing 2 lb.
 8 oz. 10 dwt. at 1 dol. 75 cents an ounce; what did the whole
 amount to?                                        $56^7 h.
    Here you reduce the pounds to ounces, to which add the 8 oz.
 and multiply the  sum by the  price;  and take parts for the 10
• dwt. which make % an ounce.
    25. A merchant bought  several casks of wine, containing in
 the whole, 147 ga's- 2 qts. 1 pt. at 1 dol. 17 cents a gallon ;
 what was the amount?                       Ans. >172.72.
    26. What will 416 yards of calico amount to, at 25A cents a
 yard?                                       Ans.M06.()8.
   27. What will 454 barrels  of flour come to, at 6 dols. 25 cts.
 a barrel?                                   Ans. $23l.:i7£.
   28. What will I84 cords of wood cost, at 4 dols. 7-> cents a
 cord?                                       Ans. $*7-87j.
    29. What will 2% tons of coal cost, at 7 dols. 50 cts. a ten r
                                             Ans. $2Q(>2\.
    30. What will 35j lbs.  of beef cost, at 8 cents a pound r
                                               Ans. $2.84.
    31. A gentleman bought a barrel of sugar, weighing 2 cwt.
 1 qr. 16 lb.,  at 11 dols. 25  cts. a cwt.; what did the \\ hole come
 tof                                        Ans. ,s26.91+.
    32. The  expenses of a young lady at a boarding school, is
 250 dols. a year ; how much will it be for two years and a quar-
 ter p                                        Ans. $562.50. 
                  TARE AND TRETT.                 203

  33. At 1.25 cents a yard, what will 7244 yards of cloth come
t0 ?                                       Ans. $905,624.
  34. If a man can travel 35 miles in a day, how far can he
travel in 12J days ?                       Ans. 428f miles.
  35. At $3.50 cts. a week, how much -will 154 weeks' board-
ing amount to ?                               Ans. $54.25.
  36. What will 125 lbs. of flour come to, at 6| cts. a pound ?
                                           Ans. $7.81,25.
  37. What will 467 lbs. of pork come to, at 6J cts. a pound ?
                                         Ans. $29.18,7,5.
  38. What  will  216 lbs. of bacon  come to,  at 64 cents a
a pound ?                                     Ans. $14.04.
  39. What will 256  lbs.  of cheese  come to,  at 124 cents a
pound ?                                        Ans. $32.
  40. What will 244 bushels of potatoes come to, at 24 cents a
bushel ?                                      Ans. $7.88.
  41. What will 59 lbs. of raw cotton coine to,  at 134 cents a
pound ?                                     Ans. $7-96,5.
  42. If a  gallon  of  vinegar cost 25 cents, what will a barrel
come to, containing 32£ gallons ?               Ans.  $8.12,5.
  43. What will 184 bushels of salt come to, at 27 cents a
bushel ?                                     Ans.  $4.99,5.
  44. If a acre of land is worth 19 dollars, what is the value of
3 acres, 2 roods, 20 per., at the same rate ?    Ans. $68.87,5.
                TARE AND TRETT.

  Tare and Tkett are allowances, made upon some articles of
merchandise by the seller.
  Tare is an allowance, made for the weight of the barrel, box,
bag, or whatever contains the commodity.
  Trett is an allowance for waste, dust, &c, on some light arti-
cles, and is 4 pounds in  every 104.  It is commonly allowed
with tare.
  Gross weight is the weight of the goods, together with that
which contains the articles.
  Nett is the weight  after all allowances are made.
  When tare only is allowed, subtract the tare from the gross •
the remainder is  the nett.                                  '
im 
204
ARITHMETIC.
                        EXAMPLES.
  What  is the nett weight of 10 barrels of sugar, weighing in
the whole, 24cwt. 2 qrs. 17lbs. gross; tare in the whole, 2 qrs
16 lbs. ?                            Ans. 24 cwt. 0 qrs. 1 lb.
                cwt. qr.  lb.
                24  2  17
                00  2  l6=tare.
                24  0    l=nett weight.
  2. What is the nett  weight of 12 hhds. of tobacco, weighing
8 cwt. 1  qr, 6 lbs. each, gross; tare in the whole, 14 cwt. 2 qrs.
14 lbs. ?                            Ans. 85 cwt. 0 qrs. 2 lbs.
                cwt. qr.  lb.
                 8  1    6
                ________12_
                99  2  l6=the gross.
                14  2  14=the tare.
                ~85  (5    2=the nett weight.
  3. A gentleman in Virginia sold 25 hhds. of tobacco, weigh-
ing in  the whole, 215 cwt. 3 qrs. gross; tare, 14 lbs. per cwt.;
what is the nett weight of the whole? Ans. 188 cwt. 3 qrs. 341b.
               cwt.  qr.  lb.
         14=4)215  3  00
                26  3  244=the tare.
               188  3    §4=the whole nett.
   Here 14 pounds being the 8th part of 112 lbs., you may divide
the gross by  8, and subtract the quotient from the gross ;  the
remainder will be  the nett.   In  like manner, proceed in any
other example, where the tare is an aliquot part of 112 lbs.

   4.  A merchant bought 16 barrels of rice; weight in the whole,
42  cwt. 3 qrs. 16 lbs. gross ;  tare per cwt., 8 pounds ; what is
the nett weight ?             Ans. 39 cwt. 3 qrs. 8 lbs. 11 oz.
           lb.    cwt.  qr.  lb.   oz.
           8=^)42   3   16    0
                   3  0    7    2=tare in the whole.
                 "39"  is    8  14~=nett weight.
   5. A gentleman  in Kentucky sold 50 hogsheads of tobacco,
 weighing in the whole, 711 cwt. 1 qr. 10 lbs. gross; tare allow-
 ed for the hhd., 150 lbs. each ; what is the  nett weight ?
                                  Ans. 644 cwt. 1 qr. 14 lbs.
    Here  multiply the tare by the  number of hogsheads, the pro-
 duct will be the whole tare in pounds; which, bring to hun- 
TARE AND TRETT.                  205
 dred weight, quarters, and pounds, and subtract it from the
 gross; the remainder will benhe nett.
   5. A planter of Georgia sold 100 bales of cotton, weighing in
 the  whole, gross 510 cwt. 2 qrs.; tare per bale, 7 lbs.; what is
 the nett weight ?                          Ans. 504 cwt. 1 qr.
   6. A  merchant bought 4 hhds. of sugar,  weighing in  the
 whole, 56 cwt.  3 qrs. 21 lbs. gross; tare  allowed, 16 lbs. per
 cwt.;  what is the nett  weight?      Ans. 48 cwt. 3 qrs. 6 lbs.
   7. A planter of Maryland sold 21 hhds. of tobacco, weighing
 in the  whole, 160 cwt.  2  qrs. gross, but being a little damaged,
 he consented to allow 28 lbs. tare  per cwt.; what was the nett
 weight of the whole ?             Ans. 120 cwt. 1 qr.  14 lbs.
   8. A  merchant bought  120 boxes of tea,  weighing in  the
 whole 5457 lbs.; tare allowed per box, 6 lbs.;  what is the nett
 weight ?                                      Ans. 4737 lbs.
   9. A merchant bought 16 boxes of raisins, each weighing 1
 cwt. 1 qr. 17 lbs. gross ; tare allowed per box,  21 lbs.; what
 was the nett weight ?               Ans. 19 cwt. 1 qr. 20 lbs.
   10.  A  merchant sold 45 boxes of figs, weighing in the whole,
 84 cwt. 2  qrs. gross ; tare allowed per box, 22 lbs.; what is the
 nett weight ?                       Ans. 75 cwt. 2 qrs. 18 lbs.
   11.  A  planter in  Louisiana, sold 16 hhds. of sugar, weighing
 on an  average,  12  cwt.  2  qrs. 12 lbs.  gross;  tare allowed per
 cwt., 14 pounds ; what is the nett weight of the whole ?
                                       Ans. 176 cwt. 2 qrs.
   12.  A  planter in Alabama sold 74 bales of cotton, weighing
 on an average 746 pounds gross; tare allowed per bale,  15 lbs.;
 what is the nett weight  ?                      Ans. 54094 lbs.
   13.  A merchant bought at a public sale, 15 boxes of dried
 currants,  weighing in the whole, 17 cwt. 3 qrs.  11 lbs. gross;
 tare per cwt. 15 lbs.; what is the nett weight ?
                                         Ans, 1731.28 lbs.
   Here reduce the whole to pounds, which multiply by the tare
 per cwt.; divide  the product by  112,  the quotient will be the
 whole tare; which, subtract  from the gross :  the remainder will
 be the nett weight.
   When trett is allowed with tare,  find the tare as in the pre-
 ceding  examples, and subtract it from  the gross weight;  the
 remainder will be the  suttle pounds, which divide by 26, the
fourth part of 104 lbs., a cwt. suttle, the quotient will be the
irett; subtract this from  the suttle,  the  remainder will be the
nett pounds, as in the following examples.
            s 2 
206                    ARITHMETIC.
  14. How much is the nett weight of 48 cwt. 2 qrs.  12 lbs.
gross;  tare per cwt. 11 lbs., and trett 4 lbs. for every 101  lbs.?
                                           Ans. 4710.1 lbs.
cwt. qr. 48 2 5432 4 11 =	=tare	jercwt	suttle. 26)4898.5(188.4 tr. 26
194 112)59752(5334 28 560		=tare.	229 208
1552 375 388 336			218 208
5432 =the gross. 392 533.5=tare. 336			105 104
4898.5=suttle. 56			~~I
188.4=trett.			
   4710.1 =nett weight.

  In cases, where trett is allowed with tare, it is better to reduce
the gross  weight to  pounds; from  which,  subtract the tare.
Divide this remainder by 26 for the trett, which last, subtract
from the suttle,  and the remainder will be the nett weight in
pounds.  If the answer is required in hundred weight, quarters,
and pounds, divide the nett by 28 for quarters, and these again
by 4 for cwt., as in the following examples.

  15. What is the nett weight of 177 cwt. 0 qrs. 22 lbs.gross;
tare per cwt. 9 lbs., and trett 4 lbs. for every 104 pounds ?
                                Ans. 156 cwt. 2 qrs. 22 lbs,
   16. Required the nett weight of 17 boxes of raisins, weighing
in the whole, 26 cwt. 1 qr. 18 lbs. gross ; tare allowed, 10 lbs.
per cwt.; trett 4 lbs. for every 104 ?          Ans. 2590.3 lbs.
                SIMPLE  INTEREST.

  Simple Interest is a sura allowed for  the use of money,
goods, or property.
  The sum,  for which interest is allowed,  is called  the prin-
cipal.
  The rate of allowance for 100 dollars, for 1 year, is called the
rate per cent. 
SIMPLE INTEREST.                 207
   The amount is the principal and interest, added together.*
   Every question of simple interest, is properly an example of
 the Rule of Three.  Three terms are given to find a fourth pro-
 portion.
   General Rule.—If the time given is one year, the fourth
 proportion is the answer to the question.  But,
   If the time given is less or more  than one  year; First, find
 the interest of the given sum  for one year, by the rule of three,
 or otherwise : Then  say,  as 1 year  in months, weeks, or days,
 as the case may be, is to the given time in years, months, weeks,
 or days ; so will the interest of the  given sum for one year, be
 to the interest required.
   If the amount is sought, add the principal  and  interest
 together ; their sum will be the amount.

                        EXAMPLES.
   1. What is the interest of 125 dolls, for 1 year at 6 per cent.?
                                                 Ans. $7.50
                  D.    D.     D.
                 100 ■  125  ■■• 6 ■ I prop.
   Here 100  dollars must be the  first term;  the  given sum,
 which  is  125  dollars, takes  the second place;  because the
 fourth  proportion or answer must  be greater  than the  third
 term or 6 dollars, the allowance made  for the  use of 100 dollars
 for one year.
   2. What is the interest of 350 dolls, for 1 year, at 6 per cent.?
                                               Ans. 21 dolls.
                  D.    D.     D.
                 100 :  350  : :  6 : J prop.
   3. What is  the interest of 625 dollars for  1 year, at 54 per
 cent. ?                                        Ans. $34.37,5.
                D.     D.     Ct.                        *
                100  : 625 : :  5.5 : £ prop.
   4. What is the interest of 722 dollars for 1  year, at 5f per
 cent-?                                       Ans. $41.51,5.
                 D.    D.     Cts.                 *      '
                100  = 722 :: 5.75.:  |  prop.

  * The rate of interest established by law, is 6 per cent.; but in the
 state  of New York, the rate of interest is fixed at 7 per cent.   The
 interest exacted by the banks, is 6 per cent., allowing- only 360 days
 for the year. But money is  frequently obtained at 5 per cent., and
 even at a lower rate.  The rate of interest is generally agreed upon
 between the debtor and creditor; but, if nothing is said on that sub-
ject in the obligation, the rate is 6 per cent.,  except  in the state
 ot New York, where it is 7 per cent. 
208
ARITHMETIC.
    5. A lady has her fortune in money,  at  interest, at  5| per
  cent., amounting to 11642 dollars; what is her yearly income?
                                            Ans. $611.20.5.
                 D.     D.       ct.               *
               100 :  11642 : : 5.25 :  £ prop.
    6. A gentleman has in bank stock 7446 dollars; what will
  he receive annually at 54 per cent. ?           Ans. s■  100.53.
                D.     D.      Ct.
               100 :  7446  : :  5.5 : £  prop.
    7. A young lady has  her  fortune in money,  amounting to
  $12140.50 ; what is her  income, reckoning 6£ per cent. ?
                                         Ans. $758.78,1,25.
              D.       D.       Ct.
             100 :  12140.50 : : 6.25 : J prop.
    When the rate of interest has £, %, or £ more than the integer,
  the fraction  may be changed into a decimal, and added  to the
  whole number,  as in the above examples ;  or,  |, %, or £ of the
  principal may be taken, and added to the product of the  princi-
  pal and integer ; the sum  will be the whole interest.
    8. What is the amount of $254 for one year, at 6 per cent ?
                                             Ans. $269.24
    Here add the interest, found as above, to the  principal, the
  sum will be  the amount.
    9. What is the amount of 7420 dollars,  for  one  year, at 5jj
  per cent ?                                   Ans. $7846.65.
    10. What is the  amount of 275 dols.  50 cts. for one year, at
  5| per cent ?                           Ans, $289-96,3,75.
    11. What is the amount of 624 dols. at 54 per  cent., for one
  year?                                      Ans. $658.32.
    12. What will be the interest of 240 dollars for 3 years, at 6
  per cent ?                                     Ans. $43.20.
    Here find the  interest of the given sum for one  year, as in the
  above examples.  Then say, as 1 year is to the given lime, so
  will  the interest  thus found, be  to  a fourth proportion, which
  will  be  the  interest required.   In like manner, proceed in the
  following examples.
iX^- What will be the interest of 525 dols. for 21 months, at
  54 per cent. ?                             Ans. ^J5.53,l,25.
    Find the interest for one year as above directed ; then say :
      12 ■• 21 :: 28.87,5, the interest for 1 year : \ prop.
    14. A gentleman has 478 dols. 50 cts. at interest at 5\ per
  cent.; what will it be in 44years ?      Ans. $123.81,!,>> + . 
SIMPLE INTEREST.                 209
   Here again find the interest as above; then say :
     1 ■•  4.5 : = 27.51,3,75 the interest for 1 year :  £ prop.
   15.  What will be the interest of 256 dols.  for 18 months, at
5| per cent.?                                 Ans. $21.93.
               100  •• 256 •• =  5.75 : \ prop.
Then say : 12  -. 18  : •• 14.62  the interest for 1 year : £  prop.

   16.  What will be  the amount of a bond for 150 dols. 25 cts.
at 6 per cent., for 3f years ?                Ans. $183.80,6,2.
   Here  find  the interest  of the given sum for one year ;  then
find it for the given  time as above directed.   Add  the interest
thus found, to  the principal, the sum will be the  amourrt re-
quired.
   17.  A gentleman has a note of 520 dols., at 54 per cent., on
which no interest has been paid for 2\ years ; what will be the
amount ?                                   Ans.  $591.50.
   18.  What will be  the interest on a note of 1234 dols. 60 cts.
for 27 months, at 7 per cent. ?             Ans. $194.44,9,5.
   19.  A lady has an obligation for 1576 dols. at 54 per cent, j
what will be the interest of the  same  for 7 months ?
                                           Ans.  $50.56,33.
   20.  What will be the amount of a note for 1460 dols. for 10
months, at 6\  per cent.?                   Ans. $1536.04+.
   21.  What will be  the amount of 273 dols. 45  cts. at  5\ per
cent, for 17 months ?                      Ans. $295.72,4,7.
   25.  What will 97  dols. 50 cts.  amount to, at 64 per cent, for
33 months ?                                Ans. $114.92,8.
   When the time  is the  aliquot part of a year,  or years, the
operation may sometimes  be shortened,  by taking the parts as
shown in the examples under Practice, which see.
   23.  What is the interest of 7342 dols. for 164 months, at 6
percent.?                                 Ans. $605.71,5.
   Here,  having found the interest for one year, say :
   m.   to.
   12 ■ 16.5 : = so the interest  thus found will be to the answer.

   Here I change the  4  month to a decimal, and add it to the
integral part.   It then becomes 16.5 months.
   24.  What will be the interest of 160 dols. at 6 per cent., for
15 months ?                                   Ans. 12 dols.
   Having found the interest for one year, which is  9  dols. 60
cts., I take the fourth  part  of it, and add it to the interest of the
'mi 
210                   ARITHMETIC.

given sum for  one year.   The sum thus added is the interest
required; thus:
           m.  yr.
            3=J)9.60=interest for one year.
                 2.40=interest for three months.
               $12.00=interest for fifteen months.
  25. What is the interest of 174 dols. 64 cts. for 3 years and
7 months, at 6 per cent. ?                   Ans. $3~.54,7+.
  26. What is  the interest of 370 dols. for 194 months, at 6 per
cent. ?                                       Ans.  $36.07,5.
  27. What is the interest  of 576 dols. for 21 months, at 6 per
cent. ?                                        Ans. $61 20.
  28. What is the interest  of 174 dols. for 5| months, at 6 per
cent. ?                                      Ans. $5.00,2,5.
  Find the interest of the given sum for a year ; then say :
    12 :  5.75  : =  10.44 :  the  interest for 1  year : \ propor-
tion ; or, take parts for the £ of a year, as shown under practice.
  29. What is the interest  of 3472 dols. 56 cts. for 9 months,
at 6 per cent. ?                           Ans. $156.26,5 + .
  30. A  merchant in Philadelphia owes his  correspondent in
New York the following sums  upon note ;  to wit:
               120 dol. payable in 9  months.
               210 do.    do.   15   do.
               390 do.    do.   23   do.
               874 do.    do.   27  do.
  What will  be the  interest, reckoning  6 per cent., and the
principal added together on these  several obligations ?
                                            Ans. $1777-99.
  31. What will be the interest of  339 dols. 60 cts.  for lG
weeks,  at 54 per cent. ?                       Ans. #5.7'4,7.
  Having found the interest for one year ; namely, 18 dols. 67
cts. 3 mills, say
                52 = 16 =: 18.67,8 : £ prop.
  32. What is the interest of 650 dols. 27 cts. for 146 days, at
5|  percent.?                                   Ans. 514.95.
   Having found  the interest of the given sum for one year, as
above directed, say:
      365 ••  146  = = so is the interest for 1 year, to a J prop.; or
the answer.
    33. What is the interest of 2780 dols.  75  cts. for 20 weeks,
 at  7 per cent. ?                            Ans. #74.86,6+. 
SIMPLE INTEREST.                211
  Having found the interest for one year, say:
     w.   w.                                          .
     52 :  20 :: so is the interest for 1 year, to a £ proportion;
or the answer.
  34. What is the amount of 746 dols. for 245 days, at the rate
of 6 per cent. ?                             Ans. $776.04,4.
  35. What is the interest of 152 dols. for  5 months, 21  days,
(reckoning the month at 30 days) at 6 per cent. ? Ans. $4.33,6+
  Here reduce the time given to days, and work as directed in
the preceding examples.
  36. What will be the interest of a bond for 375 dols. for 2
years, 7 months, and 11 days, at 6 per cent. ? Ans. $58.81;2,5.
  Here reduce the given time to days,  and having found the
interest  of the given sum for a year, say ; as 365 days will be to
the given time  (in days),  so will  the interest found, be to the
interest required.
  37. What  will be the amount of a note for  875 dols. 49 cts.
for 5 years, 8 months, and 20 days, at 6 per cent. ?
                                      Ans. $1176.07,4,9+.
  Proceed as in the above example, to find the interest for the
given time; to which, add  the principal,  the  sum will be the
amount.
   38.  A  rich uncle left a legacy of 1560 dols., to be paid to his
niece on her arriving at the age of 18 years ; at the time of the
grant, she was 13 years, 3 months, and 12 days old; what will the
legacy amount to  for the time, at 6 per cent. ? Ans. $2001.48.
   39. A  father, at his decease left  three daughters : the eldest,
12 yrs. 5  mo. 10 days old—the second, 10 yrs. 1 mo.  5 days—
and the youngest, 7 yrs. 10 mo. 22 days old. To each  of them,
he left the sum of 1860 dols., with directions to his executors, to
put it at interest, at 6 per cent., and to pay the  amount of their
respective legacies into their hands, as they should arrive at the
age of 18  years : what did each one receive ?
                                 f Eldest received, $2480.
                            Ans. < The second, $2741.95.
                                 r; The youngest, $2987.78.

   Bank Interest.   It is a custom with  the banks, to  allow
360 days  to the year; which they divide into 6 equal parts of
60 days each.  A note is discounted, commonly for 60  days
only.  If it is not paid at the end of that time, it is usually ta-
ken up, and another given for the same length of time, and so on. 
212                   ARITHMETIC.

   The interest upon a note of 100 dollars, for 60 days, at 6 per
cent., is 1 dollar, or 100 cents;  that is, it is 1 cent upon a dol-
lar.   Therefore,
   The interest of any sum of money in dollars, for 60 days, at
6 per cent., is exactly the same number of cents, or 1  cent in a
dollar: for 90 days, it is 14 cents in 1 dollar ; for 30 days, it is
4 a cent in 1 dollar;  and so  for any greater or  less number of
days.
                        EXAMPLES.
   1. What is the interest of 1542 dollars, for 60 days, at 6 pei
per cent. ?                                     Ans. $15.42.
   2. What discount must be  allowed at the bank, on a note for
574 dollars, at 6 per cent., payable in  60 days, after date ?
                                                Ans.  $5.74.
   3. What discount must be allowed on a note, for 1746 dol-
lars, payable in 90 days after  date, at 6 per cent. ? Ans. $26.19-
  Interest for 60 days=17.46, the face of the note.
     Do.     30days=  8.73
                     $26.l9=interest for 90 days.
   4. What will be the  interest  allowed  at  the banks, for dis-
counting a note, for 750 dollars,  payable in 30 days, after date ?
                                                Ans. #3.75.
         30 days=£)750
                     3.75 the interest for 30 days.
   5. What interest will  be taken at  the bank of the United
States, on  a note for 1576 dollars, offered for  discount, for 60
days, at 6 per cent. ?                           Ans. $15.76.

   To find the Principal, when the amount, time, and rate per
cent., are given:
   Find the amount of 100 dollars, for the time given, at the rate
per cent.;  then, as this amount is  to 100 dollars, so  is  the
amount given, to the principal required.

