UC-NRLF
.* it
LIBRARY
OF THK
UNIVERSITY OF CALIFORN
f
<!X6
Received
Accession No.
7
. Class No.
_J
JOHN SWEtT.
PRESENTJ?t> BY
ROBINSON'S MATHEMATICAL SERIES.
THE
RUDIMENTS
V-y,.<.
OP
WRITTEN ARITHMETIC:
CONTAINING
SLATE AM) BLACK-BOARD EXERCISES FOR BEGIMERS,
AND DESIGNED FOR
GBADED SCHOOLS.
EDITED BY
W. FISH, A.M.
NEW YOKK:
, PHINNEY, BLAKEMAN &
CHICAGO : S. C. GRIGGS & CO.
1866.
ROBINSOST S
The most COMPLETE, most PRACTICAL, and most SCIENTIFIC SERIES
of MATHEMATICAL TEXT-BOOKS ever issued in this country.
(Esr Tw:m:EsrTY-T~w~o
+ 7J-3*
Robinson's Progressive Table Book,
Hobinson's Progressive Primary Arithmetic, -
Hobinson's Progressive Intellectual Arithmetic, -
Hobinson's Rudiments of "Written Arithmetic, -
.Robinson's Progressive Practical Arithmetic,
Hobinson's Key to Practical Arithmetic, - - -
Hobinson's Progressive Higher Arithmetic, -
Robinson's Key to Higher Arithmetic, -----
Robinson's Arithmetical Examples, -
Robinson's New Elementary Algebra,
Robinson's Key to Elementary Algebra,
Hobinson's University Algebra, - - - - -
Hobinson's Key to University Algebra, -
Hobinson's New University Algebra, - * -
Hobinson's Key to New University Algebra, -
Hobinson's New Geometry and Trigonometry, -
Hobinson's Surveying and Navigation, - - - -
Hobinson's Analyt. Geometry and Conic Sections,
Hobinson's Differen. and Int. Calculus, (in preparation,)-
Itobinson's Elementary Astronomy,
Hobinson's University Astronomy, ------
Robinson's Mathematical Operations, -
Hobinson's Key to Geometry and Trigonometry, Conic
Sections and Analytical Geometry,
Entered, according to Act of Congress, in the year 1861,
and again in the year 1S63, by
DANIEL W. FISH, A.M.,
In the Clerk's Office of the District Court of the United States, for the Northern
District of New York.
PREFACE.
In the preparation of this work, a special object has been
kept in view by the author, namely; to furnish a small and
simple class book for beginners, which shall contain no
more of theory than is necessary for the illustration and
application of the elementary principles of written arith-
metic, applied to numerous, easy, and practical examples,
and which shall be introductory to a full and complete
treatise on this subject.
This book is not to be regarded as a necessary part of
the Arithmetical Series by the same author, as the four
books already composing that Series are believed tu be
properly and scientifically graded, and eminently adapted
to general use j but this work has been prepared to meet
a limited demand, in large graded schools, and in the pub-
lic schools of New York, and similar cities, where a large
number of pupils often obtain but a limited knowledge of
arithmetic, and wish to commence its study quite young ;
and it is also designed for those who desire a larger num
ber of simple and easy exercises for the slate and black-
board than are usually found in a complete work on writ-
ten arithmetic, so that the beginner may acquire facility,
promptness, and accuracy in the application and operations
of the fundamental principles of this science.
(iii)
IV PREFACE.
The principles, definitions, rules, and applications so far
as developed in this work coincide with the other books
of the same series. Many of the Contractions, and special
applications of the rules, particularly those that are at all
difficult, have been omitted, and also the treatment of De-
nominate Fractions, and Decimals, all of which are fully
and practically treated in the Progressive Practical, and
the Higher Arithmetic. -A few easy and practical appli-
cations of Cancellation, Analysis, Per centage and Sim-
ple Interest have been given, and a very large number ot
easv examples.
CONTENTS.
SIMPLE NUMBERS. Page,
Definitions, , 7
Roman Notation, 8
Arabic Notation, 9
Laws and Rules for Notation and Numeration, 16
Addition, ..18
Subtraction, 29
Multiplication, ., . . . 39
Contractions, 48
Division, 54
Contractions, 68
Problems in Simple Integral Numbers, 72
COMMON FRACTIONS.
Definitions, Notation and Numeration, .. 74
Reduction of Fractions, ~. . .78
Addition of Fractions, 83
Subtraction of Fractions, 86
Multiplication of Fractions, 88
Division of Fractions, ^^ 94
DECIMALS.
Notation and Numeration, 102
Reduction of Decimals, 107
Addition of Decimals, 1$)
VI CONTENTS.
Page.
Subtraction of Decimals, Ill
Multiplication of Decimals, 112
Division of Decimals, 114
UNITED STATES MONEY.
Reduction of United States Money, 118
Addition of U. S. Money, ^ 120
Subtraction of U. S. Money, 122
Multiplication of U. S. Money, 124
Division of U. S. Money, 125
Bills, ...".... 128
COMPOUND NUMBERS.
Weights and Measures, 130
Aliquot parts, 145
Reduction Descending, 146
Reduction Ascending, 148
Addition of Compound Numbers, '. 153
Subtraction of Compound Numbers, 156
Multiplication of Compound Numbers, % . . . 159
Division of Compound Numbers, 162
CANCELLATION, 167
ANALYSIS, 172
PERCENTAGE, 177
COMMISSION, 179
PROFIT AND Loss, 180
INTEREST, 181
PROMISCUOUS EXAMPLES, 186
RUDIMENTS OF ARITHMETIC.
DEFINITIONS.
1 . Quantity is any thing that can be increased, dimin-
ished, or measured ; as distance, space, weight