                         examples.
   1. What principal in 9 years, will amount to 725 dollars, at 5
per cent. ?                                       Ans. $500.
           100 doll.
            45=interest of 100 dolls, for 9 years at 5 per cent.
           145=the amount  of  100 dollars.
                 dol.     dol.     dol.
                145  :  100  :: 725  :  \ prop. 
SIMPLE INTEREST.                 213
  2. What principal will amount to 965 dols. 754 cents, in 5 years,
at 6 per cent. ?                            Ans. $742.81,1,5+
          100
           30=interest for 5 years, at 6 per cent.
          130=the amount of 100 dollars.
               130 :  100: : 965.75,5: \ prop.
  3. What sum, at 54 per cent., will amount to 1565 dollars, in 44
years?                                      Ans. $1254.5 +
  4. What sum will amount to 571 dollars, 20 cents, in 4 years, at
5 per cent. ?                                     Ans. $476.
  5. What sum will amount to 695 dollars, in 10 years, at 5f per
cent?                                         Ans. $441.27.
  6. What sum will amount to 425 dollars, 50 cents, in  24 years,
at 6 per cent. ?                                   Ans. $370.
  To find the Rate per Cent, when the amount, time, and prin-
cipal, are given :
  Subtract the principal from the amount, and the remainder  will
be the interest for  the whole time : multiply the principal by the
number of years : then say; as this product is to the whole inte-
rest, so will 100 dollars, be to the rate per cent.

                        EXAMPLES.
  1. At what rate per cent., will 500 dollars amount to 725 dollars
in 9 years ?                                  Ans. 5 per cent.
                       725=amount.
                       500=principal.
                      225=the whole interest.
         500
           9=the time.
        4500  : 225  : :  100 : \ proportion.
  2. At what rate of interest, will 600 dollars amount to 856 dols.
50 cents, in 9 years and 6 months ?            Ans. 44 per cent.
         600            856.50=amount.
  timc=  9.5            600.00=principal.
        3000           256.50=whole interest.
      5400
      5700.0
           5700 :  256.50 ::  100 :  £ proportion.

  Here I reduce the 6 months to its equivalent decimal of a year
and add  it to the 9 years, making 9.5 years.                    ; 
214                     ARITHMETIC.
  3.  At what rate will 820 dols. amount to 1078 dols. SO cts. in 4$
years ?                                       A ns. 7 per cent".
            820                1078.30=amount.
            4.5=time.           82Q.OO=principal.
  4100                 258.30=whole interest.
3280
         3690.0       3690 : 258.30 : : 100 : J  prop.
  4. At what rate per cent, will 837 dols. amount to  1029 dols. 51
cts. in 4 years ?                              Ans. 5$ per cent.
  5. At what rate per cent, will 624 dols. amount to 658 dols. 32
cts. in one year ?                             Ans. 5£ per cent.
  6. At what rate per cent, will a bond for 150 dollars, 25 cents,
amount to $183.95,6,2 in 3% years ?     Ans. 6 per cent, nearly.

  To find the time, when the principal, amount, and rate per cent.
are given :
  Subtract the principal from the amount, which will give the in-
terest for the whole time.  Then find the interest of the principal
for one year.  Then  say :  as this interest is to the interest for the
whole  time, so is  1 year to the time required.

                         EXAMPLES.
   1. In what time will 500 dols. amount 725 dols. at 5 per cent. ?
                                                Ans. 9 years.
          500= principal.           725=amount.
            5=rate  per cent.        500
        2"5XX)=interest for 1 year.   225=whoIe interest.
                  25 :  225  : :  1  : J  prop.
   2. In what time will 820 dols. amount to 1078 dols. 30 cts.  at
7 per cent.?                                   Ans. 4.1  years.
          820=principal.        i078.30=amount.
            7=rate per cent.      820.00=principal.
        5~7\40=inter. for 1 year.   258.30=whole  interest.
                57-40 :  258.30  : =  1  : \  prop.
    3  In what time will 312 dols. amount to 377 dols. 52 cts. at G
 net cent?                                    Ans  3* years.
 F  4  In  what time 746 dols. amount to $776.04,4, at 6 per cent.r
                                              Ans. 245 days.
        746=principal.          776.04,4=amount.
            6=rate per cent.      746.00  = principal.
       4X76=int. for 1 year.      30.04,4=whole interest.

              44.76 : 30.04,4 •.■365-1 prop. 
                  COMPOUND INTEREST.                215

  5. In what time will a legacy of 1560 dols. amount to 2001 dols.
)4 cts. at 6 per cent. ?               Ans. 4 yrs. 8 mo. 19 days.
  6. In what time will a bond for 150 dollars, 25 cents, amount to
$183.95,6,2, at 6 per cent. ?   Ans. 3 yrs. 8  mo. 26 days, nearly.
  7. A father left his son, at his decease, the  sum of 1500 dollars,
to be put to interest at the rate of 6 per cent.  On his  arriving at
the age of 21 years, he received, principal and interest, 2332 dols.;
what was his age at the decease of his father?
                                   Ans. 11 yrs. 9 mo. 3 days.
              COMPOUND  INTEREST.

  Compouno Interest is that, which arises from the principal and
interest added together.  When the interest becomes due, and  is
not paid, it is added to the principal, and interest is calculated upon
the amount, as if it was the original principal, or sum.
  Find the amount of the principal for the first year, at the given
rate of interest. This amount becomes the principal for the second
year.   Find the amount of this sum or principal as before; which
becomes the  principal for the third year; and so on, for any num-
ber of years.
  From the last amount, subtract the original sum; the remainder
will be the compound interest.

                        examples.
  1. What is the compound interest of 350 dols. for 3 years, at 6
per cent. ?                                   Ans.  $66.85,5,6.
          D.   D.    D.
          100  : 6  : : 350 : J prop.
                    __6
                   21.00,  interest for the first year.
                  350.00,  the principal sum.
                 371.00—amount;  which is the principal for
                           the 2d year.
         100 :  6 : : 371  = £ prop.
                       6
                  22.26=the interest for the  second year.
                 371.00=the principal for the second year.
                 393.26=amount; which is the principal for
                          the 3d year. 
216                    ARITHMETIC.
100  : 6
393.26
     6
i prop.
                23.59,5,6=interest for the 3d year.
               393.26,0,0=the principal for the 3d year.
               4l6.85,5,6=the amount for the 3d year.
               350.00,0,0=the original sum.
               $66.85,5,6=the whole compound interest.
  2. What will 500 dollars amount to in 3  years at 6 per cent.
compound interest ?                         Ans. $595.50,7,5.
  3. What will  be the compound interest for 1200 dollars, for  1
years, at 54 per cent. ?                     Ans. $286.58,9,52.
  But the  operation may be considerably shortened by reducing
the rate per cent, to the decimal  of 100,  and  adding it to I or
unit;  which will be the amount  of 1 dollar for one year: Multiply
the principal by this amount, continually, to the number ofyeais
given.
   The following table exhibits the amount of 1 dollar for one year,
at the several rates per cent.
Rates per cent.
    3
    34
    4
    44
    5
    54
    6
    64
    7
 Amounts.
  1.03
  1.035
  1.04
  1.045
•  1.05
■  1.055
■  1.06
■  1.065
-  1.07
Rates per cent.
I   3*   -
   si   -
   Ai   -
   A%   -
   H   -
   5%   -
   64   -
   6%   -
Amounts.
1.0325
1.0375
1.0425
 1.0475
 1.0525
 1.0575
 1.0625
 1.0675
  Note.—The above table is  formed by reducing the rate  per
cent, to the decimal of 100 in the following manner.
  Let it be required to reduce the rate of 6 per cent, to a decimal;
or expressed in the form of a vulgar fraction it is T^.
       100)6.00(.06, the decimal value.  Again,
           6.00
  Let it be required to reduce 64, and 5| per cent, to decimals.
        100)6.500(.065             100)5.7500(.0575
            600                         500
              500                        750
              500                        7_oo_
              ---                          500
                                           500 
COMPOUND INTEREST.               217
  These decimal values being added to 1, or unit, become 1.06—
1.065—1.0575, which are the amounts of one dollar for one year,
at their respective rates.   In like manner, the other rates may be
reduced to their value in decimals.

  4. What is the compound interest of 250 dollars for 3 years, at
5 per cent. ?                                Ans. $39.40,6,25.
      principal=250 dollars.
               J.05
               1250
              250
  262.50=the amount of the 1st year.
    1.05
 131250
26250
 275.62,50=the amount of the 2d year.
      1.05
 13781250
2756250
        289.40,62,50=the amount of the 3d year.
        250.00        deduct the original sum.
          39.40,6,25 =compound interest for 3 years.

   In like manner, let the pupil perform the following examples.

   5. What will be the compound interest upon a note of 1152 dol-
 lars, at 5 per cent., for 6 years ?             Ans. $391.79.17+
   6. What will be the amount at compound interest, of a note for
 961 dollars, 20 cents, for 3  years, at 34 per cent. ?
   _  „.,                                   Ans. $1065.69,9.
   7. What will  be the compound interest on a note  of 750  dol-
 lars, at 5 per cent., for 5f years ?       Ans. $242.88,6,383625.
   8.  What will be the amount of a note for 1000 dollars, at 6%
 per cent., for 4 years ?                     Ans. $1286.46,6,35.
   9.  Ayoung lady, just 12 years old, had a legacy left her, of 1260
 dollars, to be paid when she should arrive at the age of  18 years,
 with compound interest at 6 per cent.; what was the  amount she
 '^rTwl   •  u                         Ans. $1787.33,4,066
   IV.  What is the compound interest upon an obligation for 1796
 dollars, 25 cents, at 6 per cent., for7 years ? Ans. $904.64,5,74672.

   But compound interest, may be still more expeditiously calculat-

or dol£r f P °f thG f°ll0Win,g  tab,CS' Which Sive the ™">™* of
one dollar, from one quarter of a year,  to forty years
         T 2 
218
ARITHMETIC.
                    TABLE I.
Showing the Amount of one dollar from one Year to forty.
Fr.  4 per cent.   4^percent.   5 per cent,   bfrpercent.  6 per cent
 9
10
11
12
13
14
L5
16
17
18
19
20
21
22
23
24
25
26
27
28
29
 30
 31
 32
 33
36
37
  1.0400000
  1.0816000
  1.1248640
  1.1698585
  1.2166529
  1.2653190
  1.3159317
  1.3635690
  1.4233118
  1,4802442
  1.5394540
  1.6010322
  1.6650735
  1.7316764
  1.8009435
  1.8729812
  1.9479005
  2.0258161
  2.1068491
  2.1911231
  2.2787680
  2.3699187
  2.4647155
  2.5633041
   2.6658363
   2.7724697
   2.8833685
   2.9987033  |
   3.1186514J
   3.2433975
   3.3731334
   3.5080587
   3.6481831
34 13.7943163
35 3.9460889
   4.1030325
   4.2680898
   4.4388134
   4.6163659
   4.8010206
 39
140
1.0450000
1.0920250
1.1411661
1.1925186
1.2461819
1.3022601
1.3608618
1.4221006
1.4860951
1.5529694
1.6228530
1.6958814
1-7721961
1.8519449
1.9352824
2.0223701
2.1133768
2.2084787
2.3078603
2.4117140
2.5202411
2.6336520
2.7521663
2-8760138
3.0054344
3.1406790
 3.2820095
 3.4296999
 3.5840364
 3.7453181
 3.9138574
14.0899810
 4.2740301
 4.4663015
 4.667347S
j 4.8773784
I 5.0968604
15.3262192
 5.5658990
 15.8163645
1.0500000
1.1025000
1.1576250
1.2155062
1.2762815
1.3400956
1.4071004
1.4774554
1.5513282
1.6238946
1.7103393
1.7958563
1.8856491
1.9799316
2.0789281
2.1828745
2.2920183
2.4066192
2.5269502 I
2.65329771
2.7859625
2.9252607
3.0715237
3.2250999
3.3863549
3.5556726
3.7334563
 3.9201291
14.1161356
 4.3219423
| 4.5380394
 4.7649414
 5.0031885
 5.2533479
 5.5160152
 5.7918101
 6.0814069
 6.3854772
 6.7047511
 7-0399887
1.0550000
1.1130250
1.1742413
1.2388246
1.3069598
1.3788426
1.4546789
1.5346862
1.6190939
1.7081440
1.8020919
1.9012069
2.0057732
2.1160907
2.2324756
2.3552617
2.48 18011
2.6214652
[2.7656458
| 2.9177563
3.0782329
 3.2475357
 3.4261502
 3.6145885
 3.8133910
 4.0231279
 4.2443999
 4.4778419
 4.7241232
 4.9839469
 5.2580671
 5.5472608
 5.8523600
 6.1742398
 6.5138230
 6.8720832
 7.250047*
 7.6488004
  8.0694844
  8.5133060
1.0600000
1.1360000
1.1910160
1.2624769
1.3382256
1.4185191
1.5036302
1.5938480
1.6894789
1.7908476
1.89S29K5
2.0121964
2.1329282
2.2609039
2.3965581
2.5 103517
2.6927727
2.8543391
3.0255995
3.2071355
8.39956.16;
3.6035374]
3.80974961
4.04 89346]
4.29187071
4.549382<J
 4.S223459I
 5.11168671
 5.41838701
 5.7434912
 6.0881007|
 6.45338671
 6.8405899
 7.25102531
 7.6860S63J
 8.1472520
 8.6360871
 9.1542523
 9.7035074
 10.2857178J 
COMPOUND INTEREST.                 219
                        TABLE II.
Amount of One Dollar for a year, and for quarters, at Compound
                           Interest.*
Rate per ct.	1 year. 1 For 3 qrs.		For 2 qrs.	For 1 qr.	Simple lDt. of 1 dol. for 1 month.
3 H A 44 5 54 6 64 7	1.03 1.035 1.04 1.045 1.05 1.055 1.06 1.065 1.07	1.022416 1.026173 1.029852 1.033563 1.037270 1.040973 1.044671 1.048364 1.052053	1.014889 1.017349 1.019804 1.022252 1.024695 1.027132 1.029536 1,031988 I 1.034408	1.007417 1.008637 1.009853 1.011065 1.012272 1.013475 1.014674 1.015868 1.017058	.002500 .002917 .003333 .003750 .004167 .004583 .005000 .005417 .005833
  To  calculate compound  interest by the tables: multiply the
amount of  1 dollar forthe given years, and at the given rate, as
found in the table  for years, by the principal; the product will be
the amount for that time, and at the given rate. Subtract the prin-
cipal from  the amount thus found ; the remainder will be the com-
pound interest required.   But,
  If the time  is  years, and  quarters of  a  year: multiply the
amount for the given  term of years, by the proper tabular value for
the respective  part, or  parts of a year; the  product will be the
amount of 1 dollar for the whole time.  Multiply this by the given
sum ;  the product will be the amount required.

  * The foregoing1 tables are constructed in the following- manner :   The
amount of one dollar, for  instance, at 5 per cent.,  is 1.05, as has already
been shown. This squared, will be the amount of 1 dollar for two years ;
cubed, will be the amount of it for  three years ;  as may be seen by in-
specting the tables; and so on, for any number of years, and for any rate
per cent.
  The table for quarters is formed upon the same principles.  But be-
cause the parts of a year, are fractions of a year : for a quarter of a year,
extract the biquadrate root of the amount of  1 dollar at the given rate',
for  the  amount of 1 dollar for that  time,  which is 1.014764, the'
amount for i of a year, at 6 percent: for half a year, take the square
root of the amount, 111^:^1.06=1.029563, the amount for halfayear ■
for three quarters of a year, the product of a half, and  a quarter, thus •
1.029563X1.014764=1.01467078+, theamuuntfor £ of ayear; and so
for any other rate. 
220                    ARITHMETIC.
  11. What will be the  compound  interest of 450 dollars, for 3
years, at 6 per cent ?                         Ans. $85.95,7,2.
      1.191016 =tabular amount of 1 dol. for S years at 6 per ct.
            450=the principal.
      59550800
     4764064
    535.95,72 =the amount.
    450.00
  $985.95,72 =the compound interest required.
  12. What will a bond for 750 dollars amount to, in 5 years, at
54per cent., at compound interest?            Ans. 980.21,9+
    1.306959=tabular amt. of 1 dol. for 5 years at the given rate.
   ______750
  980.21,9,250 =the amount for 5 years.
  13. What will 786 dollars amount to,  in 3 years and a quarter
at 6 per cent. ?                          Ans. $949.87,5,47+
            1.1910l6=amount of 1 dol. for 3 years.
            1.014674=amount of 1  dol. for 3 months, or 1 qr.
            4764064
           8337112
          7146096
         4764064
        1191016
      1191016
     1 208492968784=the amount of 1 doll, for the given time.
                 786=the given sum.
       7250957812704
      9667943750272
     8459450781488
    949.87,5,473464224=the amount for the time required.'
   14. Wh'at'is the compound interest of 629 dollars, for 7■£■«,«
 6 per cent ?                            Ans' *31f)T8,3,3958.
   15. What will be the compound  interest on a bond, for■ 750Idol-
 lars, for 4| years,  at 54 per cent. ?          Ans. $217.18,67^+
   16.  A gentleman at the birth of his son, put out  on compound
 interest, at 6 per cent. 250 dollars,  to be paid to him with  interest,
 when he arrived at the age of 21 years; what sum chtl he receive:

    17.  A gentleman had  an only  daughter,  and wishing to make
  nrovision  for her, when she should arrive at the age of 18 years
  on thTday  that' she  was  5  years old, put out  at compound 
COMPOUND INTEREST.
221
interest at 6 per cent., the sum of 1500 dollars : what was the for-
tune of the young lady ?                     Ans. $3199.39,23.

   When the amount, time, and rate per cent, are given to find the
principal: Divide the amount given, by the amount of 1 dollar at
the rate, and for the given time; the quotient will be the principal.
   18.  What sum at compound interest, in 3 years will amount to
$535.95,7,2, at 6 per cent. ?                     Ans. 450 dols.
                1.191016)535.957200(450
                         4764064 • •
                          5955080
                          5955080

   19.  What principal at compound  interest, in  3f years  will
amount to 5949-87,547346, at 6 per cent. ?        Ans. 786 dols.
   20.  What principal  at compound interest, in  13 years will
amount to $3199.39,23, at 6 per cent. ?          Ans. 1500 dols.
   21.  What sum  at  compound interest, in 4| years will  amount
to #967.18,6751814, at 5% per cent. ?   '          Ans. 750 dols.
   22.  What sum at compound interest, in 21 years will amount to
 5849.89,09, at 6 per cent. ?                     Ans. 250 dols.

   When  the principal, rate per cent., and amount are  given to
find  the time: Divide the amount by the principal, the quotient
will  be the power of the ratio, denoted by the number of years:
involve the ratio till it equals this quotient, the power of the ratio
will  be the time:  Or by  the tables; find the  quotient  under the
rate per cent.,  and in a line with it, in the first column, you will
have the number of years.
  23. In what  time will 450 dols. amount to  $520.93125, at 5 per
cent- •                                         Ans. 3 years.
                450)520.93125(1.157625=the 3d power of the
                    450-----             ratio, or 1.05.
                     709
                     450
                     2593
                     2250
                      3431
                      3150
                       2812
                       2700
1125
 900 
222                     ARITHMETIC.
  With this quotient enter the table,  and you find  it under the 5
per cent.; and in a line with it, in the first column, you will find 3
years for the time.   Or, involve the ratio or amount of 1 dollar, at
5 per cent,  to the third power, it  will equal the quotient;  which
will, therefore, be the time as above.   In like manner, proceed
with the following examples.
  24. In what time will a bond for $750 amount to $980.219250,
at 54 per cent, compound interest ?               Ans. 5 years.
  25. In what time will 629 dollars  amount to 5945.78,33958, at
6 per cent, compound interest ?                    Ans. 7 years.
  26. In what time will 250 dollars  amount to 5849.89090, at 6
per cent, compound interest ?                     Ans. 21 years.
  27. In what time will  1500 dollars amount to 53199.39,23, at
6 per cent, compound interest ?                   Ans. 13 years.
  When the principal,  amount, and time are given to find the rate
percent.:  Divide the amount by the principal, the quotient will
be that  power of the ratio, whose index  is the same with the num-
ber of years.  Extract  such root of the quotient, as is indicated by
the time: Or, with the quotient enter the table, and in  a line with
the number of years, find it; and in the top, you will have the rate
per cent.
   28  At what rate per cent, will  $450 amount to $520.93125 in
three years?                               .  Ans.  5 per cent.
              450)520.93125(1.157625=quotient or amount of 1
                  45O.....              dollar for  the time, at
                  ~fo§                   the given  rate.
                    450       ^/1.157625=1.05=ratio, or rate
                    2593                           Per cent>
                    2250
3431
3150
 2812
 2700
                      1125
                      900_
                      2250
                      2250
  In the table, and opposite 3 years in the column of years ,you
will find the quotient under the rate per cent.; or extract the cube
Too , and you will find  the root to be 1.05, the rate per cent, 0
amount ofl dollar for one year.  In like manner,  proceed  in the
following examples. 
                      COMMISSION.                     223

  29. At what rate per cent will 750 dolls, amount to 980 dolls.
21925 cents, in 5 years ?                     Ans. 54 per cent.
  30. At what rate per cent, will 1500 dolls, amount to 3199 dolls.
39,23 cents, in 13 years ?                      Ans. 6 per cent.

  Questions.  1. What do you understand by the term Interest ?
—2. How :s it divided ?—3. What is the principal ?—4. What do
you understand  by  the rate per cent. ?—5.  What do you under-
stand by the term amount ?—6. On what principle is the opera-
tion  founded ?—7. When the  time is two, three, or more years,
how do you  perform the operation ?—8. When the given time is
less than a year, how do you find the interest ?—9. Will you ex-
plain this by examples ?—10.  When the time is years, and parts
of a year, how do you proceed to find the interest ?—11. Will you
explain this by examples ?
  12.  In what does compound interest differ from simple interest ?
—13.  How  do you  proceed to find the compound interest arising
from a given sum of money, for two or  more years?—14. Can you
explain the  process by an example ?—14. May the operation be
considerably shortened by decimals ? —15.  By what would  you
multiply the principal, or given sum ?—16. How often is the mul-
tiplication to be repeated ?—17- What would be the last product ?
—18.  How  then would  you find the interest?—19. Can you ex-
plain this by an  example ?
   20.  What other method is there, still more expeditious, to calcu-
late  compound interest ?—21. What do the tables give you ?—22.
By what do you multiply the tabular amount?—23. What will
the product  be ?—24.  Will you explain this by an example ?—25.
How are the tables constructed ?—26. What  then do the several
columns contain ?
   27.  How  is the table for parts of a year constructed ?—28. How
would  you find the  amount of 1 dollar, for 1 quarter of a year, at
any  given rate of interest ?—29. How would you find it for half a
year ?—30.  How would you find it for three quarters of a year ?
  31.  How  would you proceed to find the amount of a given sum
of money, at a given rate per cent, for any  number of years, by
the help of the tables ?—32.  How would you find the compound
interest of any sum ?—33. How would you proceed  to find the
amount, for  any number of years, and a quarter, half, or three
quarters of a year ?—34. How would you  obtain the interest only ?
                    COMMISSION.
  Commission is an allowance of a certain Per Cent, to a person
for buying or selling goods, or for transacting any business.
  The operation is the same as the Rule of Three. 
 224                    ARITHMETIC.

                          EXAMPLES.
   1. What will  be the commission for  the  purchase  of goods,
 amounting to 1560 dols. at 2 per cent. ?            Ans. S31.20.
                 100 : 1560  ■■■  2 ■.  I prop.
   2. What will be the commission on the purchase of 5675 dols.
 bank stock, at 14 per cent. ?                     Ans. $bJA2,5.
                 100 = 5675  ••= 1.5  : £ prop.
   3. What will be the commission on the sale of goods amounting
 to $1260.50, at 24 per cent. ?                  Ans. $31.59,2,5.
                 100 •• 1260.50  :: 2.5 -. % prop.
   4. What will be the commission on the sale of goods, nniounting
 to 5746.25, at 3| per cent.?                   Ans. $27.98,4+.
              100 : : 746.25 : : 3.75 :  £ prop.

   When the rate per cent, has a fractional  part  connected with
 it, reduce it to a decimal, and annex  it to  the integer, as in the
 above examples.   When it is a fraction, reduce  it to a decimal,
 and  work with it according to the rules of decimals ; as in  the fol-
 lowing example.
   5. What will be the commission on  the purchase of 1140 dols.
 of bank stock, at f per cent. ?                      Ans. $8.55.
                 100 =  1140  = : .75  : ^ prop.
   6. A commission merchant sold goods amounting to 1675 dols.;
 what is his commission at 34 per cent. ?          Ans. $58.62,5.
   7.  A planter consigned his  ciop of tobacco, amounting, when
 sold, to 2174 dols. to a merchant  in  New  York ; what must he
 allow for the sale, commission and other charges amounting in the
whole lo5£ percent. ?                         Ans. $114.13,5.
   8. A manufacturer consigned goods amounting to 1825  dols. to
a merchant in Philadelphia ; what must he allow for the sab', at 2|
per cent. ?                                    Ans. $50.18,75.
  9. A merchant in the country directed his correspondent in New
York to purchase for him sundry articles of merchandise, amount-
ing in the whole to 1165 dollars ; what will  his commission be, at
2J per cent.?                                 Ans. $26.21,25.
   10. What will  be the commission on the  purchase of  50 shares
 of bank stock, at 112 dols. 50 cts. per share, at 4 per cent, r
                                              Ans. $28.12,5.
   Here find the amount of the stock, and calculate  the commission
on that sum.
   11. What will be the commission on the  sale of 125 share* of
stock of the United States Bank, at 1164 dollars a  share, at £ per
cent.?                                     Ans. $109.21,8,75.
f
 
INSURANCE.
225
                   INSURANCE.

  Insurance is the security against loss or damage in houses, ves-
sels, merchandise, &c, procured by paying a certain sum of money
to another person, for taking the risk or hazard upon himself.
  In case of any loss  or damage,  in the property so secured, the
insurer is bound to make it good, by paying the sum  mentioned in
the  policy, or instrument of writing, in  which the contract is made
between the parties.
  The operation  is the same, as in commission.

                        EXAMPLES.
  1. What will be the insurance on the sum of 575 dols. at 34 per
cent.?                                        Ans. $20,124.
                100 : 575  : : 3.5 :  %  prop.
  2. What sum must be paid for the insurance of property against
fire, valued at 2150 dols, at 24 per cent.  ?         Ans. $53.75.
  3. A gentleman  has a  house  in Philadelphia, valued at  7890
dols.;  what sum must he pay, to be insured against loss or damage
by fire, at 1J per  cent. ?                         Ans. $98.62£.
  4. A merchant has goods  in his store, valued at 25675 dols.;
what sum must he pay, to be secure against loss or damage by fire,
the premium asked being If per cent. ?          Ans. $449.31*!
  5. A merchant having a ship at sea,  valued at 12650 dols.; what
must he pay, to be insured against the dangers of the sea, the pre-
mium demanded  being 2| per cent. ?            Ans. $347.87*.
  6. A merchant having a ship on a voyage to Canton, valued at
15764  dols, and desiring to be secure against the dangers of the
sea, both  outward and homeward bound; what must he pay the
premium demanded being 54 per cent.  ?          Ans. $867.02.
  7. A merchant shipped on  board a brig bound to New Orleans
goods amounting to 2464 dols.;  what must he pay, to be insured
against the dangers of the sea, the premium being 2\ per cent. ?
                                            Ans. $55 46 25
  8. A merchant having a ship, valued at 11750 dollars, put'on
board her on a voyage to Liverpool,  goods amounting to 14754
dols.;  what must he pay, to be insured against the dangers of the
sea, on both vessel and cargo, the premium being 24 per cent. ?
  o a    .1     L  •         .,              Ans- $662.60.
  9. A gentleman having several  houses in the city of New York
valued  in  the  whole at 25760 dols.;  what must he pay to be in-
sured aga.nst fire, the premium demanded  being \% per cent. ?
          j,                                  Ans. $450.80 
226
ARtTfiMETIC.
  10. A merchant having a house, valued at 10650 dols, and goods
in his store amounting to 6740 dols. ;  what must he pay to be in-
sured against the danger of fiie,  the premium being 1A per crnt. ?
                                              Ans. Ss26().S5.
  11. What must be paid to insure a  steam  boat, running from
New Orleans to Louisville,  valued at  25675 dols.;  the premium
being A\ per cent. ?                           Ans. $1091.183.
  12. A  merchant of Louisville shipped a quantity of tobacco for
New  York, valued at  11790  dols.; what must  he pay for the
insurance of the same to the place of destination ; the premium to
New Orleans  being £ per cent, and from New Orleans to New
York 24 per cent. ?                           Ans. $383.17$.
  13. The fortune of a young lady consists in houses in 'he city of
Philadelphia, valued  at 14600 dols. ;  what must  be paid by her
guardian, to be secure against the risk of fire, the premium being
3 per cent. ?   And how much  must  be paid in the whole for in-
surance, until  she  shall  arrive at the age of 18, she being at the
decease of her father just 8 years of age ?
                                   S $109-50 annual payment.
                              Alls" I $1095 paid in the whole.
  To find the sum, for which  a policy is to be taken, in order to
secure a given amount : take the premium from 100 : then say, as
the remainder is to 100, so will the sum adventured be to the policy
to betaken out.
  14. A gentleman wishes to secure 750 dols. on an adventure to
London ; what sum must be mentioned in  the policy to cover that
amount; the premium being 4 per cent. ?         Ans. $781.25.
                 96  : 100  : =  750 : £ prop.
   1',  A merchant shipped goods to Havanna, amounting to a cer-
 tain sum ;  for his  better security against tin: dangers of the sea, he
 would insure  1560 dols. :  what must  be the policy, the  premium
 being 2i per cent.?                           Ans. 51600 dols.
   16. A merchant shipped goods to South America,,amounting to
  10750 dols ,  and  would insure to the amount of  5680 dols.; how
  much must be the policy to cover that sum, the premium being S£
  per cent. ?                                 Ans- $5886+ dols.
   17 A gentleman has perishable property, amounting to 12790
  dols '• and wishes to secure a part of this, amounting to 80G4
  doU.l what must be the policy, the premium being 5$ per■cent, r
                                               Ans. $oD>iii.00.
    18 A ladv has property, on which she wishes to secure the sum
  of 7500 dols.'; what must be the policy to cover that .urn,.the  pre-
       h in? 6i percent ?                    Ans. SeOzl 39 + -
  'n'l9.  A  planter  in Virginia effected insurance  on his crop of 
REBATE OR DISCOUNT.               227
tobacco, to the amount  of 1250 dols.; what  must the policy be,
the premium being 3f per cent. ?             Ans. $1298.90+.
  20. A  gentleman having houses, valued  at  25000 dols, would
secure against the risk of fire, the sum  of 16000 dols.; what sum
must be mentioned in  the policy, to cover  that amount;  the pre-
mium being 3\ per cent. ?                 Ans. $ 16537.46,6+.
               REBATE OR DISCOUNT.

   Discount is an allowance made for  the payment of money
  before it is due;  and is generally at a given rate per cent.
   The operation is the same, as the Rule of Three.
   To find the present worth  or sum to be received in hand: As
 the amount of 100 dols. at the rate,and for the time given, is to the
 given sum, so will 100 dols. be to the present worth, or payment.
   Subtract this from the given sum, and the remainder will be the
 discount.
                         EXAMPLES.
   1.  What is the present value of 750 dols. dye 12 months hence,
 at 7 per  cent. ?                            Ans. $700.93.45+.
                 107 :  750 : :  100 : \ prop.
   Here  107  dols. is the amount of 100 dols. for 1 year at 7 per
 cent.; for  if it were put at interest for 1 year, at that rate, it would
 produce  7  dollars.  Therefore 100 dollars in hand is worth 107 at
 the end of 12 months; and so for a longer or shorter period of time.
 The rebate or discount is considered in business transactions, equal
 to the interest  that would arise from the present worth, if put to
 interest at the given rate, and for the given time.

  cL^liatJdiscount must be made for the Presei>t payment of
 1500  dols.  due 12 months hence, at 6 per cent.?
                                           Ans. $84.90,6+.
                106  : 1500 ::  100 :  \  prop.
  Here, subtract the present  worth, as found by the operation, from
the given sum ;  the remainder will be the discount or sum allowed
lor the present payment.

cre3HitAofmoerChar;i S01k g°°?S l°  'he am0Unf of 126°  d°'s- °" a
credit  of  9  mouths;  but the purchaser  proposed to  pay him in

hea;Vinn;Vn°dUld ""* ' ^ °f 8 ^ ?*' V""' 7"Z
  H J       '                           Ans. $1188.67,9+.
               106 : 1260 : :  100 : J prop. 
228                    ARITHMETIC
  Here the interest of 100 dols. for 9 months at 8 per cent,  is 6
dollars, which add to 100, making 106 for the amount. Then  pro-
ceed as directed above.
  4. What sum must be discounted for the present payment of 760
dollars, due 18 months hence, allowing 54 per cent. ?
                                           Ans. $70J.(>r,*,5.
             m.   m.     D.   D.
            12  =  18 : :  5.5 : 8.25.  Then say,
               108.25 :  760  : : 100 : £ prop.
  Here the interest of 100 dollars, for the time given, is 8 dollars,
25 cents, which add to 100, making 108.25, for the amount of 100
dollars for 18 months. Then proceed as above, to find the present
worth : which subtract from the given sum, the remainder will be
the discount.
  5. A gentleman has due to him, on bond, the sum of 796 dollars,
payable in  10 months;  but being in  want of money, will  make a
discount for present payment of 8 per cent; what sum will he re-
ceive ?                                       Ans. $746.25 +
  6. A merchant bought goods amounting to  3500 dollars,  on a
credit  of 12  months; but  proposed to the seller to pay him  in
hand, if he would make a discount of 84 per cent.;  how much did
he nay; and what was the discount ?   Ans. $3225.80+sum paid.
                                      $274.20+the discount.
   7'. A gentleman bought a plantation, for  5760 dollars, payable
one third in 12 months, after the signature of the deed; one  third
in 24  months, and the remainder in 36 months, without  interest:
but proposed to the seller to pay him all in hand, if he would make
a discount of 74  per cent.—what was the sum actually paid, and
what was the amount of the discount ?
                        Ans. .^5022.95,7, the sum paid in  band.
                             " $737.04,3, the whole discount.
   8. What is the difference between the interest of 560 dollars, for
5 years, at 6 per cent, and the discount for the same time, and at
the same rate?                           .        Ans. $3877.
   Here you calculate the discount,  on the given sum, for the rate
and time, as directed above; also the interest upon the same sum,
for the time and rate.  The difference of these sums will be the
 amount  required.  The interest  is the greater of  the two  sums,
 which shows, that the discount of any sum, and the interest  of the
 same sum  at the  rate, and for the time given,  are considerably dif-
 ferent   But in ordinary transactions, the interest of the given sura
 for the time, is taken as the discount; and in  small sums, for short
 periods, the difference is not very great.
    9.  A person has a note for 750 dollars, due 15  months hence; 
                         BARTER.                       229

but being in want  of money, is willing  to give 74  Per cent- ms"
count for ready money;  what will be the sum received ?
                                            Ans. $685.71 +
  10. A merchant bought a quantity of cotton, amounting to 1475
dollars, on a credit of 9 and 15 months;  that is, one half payable
in 9 months, the other in 15 months:  but  proposes to pay the
whole down, if the planter will make a discount  of 84 per cent, on
the  whole; what will be the  sum paid—and  what will  be the
amount  of the discount ?            Ans. $1359-97+sum paid.
                                     $115.02,4, the discount.
                         BARTER.

    Barter is the exchange of one commodity for another, by duly
  proportioning their quantities and values.
    The operation is the same as the rule of Three;  and, generally,
  two statements are required to determine the question.


                          EXAMPLES.

    1. How  much sugar, at 9 cents a pound, must be given  in ex-
  change for 6 cwt. 2 qrs. of tobacco, at 14 cents a pound ?
                                     Ans. 10 cwt. Oqr. 12|Ib.
                   lb.  cwt. qr.   cts. ■
      1st statement.  1:6  2  .•: 14  : £ prop.
                        4
                       26
                      28
                     208
                     52

                    "728xl4=l0192=valueoftobac. in cents.
                  cts.    cts.     lb.
     2d statement.  9 •• 10192  .- : 1  .- |  pr0p.


                     9)T0l92 = 1132|lb.=10cwt.0qr.l241b
   Here, in the first operation, find the Value of the tobacco, at 14
cent, a pound, which ,s  10192 cents.  The next operation is to
find the quant.ty of sugar, at 9 cents a pound, that would equal  n
value, the tobacco as found above.  The tobacco  must be reduced
to pounds, because the first term  is 1 pound.
i 
230
ARITHMETIC.
  2. How much tea, at 1 dollar, 25 cents a pound, must be given
for  112 lbs. o f chocolate, at 50 cents a pound ?      Ans. 44.8 lb.
                   lb.   lbs.    cts.
     Iststatement.  1  : 112  :: 50 :  \ prop.
                         50
                       56.00=the value of the chocolate.
                    dol.     dol.    lb.
     2d statement.  1.25  = 56.00  ■ ■ I ■ % prop.
                              1
                     1.25)56.00(44.8=the quantity of tea.
                         50 0
                           600
                           500
                           Tboo
                            1000

   3.  How many yards of muslin, at 16 cents a yard, must be given
in exchange for 125 yards of linen, at 374 cents a vai"d ?
                                            Ans. 292.968 yds.
                   yd.  yds.      cts.
     Iststatement.   1 :  125  : =   37-5 = £ prop.
                        37-5
                    $46.87,5 =the value of the linen.
                   cts.   D. cts.    yd.
     2d statement.   16 = 46.87j5 ■ ■  1 ■ % prop.
   4.  Two merchants barter.  One has tea, which he will sell at
$1.20 cash, but in exchange, he will have  $1.25 : the other has
sugar, which he will sell at 10 cents a lb. cash; how must he  rate
his sugar, to lose nothing by the exchange ?   Ans.  10.4160 cents.
                 1.20 :  1.25 ::  10 : ^r prop.
   5. How many yards of silk, at 874centsayard> must be chang-
ed for 25 yards of broadcloth, at 4 dollars, 624 cents a yard ?
                                            Ans. 132.14+yds.
                    yd.   yd.      D.
      1st statement.   1  :  25  :: 4.62,5 : £ prop.
                     cts.       cts.     yd.
     2d statement.   87-5 :   115.62,5  ■ ■ 1  ■  i prop.
   6.  A merchant in Philadelphia sold goods to the amount of 450
 dollars;  for which  he received  $187-50, in cash, and  the rest in
 flour, at $5,124 cents> a barrel 5 how much flour dl(J 5t ,ake ,0 pay
 the balance ?                         Ans. 51.22 barrels nearly.
             $450=the whole debt.
               187«50=cash paid.
              . 262.50=the remaining sum.
                   5.12,5 :  262.50 ■ ■ 1  •■ %  prop. 
EQUATION OF PAYMENTS.               231
  7. A merchant has cotton, at 11 cents a pound, which he would
exchange for 560 pounds of candles, at  8 cents a pound; how
much cotton must he give ?                      Ans. 407T3T lbs.
  8. Two merchants barter.  One has wine at $2.25 a gall, cash;
but in exchange, he will have $2,374 : the otber ^as c°ffee at ^
cents cash ; how must he rate the coffee to be no loser in the ex-
change? And how much must he give for ahhd. of wine ?
                        Ans. 16.888 cents, the coffee in barter.
               886+pounds must be given for 63 galls, of wine.
                      cts.    dol. cts.     cts.
      1st statement.   2.25  : : 2.37,5 :: 16 : £ prop.
      2d  statement.  16.888 = 2.37,5  :  : 63 : £ prop.
  9. Two merchants barter: One has 556 lbs. of sugar, valued
at 10 cts. a pound ;  for which the other gave him 860 pounds of
cheese; at what was the cheese rated ?      Ans. 6.4,65 cts. a lb.
                   lb.   lbs.    cts.
     1st statement.   1 : 556 : : 10  = ^  prop.
                   lbs.   lb.    dols.
    2d statement.   860 : 1 •■: 55.60 •• £  prop,
  10. Two merchants barter: The one has 40 yds. of cloth, which
he rates at 98 cts. a yard ; and the other 30 lbs. of tea, at 1 dol. 50
cents  a pound; which  party pays a balance; and what is that
balance ?       Ans. $5.80, the bal. to be paid by the first party.
  11. Two persons barter: One has 250 bushels of rye, which
cost him 75 cts. a bushel; but he rates it at 85 cts.: the other has
wheat, which cost him 95 cents a bushel; at what price must the
wheat be rated, to  be equal to the profit upon the rye?  And how
many bushels of wheat must be given for the 250 bushels of rye ?
                    Ans. $ $l°7.6,6fi, the price of the wheat.
                       ' 1  197-36+,  bush, given for the rye.
            EQUATION OF PAYMENTS.

   Equation of  Payments is the  method of reducing several
stated times, at which money is payable, to one mean or equated
time, when the whole is to be paid at once; so that neither party
may sustain loss.  Rule :
   Multiply each payment by the time when it is to be paid, and
divide the sum of the products by the whole amount;  the quotient
will be the equated time, or the time when the whole sum is to be
paid. 
232                    ARITHMETIC.
                        examples.
  1. A merchant has due to him 1500 dols, payable 500 dols. in
3 months ; 500 dols. in 6 months ; and the remainder in 9 months;
but he agrees to receive the whole in one  payment; at what lime
must it be made ?                              Ans. 6 months.
             500x3 = 1500
             500x6=3000
             500x9=4500
                      9000=the sum of the products.
               9000-1-1500=6 the equated time;
  Or time  when the whole sum is  to be paid.  In like manner
proceed with the following examples to find the equated time.
  2. A planter sold a tract of land for 2500 dollars, payable in five
equal payments of 500 dols. each ; the first to be made in 6 months,
the second, in  12  months,  and so on; but afterward  agreed to re-
duce them to one payment: what must be the time of payment ?
                                            Ans. 18 months.
  3. A gentleman sold a house for the sum of 2000 dols., payable
600 dols. in hand, 500 dols. in 6 monihs,  400 dols. in 12 months,
and the remainder in  20 months ;  but he afterward agreed to re-
duce them all to one payment: when must it be made ?
                                            Ans. 8.9 months.
   4. A merchant sold goods amounting to 1800 dols.; payable,
600 dols. in 6 monihs, 600  dols. in 12 months, and the balance in
18 months; but the purchaser agreed to pay him  500 dols.  in
hand,  provided he would extend the payment of the remainder,
and reduce them to one : required ihe time of payment ?
                                          Ans. l6|8T months.
              600 X 6=  3600
              600x12=  7200
              600X18 = 10800
                       2i600=the sum of the products.
   Which divide by 1800 — 500=1300 dols, the sum remaining to
 be paid at the equated time.
   Therefore, when a sum of money is to be paid at  a given time,
 and a part is paid before the time: to find the time for the payment
 of the remainder; multiply the whole sum by the time when it is to
 be paid ; divide the product by the different of the whole sum and
 the payment made in hand, or at the particular time agreed upon;
 the quotient will  be the  equated time  for the payment of  the
 remainder.
    5.  A  lady sold a property  amounting  to 1200 dollars, payable
 at the end of 12 months;  but the purchaser agreed  to pay her
£1 
LOSS AND GAIN.
233
400 dols. in hand, if she would consent to wait a longer time for
the balance ; when must the remainder be paid ? Ans. 18 months.
  6. A young lady had a legacy left by a relation, amounting to
1850 dols, payable in 18 months, but  wishing to have a part of it
in hand, agreed with the executor, if he would advance her 500
dols. to defer the payment of the remainder proportionally : when
must she receive the balance ?                Ans. 24§ months.
  7. A  gentleman has owing to him 4500 dollars, payable in 8
months, but being in want of money, proposes to his debtor to pay
at the end of three months, the sum of 1600 dols.; and he will defer
the  payment  of  the balance proportionally ;  when  must the
remainder be paid ?                    Ans. 12.413+ months.
  8. A young gentleman has a sum of money due on an obligation,
amounting, principal and interest, to 1500 dols, but is not to be
paid for  16 months; being in want  of money to spend in his
amusements, offers to defer the balance, provided he can have 500
dols. in hand :  how long must the  balance be deferred ?
                                            Ans. 24 months.
                   LOSS AND GAIN.
   Loss and Gain  is the method of ascertaining the profit or loss
 upon the purchase and sale of goods, merchandise, or property of
 any kind.
   The operation is commonly by two statements of the Rule  of
 Three.
                        EXAMPLES.
   1. A merchant bought 116 yards of cloth at $2.50 a yard, and
 sold the same at $2.87%; what did he gain on the whole ?
                   yd.   yds.     cts.              Ans. $43.50.
     Iststatement.  1  : 116 : =  2.50  : | prop.
                         2.50
                        5800
                       232
                     $290.00=the whole cost of the cloth.
     2d statement.  I  : 116 : :  2.87,5 = £ prop.
   This will  give the amount for which the cloth was sold.  From
 which subtract the cost; the remainder will be the sum  gained.
 Or, take the difference of the cost per yard, and the sale per yard,
 which, in this case, is 374 cents, 'he gain per yard, and multiply it
 by the  number of yards, the product will be the whole gain.  This
is an expeditious method, and may be practised in questions simi-
lar to the one above. 
234
ARITHMETIC.
  2. A merchant bought 560 yds. of calico, at 174 cenls a vard;
and retailed the same at  21 cts. a yard ;  what did he gain in the
whole?                                         Ans. $19.60.
                   yd,  yds.     cts.
     1st statement.   1  : 560 • i 17.5 :  £ prop.
     2d statement.   I  : 560  : : 21 -. % prop.
  3. A grocer bought 5  bbls. of sugar, weighing in the whole, 950
pounds at 84 cts, and retailed  the same for 11 els. a pound ;  what
did he gain in the whole?                         Ans.  $23.75.
                    lb.    lbs.     els.
     1st statement.   1  : 950  : :  85  ■ % prop.
     2d  statement.   1  : 950  : =  11 : £ prop.
  4. A merchant bought 2 hhds. of wine, containing 126 gals, ut
1 dol. 75 cts. a gal,  and retailed (he sa ne, at 2 dols. 124 cents. a
gal.; what did he gain in  the whole ?             Ans. $47,124-
                   gal.   gals.      cts.
     Iststatement.   1  = 126  = =  1.75 .• £  prop.
     2d statement.   1  :  126  : =  2.12,5 : \  prop.
   5.  A merchant bought  2 pieces  of broad-cloth, containing 56
yds, at 4 dols. 75 cts. a yd.; but  upon  examination,  found  ihem
damaged.   He was, therefore, obliged to sell them for 4 dols. 11\
cts. a yard ;  how much did he  lose by the bargain ?    Ans. $35.
                    yd.  yds.     cts.
     1st statement.  1  ••  56 : :  4.75 '•  \ prop.
     2d statement.  1  :  56 : :  4.12,5  : £  prop.
   6. A merchant bought  1000 yds. of muslin for 116 dols.; how
must he retail the same to gain 25 dols.  by the bargain ?
                                Ans. 14Tlv cts. or 14 cts.  1 mill.
   Here you add the gain 25 dols. to the cost, 116 dols,and  divide
the sum by 1000, the number of yards; the quotient will be the
answer in  cents.  In  like manner all similar  questions may be
 readily solved.
   7. A gentleman  purchased  1500 lbs. of coffee for 172 dols. 50
 cts. how must he sell the  same to gain 32 dots, by his bargain ?
                                           Ans. 13 cts. 6 m. 33.
    8. A merchant bought 250 bbls. of flour at 4 dols. 50 cts. abbl;
 how must he sell the same to gain 55 dols. by the bargain ?
                                                   Ans.  #4.72.
    9.  A merchant bought  12 boxes of tea, in the whole 165 pounds,
 at 1  dollar, 45 cents a pound  ; but upon examination, it was found
 to be of an inferior quality, and he sold the same for 217 dollars, 80
 cents; what did he lose on a pound ?              Ans. 13 cents. 
LOSS AND GAIN.                   235
  Here find the cost of the tea by the rule of proportion; and from
it take the sum for which he sold it:  the  remainder will be the
whole loss : which divide by the number of pounds, the quotient
will give the loss on a pound.
   10.  A lady purchased a quantity of millinery, for which she gave
184 dollars ; and sold the same for 210 dollars; how much did she
gain per cent. ?                          Ans. 11.41+per cent.
                 D.    D.      D.
                 184  : 210  ::  100  : J prop.
  A merchant bought 100 yards  of broad-cloih,  for 425 dollars;
how must he retail it a  yard, to gain 10 percent.?   Ans. $4,67,5.
   Here ten per cent, on 425 dollars, is  42 dollars, 50 cents; which
added to 425 dollars, the  cost, will be 467 dollars, 50 cents.  This
sum divided by the  number of yards, will g*n e the answer.  In like
manner proceed with all questions of the same kind.
   12. A  merchant bought 4 chests  of tea, weighing nett 216
pounds, at 1 dollar, 16 cents a pound, and retailed  the same at 1
dollar, 31 cents; and upon  examining his account sales, found that
he had lost 8 pounds  in the weight;  what was the  gain on the
 whole—and what was the gain per cpnt. ?
                                  Ans. $21.92, the whole gain.
                                8.748, the gain per cent, nearly.
                   lb.  lb.     cts.
      Iststatement.  1 :2l6 ::  1.16 = \ prop.
     2d statement.  1 : 208  ::  131  '■ \ prop.
                      cost,  whole gain.   dol.
     3d statement. 250.56 :  21.92   :: 100 : £ prop.
   13. A  merchant sells silk at  1 dollar, 12£ cents a yard, which
 cost him only 95 cents;  how much does he gain per cent. ?
                                        Ans. 18.42 +per  cent.
   14. A  merchant bought  125 yards of broad-cloth, at 2 dollars,
 124 cents a yard ready money, and sold the same immediately, for
 2 dollars, 35 cents, on a credit of 8  months; how  much did  he
 gain in the whole—and how much per cent., allowing him interest
 upon his  money, at 6 per cent. ?        Ans. $17.50, whole  gain.
                                        6.588 per cent, nearly.
   In this question, find the amount for which the cloth was sold,
 and from  it subtract its  cost ;  the remainder would  be the  gain,
 provided  the money was paid in hand.  Then find the interest on
 the  cost  of the cloth  for 8 months,  the  credit given on the sale;
 which take from the above remainder, the balance will  be the true
 gain,  arising from the  bargain.  Having found the  true gain, find
 the gain per cent, as in the 3d statement, in the 12th question.
   15. A  merchant purchased a quantity of coffee, and found upon 
236                     ARITHMETIC.
examination, that it was damaged;  which obliged him to sell it, at
10 cents a pound, and by so doing, he lost just  15 per cent.; what
was the cost?                            Ans. 11.764 + cents.
  16. A merchant bought a quantity of goods on a credit of 6
months, but being pressed for money, sold them at  auction  for
1650 dollars, by which he sustained a loss of 15 per cent,  on the
original cost; what did the goods  cost him—and what was  his ac-
tual loss ?                  Ans. #1941.17,647, the cost nearly.
                                       $2 il.67,647, true loss.
  Here the loss would have been  $291.37,647, if he had purchased
for cash.  But  as he purchased on  a credit of 6 months,  he  had
the sum of 1650 dollars for that time; the interest of which at 6 per
cent, is 49 dollars, 50  cents, which must be  deducted from the
above sum; the remainder will be the actual loss on the bargain.
  27- A grocer purchased a hogshead of molasses, containing by
the guage 116 gals, at 40 cts. a gal, and retailed it at 46 cents a
gallon ;  but examining his account sales, found that it fell short 10
gallons :  what was his gain in the whole ; and the gain per cent. ?
                                 Ans 5 #2.36 whole gain.
                                    '( $5.08+ t^iin per cent.
  18. A merchant is offered 500 yds. of cloth at  55 cts. cash, or
60 cts.  on a  credit of 6 months.   Now he wishes to know, which
will be the better bargain, to purchase for cash, provided he pays
at the rate of 7 percent, for money, or at a credit of 6  months ?
                           Ans. $15,374 saved by paying cash.
  Here find the amount of the cloth at the cash, and credit prices,
and take their difference. Then find the interest of the sum at cash
price, for 6  months  at the given rate.  Take this fiom  the above
difference; the remainder will be the sum saved  by paying cash,
and borrowing the money.
                     FELLOWSHIP.

  Fellowship is the rule or method of dividing the gain or loss of
a joint adventure, or partnership ; so that each one concerned may
have, in proportion to his stock or capital invested: also, of divid-
ing the estate of a bankrupt among his creditors, so that each one
may share according to the  sum due to him.
  The operation is the same as the Rule of Three : as, the whole
sum or stock is to the stock  of each partner, so is the whole gain or
loss to the share of each. 
FELLOWSHIP.                    237
                        EXAMPLES.
  1. Three persons entered into partnership in trade.  The first
put in 250 dollars; the second  put in 350 dollars ; and the third
put in 500 dollars; and in 12 months, they found, by examining
their books, that they had gained 460 dollars ; how must the gain
be divided between them, so that each may have his due proportion ?
                           fj" $104.54,545+ share of the 1st.
                       .    '  146.36,363+       do.   2d.
                      Ans,<*.  209.09,09 +       do.   3d.
L $460.00,00   proof.
460
460
460
I prop.
1 prop.
I prop.
    250=stock of 1st.     1100 :  250
    350=stockof2d.      1100 :  350
    500=stock of 3d.      1100 =  500
   H00=whole stock.
  2. Two merchants entered into partnership for 12 months.  The
first put in  1500 dols, and the other, the sum of 2000 dols.; and
by misfortune, they lost 750 dols.; how must it be divided between
them?                         ^ $321.42,85+ loss of the 1st.
                               I $428.57,15   loss of the 2d.
  1500=stockof the 1st.
  2000=stock of the 2d.     3500 : 1500  = = 750  : £ prop.
  3500=whole stock.
  Here, having found the portion of loss, that is  to be sustained by
one of the partners, subtract it from the whole loss ; the remainder
will be the loss to  be sustained by the other.
  3. Three persons unite in an  adventure.   The first put in 500
dols.—the second, 400 dols.—and the third, 350 dols.;  and they
gained 850 dols.;  how must the gain  be divided between them ?
                                  C  $340 the share of the 1st.
                             Ans. ■}   272 the share of the 2d.
                                  (   238 the share of the 3d.
  4. A merchant failing in business,  was indebted to A. the sum
of $1728.24 ; to B. $1540.76 ;  and to  C. the sum of $2371.17 •
and his estate was valued at #3361.74. How must this sum be
divided among his  creditors ?        ($1030.09,21+ A's share.
                             .   j   918.34,71+ B's share.
                            rt"s'1  1413.30,00+ C's share.
                                 (.$3361.73,91  proof.
  $l728.24=A'sdebt.
   1540.76= B's debt.   5640.17 :1728.24 : •• 3361.74  .1 prop
   237l.l7=C?s debt.   5640.17 ■ 1540.76 :: 3361.74  . j prop.
  $5640.l7=sum due.  5640.17 : 2371.17 = = 3361.74  : \ prop!
  In questions of this kind, the whole sum or amount of debt is to
             A 
238                    ARITHMETIC.
any particular debt, as the whole estate is to it« particular share or
part; as appears from the above question.
   5.  The estate of a deceased person amounted  to  the sum of
15500 dols.; but A. had a claim for 2500 dols.; B. had a  claim
for 5000 dols.;  C. had a claim for 10000 dols.;  and D.  had a
claim for 6550 dols.; how much will be each one's share ?
                                    C $1611.22,6  share of A.
                                     I  3222.43,3  share of B.
                                Ans.<J   644 1.90,6  share of C.
                                       4221.41,3  share of D.
                                     I 15500.00,0  proof.
   When the  stocks are continued in company for different
 periods of time :  Multiply each partner's stock by the time of its
 continuance in the company ; and  add all the products together.
 Then it will be:  As the sum of the several products is to each
 particular  product; so  is the whole gain or loss,  to its share of
 it.
   6. Three persons  engaged  in commerce: The first advanced
 600 dols. for 6 months  ; the  second advanced 1000  dols.  for 12
 months, and the third advanced 1200 dols. for 15 months ; when
 they dissolved partnership, and found that they had gained the sum
 of 500 dols.; how must it be divided ?
                                   C $76.27,11  share of the 1st.
                              Ans. 3 169.49,18  share of the 2d.
                                   ( 254.23,71  share of the 3d.
    1st stock 600 X 9= 5400
    ^d stock 1000X12 = 12000    35400 :   5400  : : 500 :  \ prop.
    3d stock 1200X15 = 18000    35400 : 12000  = ■ 500 ■  \ prop.
  Prod, of stock &  time=3-5400    35400 = 18000  = = 500 4 prop.
    7  Two persons commenced a partnership to continue for 12
  months:  One  advanced 1500 at  the beginning; but it not being
  convenient for the other to advance at that  time,  4 monihs  after-
  ward he put into the common  stock, 2500 dollars.  At  the end of
  the  12 months, on settling their books,  they found a loss of 400
  dols.; what part must each one sustain of it ?
       9                   .    $$189.47,36+ the part of the 1st.
                         Ans. |   21o.52,64   the part of ihe^d.
    8  Three persons entered into  company for 12 months.  The
   first put in, at the beginning, 800 dols, and 5 months afterward,
  put in 100 dols.  more] the second put in,  at first, 1000 dols, but
   at the end of 6 months, took  200 dols. for his private concerns:
   The third put in, at first, 500 dols, and in three months having
   money on hand, he put in  250 dols.  more :  Being successful in 
FELLOWSHIP.                    239
business, on settling their books, they found they had gained 1200
dols.; what must each one's share of the profit be ?
                           f $421.12,4,3+ the part of the 1st.
                           j  441.56,7,3+the part of the 2d.
                      Ans,1  337.30,8,3+ the part of the 3d.
                            ' 1200.00,0,0    proof.

     800x12=9600? _stock put in  by the 1st partner multi-
     100 x 7— 700 $ ~"   plied by the time.
    1000 X 6=6000 )   stock of the  2d partner multiplied by
     800x6=4800$     the time.
     500x12=6000 1 _stock of the  3d partner multiplied by
     250X 9=2250 $ _   the time..
             29350=sura of the products.   Hence the  respec-
tive answers may readily be found by the  rule.

  The following is a  short  and expeditious method of solving
cases of partnership.
  Divide the whole gain or loss by the whole stock in trade.
  Multiply the quotient  by each partner's stock, and the several
products will be the respective gain or loss of each one.

  10. Three persons traded in company.   A.  put into the com-
pany 500 dols.; B. put in  800 dots, and C. put in 1000 dols.;
and they gained 500 dols.; what is the share of each ?
                                      f $108.69,5 + .
                                 Ansi   17-3-91.2+-
                                 Am'}  217.39,2 + .
                                      C  500.00,0   proof.
  11. Two persons entered into partnership for 12 months.  The
first put in 1200 dols, but it not being convenient for the other at
the  commencement;  4  months afterward, he put into the stock
2000 dols, and, on closing their books, they found they had gained
500 dols.; how must it  be divided between them ?
                         A   J  #236.84,20+ share of the 1st.
                         A^!,• I   264,05,78+ share of the 2d.

       1200X12=14400)   ,,     ,    ,,,.•.
       2000x  8=16000 S      product of stock and time.

                 30400=the sum of the products.
            500-f- 30400=16.4473 + cents.
        16.4473 Xl4400=236.84,l 1 +
        16.4473 X16000=263.15,68 +
                       $499.99,80  proof. 
240
ARITHMETIC.
  12. A legacy of 1200 dollars was left to three sisters, Ann,
Mary, and  Susan, to be divided in the following manner:  As often
as Ann  had five dollars and five-tenths, Mary had four, and three-
tenths, and Susan had three, and two-tenths ;  what was the share
of each?                    C $507.69,22 + the share, of Ann.
                       Ans. <  396.92,31+   do.   of Mary.
                            (  295.3S,1()+   do.   of Susan.
                             $1199.99,99  proof.
  Here, add 5.5 dollars, 4.3, and 3.2 together.   By which  divide
the given legacy.  Multiply the quotient  thence arising  by these
sums respectively, the  product will be the share of each.  In like
manner, proceed with the following question.
  13. Three persons enter trade,  and advance different sums of
money ; and at the end of 12 months,  find they have gained  600
dollars; which they divide in the following manner, according to
the stock advanced :  as often as the first has  6 dollars, the second
has 10 dollars, and the third has 12 dollars.  What is the share of
each ?                         ( $128.57,07+share of the 1st.
                          Ans.-}   214.28,44+      do.     2d.
                               (   257.14,13+      do.     3d.
                     ALLIGATION.
  Alligation is the rule; by which we determine what quantity of
any number of simples, whose prices are given, will be required to
form a compound of any proposed value.   And also, having the
several quantities and rates given, we  determine the value or mean
rate of the mixture, composed of those several quantities.

                           case  1.
  When several quantities and their rates are given, to find the
mean rate of the mixture, or  compound:
  As the sum of the several quantities is to any part of the mix-
ture, so is the total value of the compound to its value.
  1. A grocer mixed 6 gallons of wine, at 67 cents a gallon; 7
gallons, at 80 cents; and 5 gallons, at 1  dollar, 20 cents ; what is
the value of one gallon of the mixture ?       Ans. 86.77+tents.
       gall.        cts.   gall.  gall.    cts.
         6x 67=  402     18  : 1 :: 1562  : \ prop.
         7X 80=  560
         5X120=  600
        18~        1562 
                        ALLIGATION.                     241

  2. A goldsmith has 5 lbs. of silver bullion, of 8 oz. fine; 10 lbs.
of 7 oz. fine; and 15 lbs. of 6 oz. fine;  what is the value of 1 lb.
of this  mixture ?                             Ans. 6f oz. fine.
  3. A grocer would mix 12 pounds of tea, at 1 dollar, 75 cents a
pound; 10 pounds, at 1 dollar, 50 cents; 8 pounds, at 1 dollar, 25
cents;  and 5 pounds, at 95 cents a pound ; what will  be the value
of 2 pounds of this mixture ?                        Ans. $2.90
  4. A farmer mixed 20 bushels of wheat, at  90 cents a bushel;
36 bushels of rye, at 75  cents; and 30  bushels of barley,  at  50
cents a bushel; what is the value of 1 bushel of this mixture?
                                        Ans. 697662+cents.
  5. A goldsmith melted 6 oz. of silver, worth 90  cents an ounce,
with 9  oz. worth 86 cents an ounce; what is the value of the mix-
ture?                                         Ans. 87.6 cents.
                           case  2.
  When  the prices of several simples are given, to find how much
of each, at its respective price, must be taken to make a compound
of any  given rate, or price :
  Put  the  prices under each other; and connect one, which  is
greater than the  proposed price or rate with  another, that is less.
Take the difference of ea-ch price, and the mean rate, and place it
opposite  the  price with  which it is connected: these differences
will be the quantities required.

                         EXAMPLES.
  6. A farmer would mix rye worth 75 cents a bushel, with barley
worth 90 cents, and oats worth 48  cents, so as to  have a mixture
worth  65 cents a bushel; what quantity  must he take  of each?
     Ans. 35 bushels of oats, 17 bushels of rye, and 17 of barley.
                          C 90x         17
              mean rate, 65 <? 75))         17
                          ( 48X25 + 10=35
  7- A grocer would mix wine, at 80 cents a  gallon, at 90 cents,
and at  120 cents; so that the mixture could be sold, at 95 cents a
gallon; how much of each sort must he take ?
                                       C 25 gals, at 80 cents.
                                  Ans. ^ 25   do.   90  do.
                                       ( 20   do.   120  do.
  S. A grocer would mix sugar at 12 cents a pound, at  10 cents a
pound,  and at 8 cents a pound ;  so that the mixture may be worth
9 cents; what quantity must he take of each ?
                                       C 1 pound at 12 cents.
                                   Ans. < 1  do.    10   do.
                                       (4  do.      8   do.
            x 2 
242
ARITHMETIC.
  When there are two or more rates, both greater, and both less
than the mean rate, the question is susceptible of several answers,
all equally true ; as in the following examples.  This arises from
the various ways, in which the rates may be linked or connected
together.
  9. A  person  has four  sorts of sugar:  one sort, at  15 cents;
 another, at  12   cents; another, at  10 cents; and a fourth  at  S
cents a pound: he would  mix them in such  proportion, that the
mixture might be worth 11 cents; what quantities of each must he
take?
                 Answer.
                =3 at 15 cents.
                = 1   12  do.
                =1   10  do.
                =4     8  do.
      15 \  =1+3=4 at 15 cents.
      12 A =       3    12  do.
      10'];=       4   10  do.
    (  8V  =4+1 = 5    8  do.
   10. A grocer has several sorts of teas: one at  1  dollar, 50 cents
 a pound;  another, at 1  dollar, 25 cents; a third, at 1  dollar, 15
 cents ; and a fourth, at 100 cents ; and he would  make a mixture
 worth 1 dollar 20 cents a pound; what quantity of each must  he
 take?
      ^"1.50


       [l.OO
11
         Answer.
= 5 at 1.50 cents.
=20   1.25 do.
=30   1.15 do.
= 5   1.00 do.
1.20-
               Answer.
1.50\ =20atl.50 cents.
1.25x\= 5   1.25 do.
1.15'] = 5   1.15 do.
1.00/ =30   1.00 do.
  Other answers may be obtained, by connecting the rates in dif-
ferent ways, as in the preceding examples ; but these are sufficient
for all practical purposes.

                           case  3.
  When the price of several simples, the quantity of one of them,
and the mean rate of the mixture, are  given, to  find the several
quantities of the rest:  proceed as in the preceding case to find the
differences of the several simples, and the  mean rate;  then, by the
rule of  three: As the difference of the same name with the given
quantity is to the remaining differences, so is the given quantity to
the quantities required.
                         examples.
   11   \  grocer would  mix  30 pounds  of sugar, at 14 cents a
 pound, with some at 10  cents, at 9 cents,  and at 8 cents a pound;
A 
ALLIGATION.
243
how much of each sort must be mixed with the 30 pounds, that he
may afford the mixture at 11 cents?  Ans. 15 pounds of each sort.
    14\\\  =1+2+3=6
                                          i prop.
3	6:3:	: 30
3	6=3:	: 30
3	6:3:	: 30
                                         I prop.
                                         4 prop.
  Here 6 is the difference standing against the price of the given
quantity ; which is therefore the first term of the proportion, by the
rule.  The difference standing against each of the other quantities
is  the same, which shows that  equal quantities  are required.  In
like manner proceed with the following examples.
  12. A merchant has several kinds of tea, and he would mix 25
pounds at 1 dollar, 75 cents, a pound, with some at 1  dollar, 60
cents, at 1 dollar, 40 cents, and at 1 dollar,  30 cents a pound; so
that he may afford the mixture at 1 dollar, 50 cents; how much of
each sort is required ?                 C 31^ lbs at $1.30 cents.
                                Ans. 3 12} do.    1.40 do.
                                     ( 12% do.    1.60 do.
   13. A goldsmith would melt 10 oz. of silver of 18  carats fine,
with some at 16 carats, 20  carats, and 24  carats fine, so that the
mixture may be 22 carats fine; required the quantity of each?
                                   ^  10 oz. of 16 carats fine.
                               »   J  10 oz. of 20 carats fine.
                                    5  10 oz. of alloy.
                                    1170 oz. of 24 carats fine.
                  =     -    -     2
                  =     -    -     2
                  =     -    -     2
                  =     -    -     2
                  =22+4+6+2=34
   Alloy is considered of no value.  It is therefore represented by
a cypher.  This  question is susceptible of  one answer only, there
being but one rate greater than the mean rate.
                           case 4.
   When the prices of the several simples,  the quantity to be com-
pounded,  and  the mean rate, are given,  to find  the quantity of
each ;  proceed, as in the preceding cases to find  the differences of
the several simples and the mean  rate ; then, by the rule of three :
As the  sum  of the differences is to the difference opposite each
rate or price, so is the quantity to be compounded, to the quantity
required.
                         examples.
   14. How much sugar at  10 cts. a  pound, at 12 cts, and at 15
 
244                     ARITHMETIC.
cts. a pound, will make a mixture of 20 pounds, worth 13 cents a
pound?                                    f  5 lbs. at 10 els.
                                      Ans. <  5 lbs. at 12 cts.
          C 10  =   -    2                 (10 lbs. at 15 cts.
       13^120|=   -    2
          ( 15^=3 + 1=4     lb.  lb.   lbs.
                       ~~8     8  : 2  : : 20 : £ prop.
   15. A merchant has several sorts of coffee ; namely, at 10 cts,
at 12 cts. at 14 cts, and at 20 cts. a pound ; and he would make a
mixture of 40 pounds, which he could afford at 15 cts. a pound ;
what quantity of each must he take ?          C 15  lbs. at JO cts.
                                      .    J  84 lbs. at I  t cts.
                                     Ans"  ) 8$ lbs. at 1 2 cts.
                                          I 84 lbs. at 10 cts.
   16. A merchant  has three sorts of wine, namely at 75 cts, at
90 cts, and at 1 dol. 20 cts. a gal.; and he would fill a cask con-
taining  70 gals, so that he may afford to sell the mixture at 100
cents a gal.; what quantity of each sort must he take ?
                                        f 18§ gals, at 75 cts.
                                    Ans. < I8§gals.at 90 cts.
                                        (32§ gals, at 120 els.
                 SINGLE POSITION.

  Single Position is the Rule  for solving questions by making
use of a supposed number in the room of the true one ; and  work-
ing with it, as if it was the true number.
  Having proceeded with the  supposed  number, according  to the
conditions  of the question ; then, by the Rule of Three ■ as  the
result of the operation is to the supposed number, so is the given
number to the one sought.
                         EXAMPLES.
   1  A certain sum of money is to be divided among three persons
in the following manner : The first is to have 4 of it, the second \,
and the third, The remainder, which is 250 dols.; what is thejuin^?
l20=supposed number.
40
30
                70
                50=result of the operation.  Now to find the
true number,say as;  50 .120  -250 :  | prop. 
SINGLE POSITION.
245
  Here you suppose 120 to be the number.  Then by the condi-
tions of the question, take % and ^ of it, and add them together;
and take their sum from the  supposed number.   This will be the
result of the operation, and the first term  of the proportion, by the
rule.  The fourth term will be the answer.
  2. A person having a sum  of money, spent % and £ of it; and,
upon examination, found that he had only 60 dols. left;  what sum
had he at first ?                                 Ans. 144 dols.
  3. A person lent a sum of money at 6  per cent, and in 10 years
it amounted to 1600 dollars;  what was the original sum?
                                             Ans. 1000 dols.
  5. A lady being asked her age, replied ; if T67 of my age is mul-
tiplied by 7, and § of my age is added to the product, the sum will
be 292 ; what was her age ?                     Ans. 60 years.
  Suppose her age=30 yrs.; then by the conditions of the ques-
tion T6^=18, which multiplied by 7=126 ;  to which add § of 30,
and the sum will be the result of the operation. For the true num-
ber proceed according to the rule.
  5. A lady having a certain sum of money, told her  daughter,
just going to school, that £, £, and A of it made 74 dols, and
desired her to tell* on returning home,  what sum she had; what
was the sum ?                                  Ans. 120 dols.
  6. There  is  a number, which multiplied by 7, and the product
divided by 6, the quotient will be 14; what is the number ?
                                                    Ans. 12.
  7. A teacher being asked how many scholars he had ; to avoid
a direct answer, and to show  his knowledge of arithmetic, replied ;
if I  had  as many, half as many, and one quarter as many more,
than I now have, I should have 88 ; how many scholars had he ?
                                                    Ans. 32.
  8. Two persons have the same annual salary  or income :  One
saves % part of his, and puts  it out to interest; the other by dissi-
pation, spends  twice as much as  the former, and thus  runs in
debt 120  dols. a year ; what is their income ?     Ans. 360 dols.
  Suppose 240 dollars  the annual income.  Then the  first saves
80 and spends  160 dollars.   The second will spend 320 dollars,
by the  question, or twice as  much as the other.  The difference
between this  sum, and the supposed number, will be  the  result of
the operation.   Whence the true sum may be found by the rule.
  9. A young lady has a fortune in money,  the interest of which,
at 6 per cent, exceeds ^ part of the principal by 100 dols.; what
is her fortune ?                               Ans. 10,000 dols.
  10. Three persons live in the same family, whom I shall call A.
B. and C.  Speaking of their ages, B. said, my age is 1.5 tenths of 
246
ARITHMETIC.
A's age ; C. said my age is twice that of B.; and all together will
make 132 years;  what is the age af each one ?
                                         C 24 years=A 'sage.
                                     Ans. 2 36 yonrs = l»'s age.
                                         £ 7~ years=(,"sage.
  11. A gentleman purchased a chaise, horse, and harness for 240
dols.   The harness was half the price of the horse, and the chaise
was double  the price of both horse and harness;  what was the
price of each ?                            C $26.66,6j,> harness.
                                     Ans. ■)  53.33,33 horse.
                                         ( 160.       chaise.
   12.  Three merchants purchased 1200 dollars worth of sugar in
company, and  paid in the following  manner: The first paid 3
times as much, as the  second, and  the second paid 4 times as
much as the third; how much did each one pay ?
                            ( $847.05,8,82 + =share of the 1st.
                       Ans.}   282.35,2,94+ =share of the 2d.
                            I    70.58,8,23 + =share of the 3d.
                DOUBLE POSITION.
  Double position is a rule, by  which we determine the  true
number or numbers sought, by the use of two supposed numbers.
  Rule:  Suppose two numbers, and work with them according to
the conditions of the  question. Take the difference between the
results of the operation, and the given number ; and call these the
errors.
  Multiply the  first supposed  number by the second  error, and
the second supposed number by the first error, and note the product.
If the results are both greater, or both less than the given number,
take the difference of the errors for a divisor, and the difference of
the products for a dividend; the  quotient will be the answer: but
if one is greater, and the other less than the given number, take the
sum of the errors for a divisor, and the sum of the  products  for a
dividend; the quotient will be the answer.
                          EXAMPLES.
   1  It is required to divide 100 dollars between three persons in
such manner, that the first may have twice as much, as the second,
wanting 8 dollars; and the third, three times as much as the second,
wanting 15 dollars: what is the share of each ?
w     s                            r $20.50=share of the 2d.
                                Ans.)  33.00=do. of the 1st.
                                    f  46.50=do of the  3d. 
DOUBLE POSITION.
247
  1. Suppose the  2d person  to have 12 dollars.  Then  by the
question, the first will have           16    do.
and  the third will have              21     do.
                                   49=the result of the oper.
                                   100 = the given sum.
                                    51=1st error: less.
  2. Suppose the 2d person to have 16 dolls. Then by the question,
the first will have                 24  do.
and the third will  have           33  do.
                                73=the  result  of the  oper.
                               100=given sum.
                                27=2d error; less.
   sup. num. err.              ^ =$20.50, the share of the 2d.
       12 X 27= 324.              33.00, the share of the 1st.
       16x51 = 816.             46.50, the share of the 3d.
diff. of err. = 24    492 diff. of prod.  100.00 proof.
  2. A grazier had in his pasture 1000  head of oxen, sheep, and
horses.   There were 80 oxen more than horses, and  three times
as many sheep, as  horses and oxen taken together; how many had
he of each sort ?                          Ans. 165 oxen.
                                               8 5 horses.
                                               750 sheep.
                                               1000, proof.
  1st. Sup. he had 240 oxen.  Then by the question there must be
                  160 horses.  Three times as  many  sheep as
horses and oxen = 1200
                 1600= result of the operation.
                 1000= the given number.
                 600=lst. error : greater.
  2d. Sup. he had 200 oxen.  Then by the question there would be
                 120 horses. And there
  would be       960 sheep.
                 1280 = the result of the operation.
                 3 000 = the given number.
                 280 = 2d, error : greater.
  Hence the answer may be readily found, as in the preceding
example.                                                   °
  3. A labourer hired  for  forty days~upon this condition, that he
should  receive  20 cents for every day he worked, and forfeit 10
cents for every day he  was idle: At the expiration of the 40 days,
he received 5 dollars; how many days did he work, and how many
days was  he idle ?               A^ C 30 = days he worked.
                                  ' 110 = days he was idle. 
248                     ARITHMETIC.
  1st.  Suppose he worked 25 days.  Then by the question  he
would be  idle 15 days; for both  together,  must equal 40, the
term of his engagement.
       25x20=500
       15X10=150
               3.50=the result of the operation.
               5.00=given sum.
                1,50= 1st error: less.
  2d.  Suppose he worked 20 days. Then he would be idle, by the
question, 20 days.
       20x20=400
       20x10=200
               2.00=the result of the operation.
               5.00=the given sum.
                3.00=2d. error:  less.
   Having found the errors, the answer is easily obtained.
   4. Three ladies contributed 240 dollars for charitable purposes.
 The first gave 24 dollars more than the second, and the third gave
 as much as bath ihe others; how much did each one contribute ?
                                   C $~2 was given by the first.
                              Ans.)  48    do.    by the  2d.
                                   (  120   do.    by the  3d.
   5  Three persons reside in  the same family.  The age  of the
 youngest is 20 years; the age of the  second is equal to that ol the
 firsthand half the age of the third; and the age of the third is equa
 to the sum of the  ages of the other two ;  what are their several
  ° " 5                         $ 60 years the age of the second.
 aSes-                    .Ans7 80  do. the age of the third.
    1st. Suppose the age of the third person to be 60 years.   Then
 by the question,  the age of ihe 2d will be 50 years, being half the
 age of the third,  and the whole of the first.   Thus :
               60=ageof 3d.
               50=age of 2d.
               20=age of the first.
                70=result of the operation, or sum of the ages of 1st
     ageofSdJo                             and 2d persons.
                10=1st error: greater.
     2d. Suppose 50=the age of the third.
                45=the age of the 2d.
                20=the age of the first.
                 65=reSu.tof,heoperalion)orSUmonhe^e,ofol«

                 50                                   l  •
                 f5=2d error: greater. Having the errors their ages
                 XJ  *        b                 are easily found. 
                    DOUBLE POSITION.                 249

  6. A lady bought 15 yards of silk  and lining, for a cloak, for
which she gave #8.40 cents.  The price of the silk was 96 cents a
yard, and that of the lining 36 cents a yard; what was the number
of yards of each?                     Anc  5  5 yds. of silk.
                                     Ans'llO do. of lining.
  7. Two young gentlemen began business  with equal sums of
money.  The former gained in 12 months, a sum equal to £ of his
capital stock; the other, by bad management, lost 540 dollars;
when his stock was just half that of the former; what was the ori-
ginal stock of each ?                         Ans. 1440 dollars.
  1st.  Suppose 600 dols. the original stock. Then by the question
              150 dols.=the gain of the former;
              750 =the amount of his capital.
              600 dollars the capital of the latter.
              540=his  loss in trade.
               60=the remaining stock, which must=£ the cap.
 of the former, 375
              315=lst error: less.
  2d. Suppose 900 dols. the original capital. Then by the question
              225 =the gain in trade of the former.
             1125=the amount of capital of the former.
              900 dollars the capital of the latter.
              540=his loss in trade.
              360=the remaining stock; which equals % of the
other's capital, 562.5
              202.5=2d error :  less.  Having the errors, the an-
swer is readily found.
  8. A lady married a gentleman, whose age on  the day of their
marriage, was  to hers, as 9  is to 3.  After they had lived together
15 years, his age was to hers, as 2 to 1; what  was their respective
ages ?                            Ang  < 45 the age of the man.
                                     ' £ 15 the age of the lady.
  9. A gentleman left  his three daughters, at his decease, 10,000
dollars,  to be divided in the following manner:  the second was to
have 1000 dollars more than the  first; and the third as much, as
both the others; how much  had each one ?
                            C 2000 dollars the share of the first.
                       Ans. 2 3000  do.     do.    of the 2d.
                            ( 5000  do.     do.    of the 3d.
A 
250                    ARITHMETIC.
                 PERMUTATION.

  Permutation is a rule, by which we find the number of changes,
that may take place in any given number of things.
  Rule.  Multiply all the terms of the natural  series continually,
from 1 to the given number, inclusive; the last product will be the
number of changes required.
                        EXAMPLES.
  1. In how many  different ways, can five persons place them-
selves at table, to dine ?                             Ans. 120.
  2. There is a chime  of  12 bells; how many changes may be
rung with them ?                          Ans. 4790016000.
  It would require 45 years, 195 days, and 18 hours, to ring them,
allowing 3 seconds for every round or change.
  3.  In how many different  positions may 8 persons  seat them-
selves at dinner?                               Ans. 40320.
  It would require 110 years, and 142 days to do it.   If there had
been one more, it would require the amazing period of 993.51129 +
vears, to accomplish the changes.
                   COMBINATION.

  Combination is a rule, by which we find how many different
ways a less number may be taken out of a greater number of the
same kind.
  Rule.   Multiply the natural numbers, from 1 up to the number
to be combined, or taken at a time, and  note the product: Multiply
an equal number of terms beginning from the greatest number, out
of which the election is to be made, decreasing by unit or 1, and
note the  product.  The latter product, divided by the former, will
give the answer.
                        EXAMPLES.
   1 How many different ways may 6 letters be  taken at a time
  ,'f11?    J                                   Ans. 462.
out of 11 r
      1X2x3X4X5x6  =720 the product of the first series.
     11X10X9X8X7X6=332640, the product of the 2d series.
                     332640-7-720=462.
   2  How many combinations of 4 letters may be made out of 24
 letters of the alphabet?                           Ans.  10626.
       1X2X3X4=24 product of the first series.
      24X23X22X21=255024, product of the 2d series;  which
 divided by 24 gives the answer. 
        DUODECIMALS, OR CROSS MULTIPLICATION.     251

  3. A general,  being asked what reward he  should have for his
services, replied, that he would be content to receive one cent, for
every file often men, which he could make from a company of 90
men; how much did he receive ?        Ans. $57206454819.03.
DUODECIMALS, OR CROSS MULTIPLCATION.

  This is a rule commonly used for finding the area of superficies,
the  content of solids, and the tonnage of ships;  the dimensions
being given in feet, inches and parts of an inch.
                  The denominations are:
      12 fourths (marked thus, "") make 1 third.
      12 thirds (marked thus,'") make 1 second.
      12 seconds (marked thus, ") make 1 inch.
      12  inches                 make 1 foot.

                      ADDITION.
  This is  performed like Compound Addition, only observe to
carry or add 1 for every 12, to the next higher denomination.
                        EXAMPLES.
      Ft.  In.  "   '"   ""         Ft. In. "  '"  "."
      22   8  6   10  11         45   7  8  9  10
      16   4736         21   6527
      25   3  8    9   3         18   5  2  1  0
Sum,  64   4 10   11   8
                     SUBTRACTION.
  The operation is performed like Compound Subtraction, only
observe to add 12 to the upper number, when it is less than the
lower one;  and  add 1 to the lower number of the next higher
denomination.
                       EXAMPLES.
       Ft.  In.  "  '"  ""         [Ft.  In.  " "' ""
From  48  4   6  8   10         114  7   6 4  3
Take  27  8   9  4    7          71  8  10 7  4
Rem.  20  7"  9  4    3          42  10   8  8 11
                  MULTIPLICATION.
  Place the multiplier under the  multiplicand,  so that  feet may
stand under feet, inches under inches, &c.   Multiply the highest 
252                     ARITHMETIC.

denomination of the multiplier into the lowest of the multiplicand;
observing to place feet  under the feet, inches under the inches,
seconds under the seconds, &c. of the factors, in the several pro-
ducts ; and for every 12 of a  lower denomination, add  1 to the
next higher denomination.
     Observe that:  Feet, multiplied by feet give feet.
         Do.       Feet, multiplied by inches  give inches.
         Do.       Feet, multiplied by seconds give seconds.
         Do.       Inches, multiplied by inches give seconds.
         Do.       Inches, multiplied by seconds give thirds.
         Do.       Seconds, multiplied by seconds give fourths.

                         EXAMPLES.
  1. Multiply 4 feet 8 inches, by 3 feet 6 inches.
        Ft. In.                               Prod.  16 ft. 4 in.
         4   8
         3   6
       14   0=the product of the multiplicand by 3 feet.
        2   4  O=product of the multiplicand by 6 inches.
Prod.= l6   4  0=Ans.
  2. Multiply 9 feet 7 inches, by 6 feet 4 inches.
                                          Prod. 60 ft. 8 in. 4 ".
  3. Multiply 5 feet 2 inches  4  seconds, by 7 feet 5 inches and 8
seconds.                          Prod. 38 ft. 9 in. 9" 2'" 8"".
                Ft. In.  "
                 5  2   4
                 7  5   8
                36  4   4=product of multiplicand by 7 ft.
                 2  1 11  8"'=do of mult, by 5 inches.
                    3   5  6  8"" = prod. of mult, by 8 sec.
Product=38  9   9  2   8
  4. There is a marble slab, whose length is 7 feet 8 inches, and
breadth 2 ft. 4 in.; what is its superficial content?
                                          Ans. 17 ft. 10 in. 8".
  5. The  floor  of a parlour is 17 ft.  9 in. long, and 16 ft. 10 in.
broad ; what is its superficial content?      Ans. 298 ft. 9 in. 6".
  6. The wall on one side of a room is 18 ft. 5 in. long, and 10 ft.
9 in. broad; what is its superficial content ?
                                         Ans. 197 ft. H in. 9".
   7.  A board is 16 ft. 8 in. long, and 1 ft. 7 in. wide; what is its
superficial content ?                         Ans. 26 ft. 4 in. 8". 
        DUODECIMALS, OR CROSS MULTIPLICATION     253

  7. There is a yard, whose length is 47 ft. 10 in, and  breadth
35 ft. 8 in.; what is its area ?             Ans. 1706 ft.  0 in. 8"
  9. A partition wall is 104 ft. 8 in.  long ; and 6 ft.  8 in. high;
what is its superficial content ?              Ans. 697 ft. 9 in. A''.
  10. A carpet is 18 ft. 4 in. long, and 15  ft. 9 in. broad ; what is
its content in square feet ?                     Ans. 288 ft. 9 in.
  12. An oil-cloth  is  12 ft. 10 in. long, and 10 ft. 9  in. broad ;
what is its content in square feet ?         Ans. 138 ft. 11 in. 6".
  When the solid  contents  are  required,  there must be three
dimensions given, namely, length, breadth  and thickness  : which
must be multiplied continually into each other; the last  product
will be the answer : as in the following examples.
  13. A load of wood measures  8  ft. 9 in. long, 3 ft. 5 in. high,
and 3 ft. 10 in. wide ; what are its solid contents ?
             ft. in.                  Ans. 114 ft. 7 in. 2" 6"'.
              8  9=length.
            _3  5=height.
            26  3   "
             3  7  9
            29 10  9=the superficial content of one side.
             3 10    =width.
            89  8  3  "'
            24 10  11  6
114   7  2  6=solid content.
   14. A piece of timber is 12 ft. 10 in. long, 1 ft. 7 in. wide, and
 1  ft. 9 in. thick ; what aie the solid contents ?
                                       Ans. 35 ft. 6 in. 8" 6'".
   15. A bale of goods measures 7 ft.  6  in. long, 4 ft. 3 in. wide,
 and 1 ft. 10 in. thick;  what are its contents ? Ans. 58 ft. 5 in. 3".
   16. A block of marble is 11 ft. 8 in.  long, 4 ft. 5 in. broad, and
 1 ft. 4 in. thick ; what are its contents ?  Ans. 68 ft. 8 in. 5" A'".
   17. A gentleman purchased a pile of wood, the length of which
 was 32 ft. 6 in, its height 4 ft. 10 in, and the length of the wood
 was estimated at 3 ft. 10 in. only;  what  were the solid contents ?
    0  _.  .                              Ans. 602 ft. 1 in. 10".
   18. J he floor of a parlour is 20 ft. 8 in.  long, and 18 ft. 6 inches
wide ; what are its superficial contents? And how many yards of
carpet just 3 ft. wide will be required to cover it ?
                  Ans. \ 382 ftl 4 in-  —area of the floor.
                      '\A2 yds. 1 ft. 5% in. of carpet cover it.
   To find the number of yards of carpet to cover it; divide the
area of the floor by the square feet in a yard, because the carpet is
       Y 2                                            r 
254                     ARTTilMETIC.

supposed to be a yard or 3 feet wide.  Or, you may reduce the 4
inches, which is % of a foot, to a decimal, and annex it to the inte-
ger.  The area, or content of the floor, will then be expressed thus:
382.3333 sq. feet;  which divide by 9, and you have 42.48148 +
yds ; or 42 yds. 1 f. 5% in. nearly.
   19. The floor of a room is 16 feet 7 inches long; and 15 feet 8
inches wide.   A lady wishing to purchase carpet of 8 feet wide to
cover it, desired her daughter to tell her, how many yards she must
purchase; what number of yards were required ?
                                      Ans. 28 yds. 2 ft. 7+in.
   To fino the tonnage of merchant vessels : Multiply the length,
breadth, and depth of the hold continually together, and divide the
product by 95, the quotient will be the tonnage.
   20. What is the  tonnage of a vessel, whose length is 60  feet,
breadth 20 feet, and  depth of hold 8 feet ?       Ans. 101^ tons.
   21. What is the tonnage of a ship 7-1 feet keel, 26 feet beam,
and depth of hold, 11 feet?                    Ans. 222$| tons.
   But in most  cases, when, with the feet, there are parts of a foot
given, it will be easier  to  change those  parts to a decimal, and
annex them to the  integer, and proceed, as in multiplication of de-
cimals.
   22. What is the tonnage of a vessel, whose length is 69 feet 6
inches,  breadth 22 feet 6 inches, and depth  of the hold is 8 feet 6
inches?                                   Ans. 139.914+tons.
                69.5=length of the vessel.
                 22.5=breadth of the beam.
                1563.75
                    8.5=depth of the hold.
              1329T8754- 95=139.914+tons.
INVOLUTION, OR THE RAISING OF POWERS.

  A power  is the product arising from  multiplying any given
number into itself, continually a certain number of times.
  If the number is multiplied by itself once, it is called the second
power, or square of that number, as; 4X4=16, the second power,
or square of 4.
  If it  is multiplied into itself twice, it is called the third power, or
cube of that number ; as : 4X4x4=64, the third power, or cube
of 4.   If it is  multiplied three  times  into itself, it is  called  the
fourth power,  or  biquadrate of that number; as : 4x4x4X4=
256, the fourth power, or biquadrate of 4.
   If it is  multiplied  four  times into  itself, it is called the fifth 
EVOLUTION, OR EXTRACTION OF ROOTS.      255
power, or sur solid of that number;; as: 4x4X4X4x4=1024,
the fifth power, or sursolid of 4;  and so of all other numbers.
  The number denoting the power, is called the index or expo-
nent, of that power.
  The index or exponent of the square, is 2 .•  that of the cube, is
3 • that of the biquadrate, is 4; and so of any other powers.   It is
usually written thus: 42, or  ~D2, which  expresses the square
of 4  or 4X4*. 43> or  "4)*, which expresses the cube of 4, or
4X4X4: 46}  or "^l6, which expresses the  sixth power of 4, or
4X4X4X4X4X4; and so of any other powers.
   YVhen  two or more powers of the same number are multiplied
together,  their product is that power of the number, whose index or
exponent, is the sum of the exponents of the factors;  thus, 16, the
square of 4, multiplied by 64, the cube of 4, is 1024, the 5th power
of 4; or  42 X43 =45, and so  of all other powers.
                        examples.
   1. What is the square, or second power of 6 ?         Ans. 36.
   2. What is the square, or second power of 464 ?  Ans. 215296.
   3. What is the cube, or third power of 6 ?            Ans. 216.
   4. What is the cube, or third power of 25 ?        Ans. 15625.
   5. What is the biquadrate, or 4th power of 5 ?       Ans. 625.
   6.  What is the sursolid, or 5th power of 4 ?         Ans. 7776.
   7.  What is the square, or second power of .75 ?    Ans. .5625.
   8. What is the square, or second power of .5 ?      Ans. .25.
   9. What is the square, or second power of .25 ?     Ans. .0625
  10. What is the square, or second power of £ ?        Ans. TT£,
  11. What is the square, or second power of %?          Ans. \.
  12. What is the square, or second power of § ?         Ans. -£$.
  13. What is the cube of \ ?                           Ans.^.
  14. What is the cube of % ?                           Ans. {.
  15. What is the cube of .25 ?                    Ans. .015625
   It may be proper here to observe, that the power of any fraction,
 whether decimal or vulgar, is always less than its root;  thus, the
 square of % is only £, and the square of .25  is only .0625 ; and so
 of any other fractions.


   EVOLUTION, OR EXTRACTION OF ROOTS.

   That number, which being  multiplied  continually into  itself,
 produces the given power, is called the Root.
   Evolution is ihe  finding of that number, which multiplied into
 itself continually, will make the given sum, or power. It is,  there-
 fore, the opposite of Involution. 
256
ARITHMETIC.
  Those roots, that cannot be exactly found, are called Irrational,
or Surds.
  Those that can be exactly found, are called Rational.   The root
of any power  may be denoted or  expressed, by placing this  J
character before thejxiwer, with the exponent of that power written
over it: thus,  ^27 expresses the  cube  root of 27;  and v^8l
expresses the biquadrate root of 81; and so  of any other roots.
The square root is expressed simply by placing the character with-
out the index, thus:  ^16 denotes the square root of 16.

       EXTRACTION OF THE SQUARE  ROOT.
  Any  number multiplied into itself will produce a square.'  The
extraction of the square root, therefore, is the finding of the num-
ber, which multiplied into itself, will make  the given number.
  Rule :  Point off,  or separate the given sum into periods of two
figures each, if there are more than two figures, beginning at the
unit's place.  Place  the greatest square number in the left hand
period under the said period, and the root in the quotient: sub-
tract the square number from the said period, and bring down the
next period  to  the right hand of the  remainder,  if there is any.
Double  the quotient figure, and place it  on the left hand of the
dividend for a divisor ; find by inspection, or trial, how many times
this divisor will go in the dividend,reserving the right hand figure;
place this in the quotient, and also on the  right hand of the divi-
sor.  Multiply this  divisor  by the last quotient figure,  and sub-
tract the product from the dividend ;  and proceed, as before, until
all  the periods are brought down ; and the whole quotient will be
the root required.
                          EXAMPLES.
   1. What is the square root of 625 ?                  Ans. 25.

                   625(25            25X25=625  proof.
                   4
               45)225
                   225
   Here are three figures; which must constitute two periods.  The
two first on the  right  hand form one period, and the remaining
figure must form another.   The greatest root in 6 is 2, which I
place in the quotient, and its square under the 6,  and subtract  it
from 6.   To the  remainder 2, I bring down the left hand period
25.  I next double the root,  and place it  on the- left hand of the
dividend, and  find how often it will  be contained in 22, reserving
the 5. I find it will be contained 5 times, which I place in the quo- 
EXTRACTION OF THE SQUARE ROOT.        257
tient to the right of the root, already found ;  and also to the right
of the A, already in the divisor. The divisor being thus completed,
I multiply by the  figure last  placed in the quotient, and the pro-
duct, I place under the dividend ;  when, on trial, I find nothing
remains.  The root is therefore, exactly found,  and is a rational
root.
  In like manner proceed in the following examples.
  2, What is the square root of 65536 ?              Ans. 256.

                      65536(256
                      4 •  •
                    45)255
                      225
                    506~)3036
                        3036

  3. What is the square root of 5499025 ?            Ans. 2345.
  4. What is the square root of 470596?             Ans. 686.
  5. What is the square root of 2125764 ?           Ans. 1458.
  6. What is the square root of 23059204 ?          Ans. 4802.
  If the  given sum consists of  a  whole number, and a decimal,
point it off into periods, both ways from the decimal point:  and,
if the decimal does not form  an even  number of periods  of two
places  each,  add a cypher to the right hand.   If, after all the
periods are brought down,  there  is a remainder, decimals may be
added to it;  and  the  operation be continued, at pleasure, or till
nothing remains.
  7. What is the square root of 30138.696025 ?   Ans. 173.605.

                30138.696025(173.605
                1    ....
             27)201
                189
             343)1238
                 1029
            3466)20969
                 20796
347205)1736025
       1736025
  8. What is the square root of 14876.2357?  Ans. 121.9681+ .
  In this example, the integer will form three periods, there being
five figures.  In the decimal part, there will be two periods, there
being four figures.   But there will  be a remainder, after  all the 
258                    ARITHMETIC

periods are brought down.  To which decimals may be added, and
the operation continued at pleasure, or till nothing remains.
  9. What is the square root of 36884 ?         Ans. 192.052+.

                36884(192.052 +
                1 •  •
             29)268
                261
              382)784
                   764
             38405)20.00,00
                    192025
               384102)797500
                       768204
                        29296
  10. What is the square root of 20?             Ans. 4.472+.
  11. What is the square root of 30?           Ans. 5.4772 + .
  12. What is the square root of 40?           Ans. 6.3245 + .
  13. What is the square root of .25 ?                  Ans. .5.
  14. What is the square root of .75 ?             Ans. .866 + .

  To extract the square-root of a vulgar fraction : If the fraction
is not in its lowest terms, reduce it to its lowest terms; then extract
the root of the numerator, and the root of the denominator; which
will be the parts of the new fraction required.
  If the fraction is a surd, reduce it to a decimal, and  extract the
root, as in the foregoing examples.   And, if the given sum  is  a
mixed number, reduce it to an improper fraction,  and proceed as
above; or reduce the fractional part to a decimal, and  annex it to
the whole number, and extract the root of the whole, as in the fore-
going examples.  This latter method will be found, in general, ihe
easiest and most expeditious.
15. What is the square root of J?                    Ans. \.
16. What is the square root of %\ ?                   Ans. I.
17. What is the square root of |6r ?                   Ans. §.
18. What is the square root of §f£ ?                 Ans. -*.
19. What is the square root of \\ ?                   Ans. |,
20. What is the square root of \\\\ ?                Ans. f.
21. What is the square root of \\ ?             Ans. .9128+.
22. What is the square root of \\ ?             Ans. .9574+.
23. What is the square root of j9T ?              Ans. .832+.
24. What is the square root of the mixed number 30| ? Ans. 5\. 
EXTRACTION OF  THE SQUARE ROOT.        259
  3?*=-—.  "*---=—=5%,  by dividing by the denominator.

 "121
  Here, I reduce the given sum to an improper fraction, which
becomes ] f1.  The square root of which is y, or 5%, the answer.
It might have been very readily found  by adding the decimal value
of I to the integer, making 30.25, thus:

               30.25(5.5, the answer, as before.
               25
            105)525
                525
  25. What is the square root of 37|| ?                Ans. 6\.
  26. What is the square root of 272^ ?               Ans. 16%.
  27. What is the square root of 17|| ?                Ans. 4|.
  28. What is the square root of76if ?         Ans. 8.7649+.
                       APPLICATION.
  1. A farmer planted  484 trees  in a square; how many were
there on each side ?                             Ans. 22 trees.
  2. A gentleman formed 400 perches of ground into a lot or gar-
den, exactly in the  form of a square;  how many rods were on
each side ?                                     Ans. 20 rods
  3. An officer has a company of  400 men ;  if he formed them
into a solid square, how  many would there be on each-side ?
                                               Ans. 20 men.
  4. A square pavement contains 1600 square stones of the same
size;  how many are there in one side ?           Ans. 40 stones.
  5. A farmer would lay out a tract of land containing 140 acres,
2 roods, and 20 perches, into a square; how many rods will there be
in one side of it ?                              Ans. 150 rods.
  5. A number of persons gave $12.25 for a charitable purpose;
and  each  one gave as  many cents, as there were persons; how
many persons were there ?                    Ans. 35 persons.
  7- The area or content of a circle is 4276.5 feet; how many
feet must the side of a square be, to contain the same quantity ?
                                         Ans. 65.4 ft. nearly.
  8. The area of a triangle is 144 yds.; how many yards long must
the side of a square be to contain the same quantity ?  Ans.  12 yds.
  9. A gentleman has an  irregular fish pond, containing just 100
square rods. He is desirous to have another exactly  in the form of
a square ;  how long must each side be ?           Ans. 10 rods.
  10. A gentleman is desirious of forming a court yard, contain-
ing 160 rods or 1 acre of ground, and to have it exactly in the form
of a square; how long must one side be ?    Ans. 12.649+ rods 
260
ARITHMETIC.
  To find a mean proportion between 2 given numbers ; multiply
the numbers together, and extract the square root of the product.
  12. What is the mean proportion between 6 and 2 1 ?  Ans. 12.
     24x6=144.  The root of which will be the answer.
  13. What is the mean proportion between 64 and 100? Ans. 80.
  14. What is the mean proportion between 24 and 96? Ans. 48.
  Hence it is manifest, when three numbers are proportional, the
product of the extremes is equal  to the square of the mean, or
middle term.
  To find any side of a right angle triangle, the other two being
given : The sum of the squares of the two shortest sides will equal
the square of the longest side, or hypothenuse ; and the difference
of the squares of the longest side and either of the others, will  be
the square of the third side.  Prop. 47, 1  Book of Euclid.
   15.  There is a right angled triangle, whose base is 8 feet, and
perpendicular 6 feet;  what is the length of the hypothenuse ?
                                                Ans. 10 feet.
         6x6=36
          8x8=64
               100 the square root of which is 10.
   16. There is a right angled triangle, whose longest side is 10,
and base 8 ; what is the perpendicular ?                 Ans. 6.
          10Xl0=100=the square of the hypothenuse.
           8x 8= 64=the square of the base.
                  ~36=the square of the perpendicular.  The
square root of which will be the perpendicular sought.
   In like manner may the base  be found, the perpendicular and
hypothenuse being given.
   17.  The wall of a fortress is 17 feet high, which is surrounded
by a ditch 20 feet from the wall; what must be the  length of a lad-
der, that will reach from the outside of the ditch to the top of the
wall p                                            Ans, 26.2 ft.
   Here the height of the wall maybe considered the perpendicular
of the right angled triangle, and the breadth of the ditch, as the
base; while the ladder will be the hypothenuse.
   18.  On  the margin of a river 24 yds. wide, stands a tree;  from
the top of which a line 36 yards long, will just reach to the  other
side of the stream; what is the height of the tree ?
                                          Ans. 26.83+yards.
   Here  the hypothenuse and  base are given, to find the  perpen-
dicular; which may be done, as example  16.
   19. Two ships sail  from the same  port;  one sails due east 50
miles, and the other due south 84 miles; how far are they from
each other?                                  Ans. 97.75 miles. 
EXTRACTION OF THE CUBE ROOT.           261
  This is  solved as example 15, the hypothenuse of the triangle
being required.
  20. A ladder or pole, 40 feet long, placed  in  the  middle of a
street, will  reach a window of a house on each side of the street 24
feet from the pavement; what is the width of the street?
                                           Ans. 64 feet wide.
  In this example the perpendicular and hypothenuse are given
to find the  base; which is found as example 16, and  doubled will
give the width of the street.

        EXTRACTION OF THE CUBE ROOT.

  The cube of any number is the product of that number, multi-
plied by its square, as: 3x9=27 the cube of 3.
  The extraction of the cube root, is the operation of finding what
number multiplied into itself twice, or multiplied by its square, will
produce the given number.                       •
  Rule.  Separate the given number into periods of three figures
each, beginning at the unit's place.   Find the greatest cube in the
left hand period, and  subtract it from the  said period, placing the
root in the quotient.  Bring down the next period  to the right hand
of the remainder, if there is any.
  Square the root, and multiply it by 3, the index or exponent of
the cube, with the addition oftwo cyphers or decimals, for a divisor.
  Find how many  times this divisor is contained  in the  dividend,
and place  it in the quotient: Multiply the  divisor by this figure,
and note the product; multiply all  the figures  of  the root, except
the last one, by 3, the index of the  given power, with the addition
of a cypher or decimal; and  multiply that  product by the square oL
the  last figure, and place the product under the preceding one; then
cube the last figure in the root, and place  it  under the last ascer-
tained product.  Add these three sums  or products together, and
subtract it  from the dividend.
  Bring down the next  period to  the remainder, and proceed as
above directed, till all the periods are brought down.

                       examples.
  1. What is the cube root of 17576 ?                 Ans. 26.

         17576(26              2x2x3.00=1200 divisor.
    23=  ?___                     1200x6=7200 )
    1200)9576=dividend.      2X 3.0X 6x6=2160 C
          9576  subtrahend.        6x6x6= 2l6\
Z
9576=subt'nd. 
262
ARITHMETIC.
2. What is the cube root of 12812904 ?              Ans. 234.

     12812904(234
23= 8   •   •              2x2x3.00=1200,  1st divisor.
  1200)4812, dividend.          1200x3=3600 ~)
        4167, subtrahend        2x3.0x9= 540 C
 156700)645904, dividend.       3x3X3=_27)
         645904, subtrahend.               4l67=lst sub.

                        23X23X3.00=158700,  2d divisor.
                          158700x4=634800 )
                         23x3.0x16=  11040 \
                             4X4X4=     64)
                                       645904=2d  subtrahend.
  3. What is the cuhe root of 3796416 ?*              Ans. 156.


  *  It may be proper, here, to explain more  particularly this method of
extracting the cube root.  It  is deduced from the binomial  theorem,
with whicli, it is not expected the pupil should, in this stage of his pro-
gress, be acquainted.
  Let a+b be any given root; its cube will be expressed thus: a3+3a2b
4-3ab2+b3.  Now on this principle, let it be proposed to extract  the
root of 3796416.  Here there are three periods, and consequently there
will be three places in the root.  The greatest cube in the first period is
1, which must be cubed and subtracted from  3.  The  13 represents the
first place, or a3 in thetheorem or formula. The remainder is 2; to whicli,
bring down the next period, and we have 2796 for a dividend. Now to
form a  divisor for the next figure in the root, square the root already
found, that is, IX 1=1, or jn the formula a%,  in the second place. Multi-
ply this by 3, the coefficient of a2, with the addition oftwo cyphers, whicli
hiiswer no other purpose than  to remove  the product  two places farther
to "the left hand;  and, we  have the divisor 1X1X3.00=300.  This will
be contained in the dividend 5 times, which will be the second figure in

  Now to obtain a subtrahend, or sum to be subtracted from the dividend,
multiply the divisor 300,  by 5, the last  ascertained root, and we have
1500, represented in the formula by 3a2b.  Again  multiply 1, the first
figure in the root by 3, the coefficient of a  in the third place of the
formula, with the addition of one cypher, which only serves to remove
the product one place farther toward the  left hand.  This product is mul-
tiplied by 5 X5, or the square of the second figure in the root,  represent-
ed in the formula by b* in the third place.  To Uiese  two products, add
the cube of 5; or 5X5X5, which in the formula is represented by b3, in
the fourth place.   The sum of these three products is 2375, the first sub-
trahend.                     .....      ,       • •      -i
  The  remainder is 421; to which, bring down the  remaining period,
and we have 421416, for the 2d dividend.       ,.,,,.
  Now,  for a second divisor,  square the root already found, that is,
15X15', expressed in the second place of the formula  by a*, and multi-
ply by 300, for the reason already given,  and the second divisor will be 
EXTRACTION OF THE CUBE ROOT.         263
   In like manner may the cube root of any other number be found.
 If the given power consists of a whole number, and a decimal, you
 must point off both ways from the decimal point; and  if there are
 not decimal places sufficient to constitute one or more periods of
 three places each, as the case may be ; then add so many decimal
 places to the right hand, as will make the periods of three figures
 each ; and proceed, as in the above examples, to find the root.
   4.  What is the cube root of 846045J9?             Ans. 439.
   5.  What is the cube root of 94818816?             Ans. 456.
   6.  What is the cube root of 970299 ?                Ans. 99-
   7.  What is the cube root of 2000376 ?               Ans. 126.
   8.  What is the cube root of 15926.9725 ?        Ans. 25.16+.
   If, after  all the periods are brought  down,  there should be a
 remainder, the operation may be  continued by adding decimals,
 constituting one or more periods of three places each.
   9.  What is the cube root of 96 ?                 Ans. 4.56+.

         96(4.56+
     43=64               4X4X3.00= 4800, 1st divisor.
    4800)32.000            4800x5=24000 }
          27 125          4X3.0X25= 3000 £
    607500)3875.000         5x5x5=  125)
           3693 816                   27125, 1st subtrahend.
             181184 Rem.   45x45X3.00=  607500, 2d div.
                              607500x6=3645000 )
                             45x3.0x36=  48600 V
                                6x6x6= __216)
                                          3693816, 2d sub.
   10.  What will be the cube root of  160, the decimal  being con-
 tinued to three places ?                          Ans. 5.428+.

67500.  The quotient figure will be  6, which will complete the root
tor the subtrahend, multiply this divisor (that is, the product of 15X15
ZJV^ S7tmd P'T °f the f°rmula ^ 3, the coefficient of" *&
the addition of two cyphers) by the root  last found  tK,t • u *   u•  ,
.ill .*. 40MM.   ILU ,5. ,^1'pSSv'fo   ' tt'S
         ■rhlal\u......•>« i    ,,  u DV °° in tfie last pace of the



brought do,™, th=reiSafera.i„d",  th"  deVfa, «   bKT 
264                     ARITHMETIC.
  To extract the cube root of a vulgar fraction :
  If the fraction is not in its lowest  terms, reduce it to its lowest
terms, and extract the root of the numerator and of the denomina-
tor, for the  parts of the new  fraction.  If the fraction is a surd,
reduce it to a decimal, and extract the root.
  A mixed number may be reduced to an improper fraction,  and
the root extracted as above ; or the fractional part may be reduced
to a decimal, and  connected  with  the  integer, and  the root ex-
tracted.
   11.  What is the cube root of ^ ?                    Ans. \.

             ^8=2_
            "^/27=3~
   12. What is the cube root of ffr ?                    Ans. 4.
   13. What is the cube root of T3TW ?                  Ans- 8-
   14. What is the cube root of £ffc ?                  AnsV*-
   15. What is the cube root of f ?                   Ans. .763.
   16. What is the cube root of 13§ ?             Ans. 2.3908 +
   17. What is the cube root of 7|?               Ans- 1-966+
   18. What is the cube root of 9 J-?              Ans. 2.092+ .

                        APPLICATION.
   1.  What is the length of one side of a cubical vessel, which con-
 tains 46656 solid inches ?                       Ans. 36 inches.
   2.  A building in a cubical form contains 21952 feet; what is the
 length of one of its sides ?                            Ans- 28 "•
   Rule.—The solid contents of similar figures are to each other,
 as the cubes of their similar sides or diameters.
   3  If a'globe of silver, 4 inches in diameter is worth 200 dollars,
 what will be the value of one, 10 inches in diameter r^      ^

   4. If a globe of water 6 inches in diameter will weigh 1.5
 pound ; what will be the weight of one 24 inches ?    Ans. «,6 lbJ.
 V  5.  A ball of lead 2 inches  in  dtameter we.ghs 1.5 lb..,.what
 will be the  weight of another 10 in. in diameter ?   Ans. 187* l_os.
    6   A gentleman wishing to  build a cubical  cistern contam-ng
 729cubic feet of water ; how long must he make one of ns „ta ?

    7  \ gentleman  has an ingot of silver in  the form of a cube, one
  side'of wh.  h  is 4 inches,  which  he  values at 250idols.; wha
  nmst be the length of a  cube of the  like metal,  whjch^hjU  be
  worth 4 times as much ?
    Here you cube the side of the given cube, winch is 4, and mul-
  tiply that  by 4, the  proportion between  the given value  of the 
ARITHMETICAL PROGRESSION.            265
given cube, and that of the one sought.  The cube root of the pro-
duct will be the side sought. See the rule under the second example
of this.
  8.  A gentleman has a cubical vessel, whose side is 2 feet,  and he
is desirous to make another of the same form, which shall contain 3
times as much ; what must be the length of one side ?
                                           Ans. 2 ft. 10§ in.
         ARITHMETICAL  PROGRESSION.

   Arithmetical progression is a series of  numbers increasing
 or decreasing by a common difference.  Five particulars are to be
 observed, viz.
       The first term of the series }  ,,  ,
       The last term of the series J called extremes-
       The common difference.
       The number of terms;  and
       The sum of all the terms, or sum of the series.
   Any three of them being given, the other two may be found.
   When the first term, the common difference, and the number of
 terms are given, to find the last term, and the sum of all the terms:
   Rule  1.  Multiply the number of terms, less 1, by the common
 difference,  and to the product  add the first term ; the  sum will be
 the last term.
   Rule  2.  Add the first term, and last  term together, and mul-
 tiply this sum by the number of terms;  half the product will be
 the sum of all the terms.
   When  the  two extremes and the number of terms are given, to
 find the common difference :
  Rule 3.   Divide the difference  of the extremes by the  number
 of terms,  less 1 ; the quotient will be the common difference.
  When  the first term, sum of  the series, and number of terms  are
given to find the last term, and common  difference; or when  the
last term, sum of the series, and number of terms  are given to find
the first term, and common difference :
  Rule 4.   From twice the sum of the series, subtract  the product
of the first term or last term, (as the case may  be,) and number of
 erms; and d,vide the difference by the number of terms, the quo-
tient will  be the last term, or  first term, (as  the  case may be.)
ruteSIF        Mtren,es> the common difference may be found by

            z 2 
266
ARITHMETIC.
  When the first term, last term, and common difference are given,
to find the number of terms :
  Rule 5.  Divide the difference of the extremes, by the common
difference, the quotient+1, will be the number of terms.

                         examples.
  1. A lady purchased 20 yards  of muslin, and  agreed to give 1
cent, for the first yard, 3 cents for the second yard, 5 cents for the
third yard, increasing two cents for every yard ; what did  she pay
for the last yard, and what did the whole amount  to?
                                 .    $39 cents for the last yd.
                                Ans-£$ 1.00 the amount.
  Here 1st. term= 1 ^   By rule 1st.  20—1 = 19
  Num. of terms=20 >                        2 com. dlf.
  Com. diff.   = 2 )                       38
                                             1= 1st term add.
                                            39=last term.
   Having found the last term, the sum of all the terms is readily
found by rule 2d.
   2.  How many strokes, does the hammer of a clock strike in 12
hours, beginning at 1 o'clock ?                        Ans. 78.
  Here are given, the 1st term=  1 ^
                 the last term=12  >
          the number of terms=12  )
   The answer is found by rule 2d.
   3.  A labourer was hired to load a wagon with cabbages, placed
 in a row, at the distance of a yard from each other ; how  far must
 he travel to bring 100, one at a time, to the wagon, which is placed
 a yard from the first one ?                     Ans. 10100 yds.
   Here we have 1st term=  1  )     By rule 2d.
            the last  term = l00  V-  100+1 = 101 sum of extremes.
          number of terms=100  )           __50
                                          5050
                                             2
                                        I0l00=distance.
  Here you multiply the sum of the extremes by 50, half the num-
ber of terms; or muhiplv by the whole number of terms, and divide
by 2.  This product multiply by 2, because he travelled the  dis-
tance twice over, once in going to the wagon, and once in return-
ing after the next cabbage.   This you may reduce to  miles by di-
viding  by the vards in u mile, viz. 1760, and you will find the dis-
tance to be 5 miles, 5 furlongs, and 200 yards.
   If the cabbage had been placed at 2  yards  distance, and tire 
ARITHMETICAL PROGRESSION.            267
wagon at  2 yards  from the first one, you  must have  multi-
plied by 2,  because he travelled the distance twice; and that pro-
duct  again  by  2,  because they were placed at the distance of 2
yards from each other, to obtain the answer in yards.  In that case
the distance would have been 11  miles, 3 furlongs, and 180 yards.
  4.  A merchant sold  100 yards of cloth, at  1 cent for the first
yard, 10 for the second, 20 for the third, and so on increasing 10
cents on every  yard ; what did he receive for the last yard—and
what did the whole amount to?  .    $ $9-91, price of the last yd.
                                 I $A96, whole amount.

  Here the 1st term=l   ) By rule 1st.  100—1=99
  the num. of terms=l00 >                        10
  com. diff.       =10 j                      990
                                                  1 1st term.
                                               991=lastdo.
  Having found the last term, the answer will be easily found by
rule  2d.
  5. A merchant bought 100 yards of cloth, and, by agreement,
was to give 10 cents for the first yard, 20 cents for the second, and
so on, increasing 10 cents on every yard; what was  the price of
the last yard—and what did the whole amount to?
                                    .    ^ Last yard=tO dol.
                                       ' £  Amount=505 dol.
  Here the 1st term=l   ~) By rule 1.   100—1=99
      num. of terms=l00 >                      10 com. diff.
com. diff.    =10
990
                                             _10
                                             1000 last term.
  Now, having found the last term, the sum of the series is found
by rule 2.
   First term=  10
LaStterm = 1000CTI,esum==101°
50= % num. of terms.
505.00 cents.
  It may here be proper to observe, that the result in this example
is different from that in the  preceding, owing  to  the difference in
the price of the first yard of cloth.   To obtain a true answer, you
must add the price of the fust, yard, to wit, 10 cents in the room of
a unit, or 1 for the last term.  And, in the latter part of this ex-
ample, you must consider the^n'ce of the first term, and not a unit
or 1, as the first term. 
268                     ARITHMETIC.
  The same observation will apply to olher examples of a similar
kind.
  6. A merchant purchased  54 yards of cloth, and gave 20 cents
for the first yard, 50 cents for the second yard, and so on, increas-
ing 30 cents on every yard ; what was the price of the last yard,
and what was the amount of the whole ?
                              .    $ The last yard cost $ 16.10.
                                   I Whole amount $440.10.
  7. A lady hired a servant for 12 years, and by agreement was
to allow her 12  dols. for the first year, 14 dols. for the second year,
16 dols. for the third year, and so on, increasing 2 dols. in every
year;  what were  her wages for the  last year—what  were her
wages on an average, and what was the whole amount ?
                           ( She received the last year, 34 dols.
                       Ans. < Whole sum received,  276 dols.
                           £ Annual average, 23 dols.
   8.  A person has 10  children, whose several ages differ  alike
from each other.   The  eldest  is  48  years, and the youngest is 3
years old ; what is the common difference of their ages ?
                                               Ans. 5 years.
     48—3=45, which divide by 10—1=9:  the answer is found
by Rule 3.
   9.  A person proposes to travel from the city of New York to
a certain pUce in 19 days ; to  travel  6 miles the first  day, increas-
ing by an equal excess until  his last  day's journey shall be 60
miles ;  what is the increase—and what is the distance, which he
will travel ?                      A   $ daily increase 3 miles.
                                      \ whole distance  627 do.
   Extremes,     60—6=54                         n    ,
           '      ^      —=3 the common difference. By rule 3.
   No. of  terms,  19—1 = 18                           J
          60+6=66x19=12544-2=627-  By rule 2d.
   10. A lady being asked how many children she had; to avoid a
direct answer replied, lhat her youngest child was 4 years old, and
her eldest was just 32  years  old, and that the difference of  their
ages was  just 4 years ;  how  many children had she ?      Ans. 8.
   11. When  the extremes  are  3 and 45, and the common dif-
 ference 6; what is the  number of terms?                Ans. 8.
   Here 45—3=42-=-6=7+l=8.   By rule 5th.
   12.  A gentleman on a tour went three miles the first day, and
 increased every day by 5 miles,  till  on the  last day, he travelled
 58 miles ; how many days did he travel ?         Ans. 12 days.
   13.  A gentleman, the first year he kept house, expended $500, 
ARITHMETICAL PROGRESSION.            269
but afterward increased his expenses by an equal sum ; and at the
end of 12 years, found that he had expended altogether the sum of
12600 dollars ;  what was the annual increase of his expenses, and
how much did he spend the last year ?
                              3 He spent the last year $ 1600.
                         Ans' I Annual increase $ 100.
    1st. terras $500 )        12600 x  2=25200
sum of series=l2600 [ given.    500X12= 6000
  num. terms=    12 )                    19200-7-12=1600
last term, by Rule  4th.  Having  the two extremes, the common
difference is found by rule 3d.
   14. A gentleman found his family expenses increase in arithme-
tical progression, and, at the end of 15 years, his  whole expenses
amounted to 18,300 dollars; and his expenses  the last year were
1500 dols.; what  was the annual increase, and what did he spend
the first year ?                 A no  $ $940, sum spent 1st year.
                                  I   $40, common difference.
   I8300x  2=36600, the product of the series  bv 2, by Rule 4.
    1500X15=22500, prod, of the last term and number of years.
               14100-j-15=940, the sum spent the first year.
   Having the  two extremes, the common difference is found by
Rule 3d.
   15. A merchant began business with 3500 dols. and added 600
dols.  annually to his capital; in what  time would it amount to
 12500 dols.?                                  Ans. 16 years.
     12500 — 3500=9000-7-600=15 + 1 = 16, by Rule 5th.
   16. A young gentleman of fortune, on his arrival at the age 21
years, commenced by spending  a certain sum; and by increasing
his expenses annually  in arithmetical progression for 13 years,
found that he spent 2500 dollars  the last  year ;  and, in the whole,
that he had diminished his estate  22750  dollars ; what did he
spend the first year, and what was the common difference ?
                                      .    5 $1000.
                                     Ans'l  $125  com. diff.
   22750X 2=45500=prod. of sum of the series by 2. Rule 4th.
    2500xl3=32500=prod. of last term, and number of years.
               13000-r-13 = l000=sum spent the  first year.
   Having the extremes, the  common  difference is found by Rule 3.
   17- A young gentleman began his  career  by  spending  the first
year after he came of age, 500 dols.,  and increased every year by
an equal sum ;  and in 12 years found that he had  spent, in all, the 
270
ARITHMETIC.
sum of 9000 dols.; what did he spend the last year, and what was
the common difference ?  .    ^ $1000, the sum spent the last yenr.
                       ' "S" t $45.45,45+ the com. difference.
         9000 x  2=18000
           500X12= 6000
                     1200-7-12=1000 sum spent the last year.
Having the extremes, the common difference is found by Rule 3.
         GEOMETRICAL PROGRESSION.

  Geometrical Progression is a series of numbers, increasing
by one continual multiplier, as : 2, 4, 8,  16, &c. which increases
by the continual multiplication of 2 :  or decreases by one continual
divisor, as : 32,16, 8, A, 2, which decreases by the continual divi-
sion of 2.
  In every  geometrical series, five particulars are to be observed :
  1. The least term.
  2. The greatest term.
  3. The common ratio, or number, by which the series increases
or diminishes.
  4. The number of terms.
  5. The sum of all the terms, or series.
  Any three of these five being given, the other two may be found.
  When the first term, the ratio, and number of terms are given to
find the last term :
  Rule 1.   When the first  term and ratio are alike; raise the
ratio to that power indicated by the number of terms; the last pro-
duct is the term required.
  Rule 2.   When the first term and ratio are different; raise the
ratio to that power denoted by the number of terms less 1, and mul-
tiply this power by the first term; the  last product  is the term
required.
  When the first term, last term, and ratio are given, to find the
sum of  the series :
  Rule 3.   Multiply the last term by the ratio; from  the product
subtract the first term; and divide by the ratio, less  1  or unit;
the quotient will be the sum of the series.

                         EXAMPLES.

   1. The first term being 2, the ratio 2,  and the number of terms
 16; what is the last term of the series ?             Ans. 65536. 
GEOMETRICAL PROGRESSION.            271
                2  4
Num. terms=l6 ")
First term =  2 >
Ratio     =  2 )
  Therefore, use rule 1.
8  16   32  64=6 power of the ratio.
            64
           256
          384
          4096=12 power of the ratio.
             16=4 power of the ratio.
24576
4096
                                65536=16 power of the ratio.
  2. What will be the last term of the series of  18 terms, the first
term being 3, and the ratio three ?             Ans. 387420489.
                  3   9  27   81  243=  5 power of the ratio.
Num. terms=l8 ^
First term  =  3 >
Ratio      =  3 )
  Therefore, use rule 1.
243
59049=10 power of the ratio.
  343
                            14348907=15 power of the ratio.
                                   27=  3 power of the ratio.
                           387420489 = 18 power of the ratio.
  Here, it is obvious that all the powers of the ratio  are not pro-
duced.  That would have been  quite unnecessary.   Having pro-
duced any convenient power of the ratio, multiply it by itself, and
you will have the power of the ratio denoted by double  the former
power ; and so on, until you find the power denoted by the number
of terms.  Thus, in ihe first example, by multiplying the 6 power
of the ratio by itself, you  have the 12 power; and this again by
the fourth power of the ratio, and you have the l6lh power of the
ratio, the power sought.   In the same  manner proceed with  the
following example, and all others of the like kind.
  3. The first term being 4, the ratio 3,  and  the number of terms
10; what is the last term ?

Num. terms=10 ~)
First term  =  4 >
Ratio      =  3 )
   Therefore, use rule 2d.
                       Ans. 78732.
3  9   27  81=4 power of the ratio.
         _81
         6561=8 power of the ratio.
            3 = the ratio.
        19683=9 power of the ratio.
            4  the first term.
                                73732=lhe last term.
  4. A farmer sold 24 sheep, and, by agreement, was  to receive
for the whole the sum that the last one came to, reckoning 2 cents
for the first sheep, 4 cents for the second, and so on, in geometrical
progression ; what sum did he receive?        Ans. $167772.16.
  5. A wealthy gentleman having his family at dinner, consisting 
272                     ARITHMETIC.
of 10 children, proposed to make each of them a present of a sum
of money ;  the youngest to  have 10 cents, the next 30, and so on
in triple proportion;  how much did the eldest receive ?
                                              Ans. $1968.30.
                         3  9   27  81=4 term of the ratio.
Num. terms=10 ^                   81
First term  =10 >                   81
Ratio      = 3 )                 648
  Therefore, use rule 2d.           6561=8 term of the ratio.
                                     3   the ratio.
                                 19683=9 power of the ratio.
                                     10
                                1968,30=the last term.
  6. A merchant sold 24 yards of cloth, at 2 cents for the first
yard, 4 cents for the second, 8 for the third, and so on in duplicate
proportion ; how much did the cloth amount to ? Ans. $335544.30.
  Having found the last term by rule I, find  the sum of all  the
terms by rule 3d., which will be the answer in  cents.
  7. A lad, ignorant of progression, agreed  to  buy 18 oranges,
and to pay, for the  whole, the  price of the last  one, reckoning 1
cent for the first, 4 cents for the second, and so  on in quadruple
proportion ; what sum would he have to  pay for the oranges—and
how much would be the average price ?   .    $  $171798691.84.
                                       A   I $9544371.77.
  Here the ratio and first term are different; therefore, use rule 2d.
  Ratio=~4]l7=l7179869184=the last term of   the series;
which, divided by 18.  will  give the average  price=954437l77
cents, nearly.  In this example, the first term being unit or 1, it is
not necessary to multiply by it.
  8. A lady agreed to purchase 15 yards of silk, for a  dress for
her two daughters, and imprudently bargained  upon these terms ;
to pay for the 1st yard 4 cents, for the 2d, 16 cents,  for the  3d,
64 cents; and so on, in quadruple proportion ; what  sum did the
whole amount to ?                        Ans. $14316557-64.
   Here the ratio and first term are different; the  last term is found
by rule 2d.   And the sum of the series is found by rule 3d.
  9. A wealthy gentleman married his daughter on a new year's
day, and presented her with 5 dollars, or half  an eagle, and pro-
raised to double the same on the first day of each month during that
year, as her marriage portion, which the young bride readily ac-
cepted, being acquainted with  the  principles  of geometrical pro-
gression ; what sum  did she receive?              Ans. $20475. 
GEOMETRICAL PROGRESSION.
273
  Here the first term is 5, the ratio 2, and the number of terms 12.
The last term is found by rule 2d, and the sum of the series by
rule 3d.
  10. A gentleman having a valuable horse, proposed to sell him
upon the following terms, (to wit) 1  mill for the first  nail, 2 mills
for the second nail, 4 mills for the third nail, and so on through the
32 nails, which he had in his 4 shoes ; a jockey considering him a
great bargain, readily accepted the terms; what sum did the horse
cost him ?                                Ans. $4294967.29,5.
  Here the first term and ratio are different, the last term is found
by rule 2d ;  and the sum of the .series by rule 3d.
  11.  If the posterity of Noah, who  were 6 persons,  at the flood,
doubled in 20 years; how many inhabitants would there be in the
world 2 years before the death of Shem, who lived 502 years after
that event ?                           Ans. 201336586 persons.
  Here the  first term is  6, the ratio 2, and the number of terms 25,
or 5_o_o#  The last term  is found by rule 2d.
   12. A  gambler lost  5  dollars  at the  first game of cards, and
doubled the stake each  game  in the hope of recovering what he
had lost; but the game was  always against him, and the last one
he  played, he lost 5120 dols.; how much  did  he lose in  all, and
how many games did he play ?      .     $ He lost $10235.
                                        £ Number of games  11.
   Here the ratio is 2, the first term 5, and the last term 5120. The
 sum of the series is found by rule 3d.
    13. A manufacturer introduced a new  branch of business into
 the country, and  took  5  apprentices for 7 years; after which he
 took 5 more,  and his  former apprentices took each  five, for the
 same length of time, and so on.   Now, supposing all, who learned
 the business, were to be alive at the end of 42 years, what would
 be the number?                                    Ans. 9331.
               Here the first term=l
                  Number of terms=6
                        And ratio=6 ; for 42 years divided by 7
 gives  6.  And the 5 apprentices, with their masters, also make 6.
 The last term may be found by rule 2 ;  and the sum of the series
 by rule 3.
    14. A labourer engaged to work for a farmer for  12 days, at 1
 cent for the first  day, 3 for the second, 9 for the third, and so on,
 in geometrical proportion ; what was the amount of his wages for
 the time ?                              Ans. $2657.19 nearly.
    Here the first term and ratio are different: the last term is found
 by rule 2d, and the sum of the series by rule 3d. 
274
ARITHMETIC.
  In a decreasing geometrical  series, when the  first term, ratio,
and number of terms are given,  to find the last term :
  Rule 1.  Divide the first time by the ratio, raised to the power
denoted by the number of terms, less 1; the quotient will be the
last term.
  When the  first term, last  term,  and  ratio are  given, to find the
sum of the series :
  Rule. 2.  Multiply the first  term of the decreasing series by the
ratio: from the product subtract the last term, and divide the remain-
der by the ratio, less 1; the quotient will be the sum of all the terms.
   15.  What  is the last term of a  decreasing geometrical series,
whose first term is 128, number of terms 4, and ratio 2 ?  Ans. 16.
   16.  What  is the  last term of a  decreasing  geometrical series,
whose first term is 729, ratio 3, and number of terms 5 ?  Ans. 9.
   17.  What  is the  last term of a  decreasing geometrical series,
whose first term is 4096,  ratio  4, and number of terms 6? Ans. 4.
   These examples are performed by rule 1, and all others of the
same kind.
   18.  In a decreasing geometrical series, whose first term is 64, ratio
2, and last term 4; what will be the sum of all the terms ? Ans. 124.
   19.  In a decreasing geometrical series, whose first term is  729,
ratio 3, and last term 9 ;  what will  be the sum of all the terms ?
       '                                           Ans. 1089-
   20. In a decreasing geometrical series, whose first term is 4096,
ratio 4, and last term 16; what will be the sum of all the terms r
                                                  Ans. 5456.
   These examples are performed by rule 2 of this : and all others
 of the like nature.
   21  A certain number of persons settled in  a country, and after
 a period 220 years, their number were 206848.   Now, supposing
 thev doubled their number every 20 years ; how many were ihere
 at first?                                          An*-im'
    Here the first term of  the series  is 206848,  the ratio 2, and the
 number of terms 11, or \%°.  Solved by rule  1st of tins.
   ANNUITIES AT  COMPOUND INTEREST.
  An annuity is any sum of money payable yearly, half yearly, or
quarterly, for a number of years, for life, or for ever.  It draws in-
Kifit is not paid, when due.  If it remains unpaid  tor any
length of time, compound interest is computed.
  When the annuity, time, and rate  per  cent, are given to  find
the amount: 
           ANNUITIES AT COMPOUND INTEREST.        275

  Rule.—Involve the ratio to the power denoted by the time ;
from which take unit or 1; the remainder, divide by the ratio, less 1
or unit; the  quotient  multiplied  by the annuity  will give  the
amount, when the payments are made yearly.
                         examples.
  1. What will an annuity  of 30  dollars, amount to in 4 years,
at 5 per cent.?                             Ans.  $129.30,375.
  Here the ratio is 1.05, which, involved to  the  fourth power, is
1.21550625.  From  this  take unit,  or 1 ;  the  remainder is
.21550625.  This divide by the ratio, less 1 or .05 ; and multiply
the quotient by 30, you  will  have the amount of the annuity for 4
years, thus:
                  .05).21550625
                       4,310125
                              30
                       129.30"^3775=the amount.
  2. What will a pension  of 96 dollars a year  amount- to, in 7
years,  at 6 per cent. ?                      Ans. $805.80,8352.
  Here the ratio  involved to the  power denoted by the time, is
1.5036202 ;  with which proceed, as in the above example, or as
the rule directs.
  3. A gentleman owns a ground rent of 84 dollars  a year; what
will it amount to,  at 5% per cent., if it remains unpaid for 6 years ?
                                         Ans. $578.59,6284.
  Here the  ratio is,  1.055, which involved to  the 6 power, is
1.3788426, with which proceed, by the rule, to find the amount.
  4. A lady has  an annual income of 500 dollars, arising from
ground rent; what sum would it amount to, at 6 per cent., if  not
paid in 4 years ?                            Ans. $2187-30,8.
  When the amount, rate per cent., and time are given, to find the
annuity:
  Rule.  Involve the ratio to the power  denoted by the time,
from which take unit or 1, for a divisor: Multiply the amount by
the ratio, less unit or 1, for a dividend; the quotient arising from
the division will be the annuity.
                         examples.
  1. What annuity being unpaid  for  4 years,  will amount to
$129.3036, at 5 per cent. ?                          Ans. $30.
                  129.3036=amount of the annuity.
                        .05=the ratio less 1, or unit.
                  6.465160
  Which divide by .21550625, the ratio involved to  the time, less
unit or  1.  The quotient will be the answer, by the rule. 
976                     ARITHMETIC.

   2. What  annuity remaining unpaid for 9 years, will amount to
$1984.78,152, at 5 per cent. ?                        Ans. 180.
   3. What  annuitv will amount to $578.59,6284, in 6 years, at
5% per cent.  ?                                       Ans. $84.
   When the annuity, amount, and rate percent., are given, to find
the time :
   Rule.  Multiply the amount by the ratio, less unit or 1.   Di-
vide  the  product by the annuity;  and add unit or 1 to the quo-
tient ;  which will  be that  power of the ratio, denoted by the time.
Divide this continually by the ratio, until the quotient is unit or 1;
the number of divisions will show the time.
examples.
  1. In what time will 30 dollars amount to, #129.30,75, at 5 per
cent. ?                                           Ans- 4 vears-
                  I29.30375=the amount.
                         .05=ratioless unit or 1.
                  6.465l875-r-30=.21550625 :  to which add-
ing unit, or  1, you have 1.21550625.  This sum divided  by 1.05
continually, till the quotient  is 1 or unit, the number of divisions
will be the time, by the rule.
   2. In  what time  will a pension  of 96 dollars  amount to
$805.80,8,35, at 6 per cent. ?                     Ans. 7 years.
   When the annuity, time and rate per cent, are given, to  find the
present worth:
   Rule.   Involve the ratio to the time, and by it divide the an-
nuitv.  Subtract the quotient from the annuity, and divide the re-
mainder by  the ratio less  1.   The quotient  will  be the  present
worth.
                          examples.
    1. What is the present worth of an  annuity of 30 dollars a year,
 to continue 5 years,  at 4 per cent. ?          Ans. $133.55,4675.
    Ratio=1.04 involved to the time=1.2l66529 : by this divide
 30 the annuity, and you have 24.657813 + , which take lrom the
 annuity, and there will remain 5.3421 87-  This divide by  .04, and
 the quotient will be the answer.
    2  What is the present worth of a ground rent of $588.86,5, for
 4 vears, at 5% per cent. ?                 Ans. $2064 05,426+
    ~3   What is the present value of a pension  of 64 dollars, to con-
 tinue 5 years, at 6 per cent. ?                Ans. $269-59,104.
    When the present worth, time and rate  per cent., are  given, to
 find the  annuity: 
ANNUITIES AT COMPOUND INTEREST.        277
   Rule 5th.  Multiply the present worth by the ratio involved to
the time;  and this product by the ratio, less 1  or unit, for a divi-
dend : divide by the ratio involved to the time, less unit or 1 ; and
the quotient will be the annuity.
                         examples.
   1. What  annuity,  to  continue 5 years, may be purchased  for
$133.55,4,675, allowing 4 per cent, interest ?         Ans. $30.
                 f/)4J5 = 1.2l66529, the 5th power of the ratio:
whicli multiply by the present worth; and this again by .04, the
ratio less unit or 1.   Then proceed as the rule directs, to find the
annuity.
  2. A gentleman, having in ready money $2064.05,426+, wishes
to purchase an annuity for 4 years, allowing 5£ per cent.;  how
much can he purchase ?                      Ans. $588.86,5.
   3. A lady, having  the sum  of 5000 dollars, is desirous of pur-
chasing an annuity to continue 6 years, allowing  5  per cent, inte-
rest ; what sum can she purchase ?          Ans. $985.08,756+
   When the annuity, present worth, and ratio are given, to find
the time:
   Rule 6th.  Multiply the present worth by the ratio, less unit or
1 : subtract the product  from the annuity, and divide the annuity
by the remainder.  The quotient will be the power of the ratio in-
volved to  the time :  which divide by the ratio  continually, till the
quotient figure is  unit or 1.  And the  number of divisions will
show the time.
                         examples.
   1. How long may  an annuity of 30 dollars a year, be purchased
for $133.55,4678, allowing 4 per cent. ?          Ans. 5 years.
                  133.554675=present  value.
                          .05=the ratio less 1.
            30—6.677773375=23.32226625 : by this divide 30,
the annuity,  which will give the power of the ratio, indicated by the
time.  Proceed then by the rule to find the time.
   2. A house is let for $588.S,65 a year; but the person having
money in hand to the amount of $2064.05,4266, proposes to pay
this sum, being allowed 5% per cent, for his money;  how long may
he occupy  the premises ?                          Ans. 4 years.
   But every case  of  annuities may be much easier solved by the
help of the following tables.
            A a  2 
278
ARITHMETIC
TABLE  I."  Showing the Amount of One Dollar Annuity, from
                    one Year to forty.
yr.\ 4 per cent.
   1.
   2.04
   3.1216
   4.246464
   5.416322
   6.632975
   7.898294
   9.214226
  10.582795
  12.00610?
  13.486351
12'15.025805
I3l 16.626838
I4'l8.29l911
15 20.023588
4J per cent.
5 per cent.
  21.824531
  23.697512
  25.645413
19 27.671229
20129.778078
2131.969202
  34.247970
  36.6178S8
  39.082604
  41.645908
  44,311745
  47.084214
  49.967582
29 52.966286
30:56.084938
 1 59.328335
32162.701469
3366.20952?
34'69-857904
35J73.652225
3677-598314
|37 81.702246
33,85.970336
39 90-409150
40195.025516
 1.
 2.045
 3.137025
 4.278191
 5.470710
 6.716892
 8.019152
 9.380014
10.802114
12.288210
13.841179
15.464032
17-159913
18.932109
20.784054
22-719337
24.741707
26.858084
29-063562
 31-371423|
 33.783137
 36.833378
 38-937030
 41.689196
 44.565210
 47-570645
 50.711324
 53.993333
 57.423033
 61.007069
 64.752388
 68.666245
 72.7562261
 77.030256
 81.4966 IS
 86.163966
 91.041344;
 96.138205
 IOI.461121
 107-030329
BJ per cent.
 1.
 2.05
 3.1525
 4.310125
 5.525631
 6.801913
 8.142008
 9-549109
11.026564
12.577892
14.206787
15.917126
17-712983
19-598632
21.578563
23.657492
25.840366
28.132385
 30.539004
 33.065954
 35.719252
 38.505214
 41.430475
 44.501999
 47-727099
 51.113454
 54.669126
 58.402583
 62.322712
 66.438847
 70.760790
 75.298829
 80.063771
 85.066959
 90.320307
 95.836323'
 101.6281391
 107.709346
 114.095023
 120.799774
6 per cent.
 1.
 2.055
 3.1.6822
 4.342266
 5.581091
 6.888051
 8.266894
 9-721573
11.256259
12.875354
14.583498
16.385590
18.286798
20.292572
22.408663
24.641140
26.996402
29-481205
32-10^671
 34.868318
 37-786075
 40-864309
 44.111846
 47-537998
 51-152588
 54.965979
 58-989109
 63-233510
 67-7U353|
 72.43347«'
 77-419429
 82.667498
 88.224760
 94.077122
 100.251363
 106.76518S
 113.637274
 120.887324
 J128-536127
 136-605146
 1.
 2.06
 3.1836
 4.374602
 5.637093
 6-975318
 8.393837
 9.897468
11.491316
13.180795
14.971643
16.869942
18.882138
21.015066
23.275971
25,672528
28.212881
30.905653
33.759993
 36.7S5592
 39-992728
 43.392291]
 46.9938281
 50.815578
 54-864513
 59.156383
 63.705766
 68.528117
 73.639798
 79-058186
 84-801677
 90.889778
 97-343165
 104-183754
 111.434780
 U9-120867I
 127-2681181
 135.904206
 145.058458
 154.761966 
ANNUITIES AT COMPOUND INTEREST.       279
TABLE II. Showing the present worth of One Dollar Annuity,
                 from one Year to forty,,
Yr.	4 per cent.	4£ percent.	5 per cent.	5& per cent.	6 per cent.
, 1	0.96154	0.95694	0-95231	0.94786	0.9*4339
2	1.88609	1.87267	1.85941	1.81632	1.83339
3	2.77509	2.74876	2.72325	2.69f96	2.67301
4	3.62989	3.58752	3.54295	3.50514	3.4651
5	4.45182	4.38997	4.32988	4.27028	4.21236
6	5.24214	5.15787	5.07569	4.99553	4.91732
7	6.40205	5.8927	5.78637	5.68297	5.58238
8	6.73274	6.59579	6.46321	6.33457	6.20979
9	7.43533	7-26879	7.10782	6.95220	6.80169
10	8.11089	7-91272	7.72173	7.53762	7.36008
11	8.76048	8.52892	8.3064	8.09254	7-88687
12	9.38500	9.11858	8.86325	8.61852	8.38384
13	9.98565	9.68285	9-39357	9-11708	8.85268
14	10.56312	10.22282	9.89864	9-58965	9-29498
15	11.41839	10.73954	10.37965	10.03759	9-71225
16	11.65229	11.23401	10.83777	10.46216	9-10589
17	12.16567	11.70719	11.27407	10.86461	10.47726
18	12.65929	12.15999	11.68958	11.24607	10.8276
19	13.13394	12.59329	12.08532	11.60765	11.15811
20	13.59032	13.00793	12.46221	11.95034	11.46992
21	14.02916	13.40472	12.82115	12.27524	11.76407
22	14.45111	13.78442	13.163 12.58317		12.04158
23	14.85684	14.14777	13.48857 1 12.87504		12.30338
24	15.24696	14.49548	13.79864	13.15170	12.55035
25	15.62208	14.82821	14.09394	13.41391	12.78335
26	15.93277	15.14661	14.37518	13.66250	13.00316
27	16.32959	15.45130	14.64303	13.89810	13.21052
28	16.66306	15.74287	14.89S13	14.12142	13.40616
29	16.98371	16.02189	15.14107	14.33310	13.59072
30	17.29203	16.28889	15.37245	14.53375	13.76483
31	17-58849	16.54439	15.59281	14.72393	13.92908
32	17-87355	16.78889	15.80268	14.90420	14.08404
33	18.14764	17-02286	16.00255	15.07507	14.23023
34	18.41126	17-24676	16.1929	15.23703	14.36814
35	18.66461	17-46101	16.37419	15.39055	14.49825
36	18.90828	17-66604	16.54685	15.53607	14.62098
37	19-14258	17-86224	16.71129	15.67400	14.73678
38	19.36786	18.04999	16.86789	15.80474	14.84602
39	19.58448	18.22965	17-01704	15.92866	14.94907
40	19.79277	I 18.40158	17-15909	16.04612	14.92640
 
280
ARITHMETIC.
TABLE III.*
Rate per cent.	Half yearly pay-ments.	Quarterly payments.
4 4% 5 5% 6	1.009902 1.011126 1.012348 1.013567 1.014781	1.014877 1.016720 1.018559 1.020395 1.022257
   When the annuity, time, and rate per cent, are given to find
the amount.

   Rule.—Take the tabular number in the column, under the rate
per cent, and opposite to the  time from table 1, and multiply  it by
the annuity ; the product will be the amount.   But,
   If the payments  are half yearly,  or  quarterly,  multiply the
amount found as above directed,  for years,  by the proper tabular
amounts in table 3d ; the product will  be the true amount.


  * Table first is constructed in the following manner:  The interest, at
the given rate percent, is added to the annuity for the first year; whicli,
with the annuity for the second year, becomes the principal for the
second year.   The  interest arising on this amount, is added to the prin-
cipal, together with the annuity for the third year, for the principal for
the third year; and so on,  for any number of years at pleasure.  The
table might, in this  way, have been extended to any greater number of
years, but  these are sufficient for all practical purposes.  The second
table is constructed in the following manner: Divide the annuity by the
ratio involved to the time of its continuance, and  subtract the quotient
from the annuity: Divide the remainder by the ratio less unit or 1, and
the quotient will be the present worth, thus: Divide 1 by 1.04, the quo-
tient is,  .9615381;  which subtract from  1, and the remainder will be
.0384619, which divide by .04, and the quotient will be, 0.96154, the
present worth of an annuity of 1 dollar for one  year, as in the table
opposite 1 year, and in the column under 4 per cent.: and so for any num-
ber of years, and at any rate per cent.
  Table third is constructed  in the following manner :  Extract the
square root of the ratio; and from the root, take unit or 1; by the remain-
der, divide the ratio less  unit or 1; half the quotient will be the tabular
value for half yearly payments, thus: x/1.04=l.019803902; from which,
take 1,  and by the remainder .019801902, divide  .04, the quotient will
be  2.019804 nearly;  half of which, will be the tabular value: to wit,
1.009902, as may be seen in the column of half yearly payments, and
opposite 4 per cent.  The tabular value of quarterly payments is found
in a similar manner, only take the biquadrate root of the  ratio, and
proceed, as for half yearly payments, but observe to divide the quotient
by 4. 
ANNUITIES AT COMPOUND INTEREST.        281
                         examples.
  1. A  young lady has  an annual income  of 500 dols., arising
from ground rent; what  will it amount to in  7 years, allowing  6
per cent, interest ?                          Ans. $4196.93,85.
      8.393837=amount of 1 dol. for 7 years at 6 per cent.
     __________500
     4196.91,35,00=the amount required.
  2. A gentleman, at the age of 24 years, settled upon a salary of
800 dols. and  by economy saved  annually  250 dols., which he
regularly invested  in public stock,  as well as the interest arising
therefrom ;  what sum would the whole amount to in  26 years, or
when he would be 50 years of age, allowing 6 cents interest ?
                                        Ans. $14789.09,575.
         59-156383=tabular amount of 1 dollar.
                250
      $14789.095750=the amount required.
  3. A  gentleman was so well  pleased at the birth of a daughter,
that he determined annually, on her birth day, to  invest the sum of
300 dols. at 5 per cent, as her portion of  his estate,  and  also the
interest  arising thereform, until she should arrive  at the age of 18
years ; what did her portion amount to ?      Ans. $8439.7l>55.
  4. An officer, retired from service,has a pension of 450 dollars  a
year, which he  invests in  productive  stock, as also  the  interest
arising from it; to what  sum will it amount in 15 years, allowing
5%  per cent, interest.                     Ans. $10083.89,834-.
  5. What will an annuity of 30 dols., payable  half yearly,
amount to in 4 years, at 5 per cent. ?          Ans. $130.90,04.
  Here find, in the first place, the amount for yearly payment as in
the above examples, and multiply that amount by  1.012348, taken
from the table for half yearly and quarterly payments;  the pro-
duct will be the true amount.
  If the payments had been quarterly, you must have multiplied
by, 1.018559;  and in that case,  the  amount would have  been
$131.7035 : hence it appears, that half yearly are more profitable
than yearly, and quarterly more profitable than half  yearly, pay-
ments.
  This arises from the interest, which  becoming due at short inter-
vals, is then made a part of the principal, and draws interest also.
  6. A  gentleman,  beside  his  family expenses,  reserves the sum
of 400 dollars semi-annually, and invests the same, as well as the
interest  arising  therefrom,  in public  stocks; what sum would  it
amount to in 12 years, allowing the funds produced 6  percent. ?
                                        Ans. $13695.45,733. 
282
ARITHMETIC.
   When the amount, rate, and time are given to find the annuity:
   Rule.—Divide the amount by the number in the column under
the ratio and opposite to the time : the quotient will be the annuity.
   When the payments are half yearly or quarterly; divide the an-
nuity, found as above, by the proper tabular amount for half yearly
or quarterly payments.
                        EXAMPLES.
   1. What annuity, being unpaid  for  seven years, will  amount
$4196.93,85, at 6 per cent. ?                        Ans.  500.
                4196.93,85-H 8.393837=500.
   2. What annuity will amount  to $8439-71)55 in 18 years, at 5
percent.?                                     Ans. 300 dols.
   3. What annuity being unpaid 4 years, will amount to $130.9004
at 5 per cent, payable half yearly ?                Ans. 30 dols.
   Here divide  the amount by 4.310125 the tabular  amount  of 1
dol. for 4 years at 5  per cent.; and  the quotient thence arising by
1.012348, found in  the table for quarterly payment, opposite the
rate per cent.; this last quotient will be the true sum.
  4. The  rent  of  a  house has remained unpaid for  6 years, and
has amounted to the sum of $578.59,6,284, interest being 5% per
cent.;  how much is the rent per  annum ?          Ans. 84 dols.

  When the annuity, time, and rate per cent, are given to find
the present worth  :
  Rule.—Multiply  the number  in the column, table 2d, under the
rate, and opposite the time, by the annuity  ; and the  prod uct will
be the present worth.
  When the payments are half yearly or  quarterly,  multiply the
present worth, thus found, by the  proper tabular amount  for %
yearly, or  J payments, table 3d, and the product will be the  pre-
sent worth.
                        examples.
  1. How much ready money will  purchase an annuity of $415.81,25
to continue 4 years, allowing 6 per cent. ? Ans. $1440.83,189375.
     3.4651 =present worth of 1 dol. for 4 years at 6 per cent.
  415.8l25=annuity.
   1440.83,189375=the sum required.
  2. How much  ready money will  purchase an annuity of 500
dols. to continue 7 years at 6 per cent. ?          Ans. $2791.19.
   3. How much money will  purchase an  annuity of 450 dols. to
continue 15 years, allowing 5% per cent. ?      Ans. $4516.91,55. 
ANNUITIES AT COMPOUND INTEREST.        283
  4. What is the present value of an annuity of 20 dols., to con-
tinue 6 years, payable half yearly, at 5 per cent.?
                                           Ans. $102.76,73.
    5.07569=the present worth of 1 dol. for the time, and at the
          20                                     given rate.
    I01.51380=the present worth  for yearly payments; which
multiplied  by 1.012348 taken from table  3d, the  value of half
yearly payments, will give $102.76,73 the present worth of an an-
nuity payable half yearly fur the time, and at the given  rate.  If
the annuity had been payable quarterly, then the present worth for
yearly payments  must  have been multiplied by 1.018559, taken
from ihe same  table in the column  under quarterly payments.
The product would be  the  present value  of the annuity for quar-
terly payments.  In that case the purchaser must  pay $103.39,78.
In like manner, the present value of any annuity, payable half year-
ly or quarterly, may be found.

   When the present wokth, time, and  ratio are given to find the
annuity.:
   Rule.—Divide the present worth by the tabular value of 1 dol-
lar, table 2d, taken  from the column under the rate, and opposite
to the time ;  and the quotient will be the  annuity.
   When the payments are half yearly or quarterly, divide the
annuity as found for yearly payments, by  the proper tabular value
for half yearly or quarterly  payments ;  table 3d.

                        examples.
   1. What annuity to  continue for 4 years  will  $2064.05,42661
purchase, allowing 5% per cent. ?               Ans. $588.86,5.
    2064.0542661-5-3.50514=588.865 the annuity required.
   2. A  person has the sum of  $2791.19, and wishes to purchase
an annuity to continue  9 years, bearing 6 per cent,  interest; how
much can he purchase ?                         Ans.  500 dols.
        2791.190-^5.58238=500, the annuity required.
   3. A  lady, having the sum  of 5500 dollars in hand,  wishes to
purchase an annuity for her daughter, who is just 10 years of age,
to continue till she shall arrive at  the age  of 25 years, bearing
interest at the rate of 6 per cent.; how much can  she purchase ?
                                          Ans. $566.29,34-.
               5500.000-1-9.71225=566.2934-.
   4. What annuity to continue 6 years, payable  half yearly, may
be purchased for $102.76,73, at 5 per cent. ?       Ans. 20 dols.
     102.7673000-=-5.07569=20.246960, the annuity  in yearly
payments: which divided  by 1.012348, taken from table 3d, in 
284                     ARITHMETIC.
the column under half yearly payments, and opposite to  5  per
cent, gives 20 dollars for the annuity.   In the same manner, pro-
ceed for quarterly payments.
           ANNUITIES IN  REVERSION.
  An annuity is said to be in Reversion, when it is not to com-
mence, till after a certain  number of years.
  When the annuity, time of reversion, time of continuance
and ratio are given, to find the present worth :
  Rule.—Divide the annuity by the ratio involved to the time of
continuance, and subtract the quotient from the annuity, for a divi-
dend : Multiply the ratio involved to the time of reversion, by the
ratio, less 1, for a divisor ; the  quotient thence arising will be the
present worth.
  Or, by the tables :
  From table 2d, take the value of 1 dollar, in the column under
the rate, and opposite the sum of the time of reversion and the con-
tinuance of the annuity.  Also, take the value of 1 dollar from the
same table, and opposite  the  time  of reversion.  Take the  dif-
ference of these values, and multiply it by the annuity; the product
will be the present worth.
                         examples.
   1. What is the present worth of a pension of 96 dols. a year, to
continue 7 years, at 6 per cent.; but not to commence till after the
expiration of 3 years ?                        Ans. $449-95,872
         7-36008 =tubular value of 7-f 3 = 10  years.
         2.67301 =tabular value of 3 years.
         4.68707=difference of tabular values.
       _______96
       449.95872—the product, or present value.
   2. A young man is to come to  the possession of an annuity of
250 dols. a year, and to continue for 10 years, but not to commence
till the expiration of 4 years, when he will be just 25 years of age;
but wanting ready money, he proposes to sell it, and allow 5% per
cent, to the purchaser ; what sum will he receive ?
                                            Ans. $1521.12,75.
          9.58965=tabular value of 104-4=14 years.
          3.50514=labular value of 4 years.
          6.08451 =diflerence of tabular values.
               250
        1521 .l2750=present value required. 
            ANNUITIES AT COMPOUND INTEREST.        285

   3.  What is the present value of an estate, worth 100 dols- a year,
 to continue 25 years, but not to commence till the expiration of 15
 years, allowing 5 per cent, interest ?            Ans. $677-94,4.
   4.  A company of gentlemen propose to erect a splendid build-
 ing for a public house, which will be in 3 years completed.  They
 offer it on a lease for 10 years, at  1000 dols. a year ; what is the
 present value of the lease at 6 per cent. ?         Ans,  $6179.67-
   5.  A lady has a ground rent of 100 dollars a year, but which is
 so situated that she  cannot come into possession till the expiration
 of 2 years : being in want of ready money, she proposes to sell it
 for 4 years, and allow at the rate of 5  per  cent.;  what sum must
 she receive for it ?                           Ans. $321.77,723.
     1.05)4=1.215506, the fourth power of the ratio, by the rule.
     1.63) 2 = 1.102500, the  2d power of the ratio, by the rule.
           100.000000-^-1.215506=82.262034-:  which subtract
 from 100, the remainder is 17-73797-  Divide this by the square of
 the ratio, multiplied by the ratio, less unit or 1 ;  and the quotient
 will be the present  worth, by the  rule; thus: 1.102500x.05.=
 .055125. By this divide 17.73797, and you will have $321.77,24-
 the present value.
   When the present worth, time, and ratio are given to find the
 annuity:
   Rule.—Involve the  ratio to a power, equal to the number of
 years from the time of purchase lo the expiration of the annuity,
 for a dividend  ; and involve the ratio to the time of continuance,
 and  take from the product unit or  1, for a divisor.   Multiply the
 quotient thence arising by the present worth ; and this product by
 the ratio,  less unit or 1, and you have the annuity.
  Or, by the tables : From table 2d, take the value of 1 dollar in
 the column under the  rate, and  opposite the sum  of the time of
 reversion and the continuance of the annuity: also take the value of
 1 dollar from the same table, and opposite the time of reversion •
 take the difference of these values, and divide the present worth by
 it; the quotient will be the annuity sought.

                         examples.
  1. What annuity to continue 7 years, but not to  commence till
 after the expiration of 3 years, can be purchased  for the sum of
$449-95,872, allowing 6 per cent, interest ?        Ans. 96 dols.
    7.36008=value of 1 dol. for 7+3=10 years, table 2d
    2.67301 =value of 1 dol. for 3 years.
    4.68707)449-95872(96.
             4218363
E 
286
ARITHMETIC.
  2. What annuity to  continue 25 years, but not to commence
till  the expiration  of 15  years, may  be purchased for the sum of
$677-94,4, allowing 5 per cent, interest ?      #    Ans. 100 dols.
  3. A gentleman  purchased a lease of a house for ten  years, but
not to commence till the expiration of 3 years ; for which he gave
the sqm of ?6l79.67j allowing 6 per cent, interest; what was the
rent per annum ?                              Ans.  1000 dols.
  4. A farmer purchased a lease of some land, to continue 7 years,
but not to commence till the expiration of 4 years ; for  which, he
paid in hand £365.60,5; what was the annual rent, allowing 5 per
cent, for his money ?                              Ans. #76.8.
                   PERPETUITIES.

  An annuity to continue for ever, is called a Perpetuity.
  When the annuity and ratio are given, to find the present worth.
  Rile.  Divide the annuity by the ratio, less unit or 1.   The
quotient will be the present worth.
  If the payments are quarterly, or half yearly, multiply the pre-
sent worth for yearly payments, by the tabular value in table 3d ;
the product will be the present worth.

                          examples.
   1. What is the present worth of a perpetual annuity of 100 dol-
lars, allowing 5 per cent, interest ?                   Ans. 2000.
              100-^0.5=2000 the present value.
   2. What  is  the present value of a perpetual annuity of 250 dol-
lars, allowing 6 per cent, interest ?             Ans. $4166.66,6.
   3  What is  the value of a perpetual annuity of 550 dollars, pay-
able half yearly, at 5% per cent. ?               Ans. $10135.67.
                 550.000—.055 = 1000 present worth for yearly
 payments;  which multiplied by 1.013567, the  tabular value  in
 table 3d, for half yearly payments,  gives 10135.67 dollars, the pre-
 sent value required.
    4. What sum may be given for the purchase of a freehold estate
 of the annualincorae of 300 dollars, payable quar<er,X> ^°*m£ J
 per cent, interest?                           Ans. $5111.28,5.
         300.00-4-06=5000x1.022257=5111-28,5,000.
    When the present worth and ratio are given to find the an-

 ""rule.   Multiply the present worth by the ratio, less 1;  the
 product will be the annuity. 
PERPETUITIES.                    287
                         EXAMPLES.
  1. A gentleman purchased a freehold estate for the sum of 5000
dollars  ready money, allowing  5 per cent, interest; what is the
annual income?'                                   Ans. $250.
  2. A  lady of" fortune,  having 10000 dollars  ready money,
wishes to purchase a perpetual annuity; what annuity can she pur-
chase with the above sum, allowing 6 per cent, interest? Ans. $600.
  3. A  young gentleman has a  ground rent of a  certain annual
value ;  but wishing to go into business, offers to sell it for the sum
of 6540 dollars, and  allow 5% per cent, interest; what  is the an-
nuity ?                                         Ans. $359.70.
  When the present worth and annuity are given to find  the ratio :
  Rule.  Add the present worth and annuity together, and divide
their sum by the present worth; the quotient will be the ratio.

                         EXAMPLES.
   1. A freehold estate of the annual value of 250 dollars, can be
purchased for 5000 dollars;  what is the rate per cent, allowed ?
                                           Ans. 1.05 the ratio.
                  5000=the present  worth.
                    250=the annual value on annuity.
                  525~0~-r- 5000=1.05.
   2. A gentleman bought a ground  rent of 550 dollars, for the
 sum of 16450 dollars ;  what is the rate  per cent. ?
                                         Ans. 1.03,343 nearly.
   3. A person, for the sum of 6540 dollars, purchased an annuity
 of $359.70; what is the rate per cent. ?                Ans. 5%.

            PERPETUITIES IN  REVERSION.
   When the perpetuity, time of reversion, and rate per cent.,  are
 given to find the present worth.
   Rule.   Multiply the ratio, involved to the time  of reversion by
 the ratio, less unit or  1, for  a divisor :  by this divide the annuity;
 the quotient will be the present worth.

                          EXAMPLES.
    1. What is  the value of a  perpetuity of 250  dollars, to com-
 mence 4 years hence, allowing 6 per cent, for  present payment ?
                                         Ans. $3300.39,0424-
             J7oq4=1.2624769X.06=.0757486l4.   By which
 divide  5250, the annuity, and the quotient will be, 3300.39,042 +
   2. A gentleman has a perpetuity of 696 dollars; but not to com-
 mence  till  the expiration  of 4 years; which he  wishes to sell, 
288                     ARITHMETIC.
allowing 4 per cent, to the purchaser; what is it worth  on those
terms?                                    Ans. $14873.59,5.
   3. A young lady received at the decease of her father, a ground
rent of 200 dollars annual income, as her part of his estate;  but it
was so situated, that she could not come into possession of it, till
after the expiration of 2 years; what is it worth in ready money,
allowing 5% per cent. ?                      Ans. $3267.09,9 +
            f^5"|2=1.113025x.055=.06l2l63750; by which
divide  the annuity, 200 dollars,  and you have  3267-099+the
present value of the perpetuity required.
   When the present worth of the perpetuity, the time of rever-
sion, and the rate per cent., are given to find the annuity:
   Rule.  Multiply the  present worth, the ratio, less 1, and the
ratio involved to the time of reversion together;  the product will be
the annuity.
                          EXAMPLES.
   1. The present worth of a perpetuity is 5889.99,65, which does
not commence until the expiration of 2 years, allowing  6 per cent.;
what is the annuity ?                                 Ans. 560.
           Xo6|2=1.1236x.06=0674l6; which multiplied by
the present worth, will give the answer.
   2. A gentleman paid  the sum of 53300.39,042+ for the pur.
chase of a perpetuity, not to commence until the expiration of 4
years ; what is the annuity, allowing 6 per cent. ?     Ans. $250.
THE END 
 
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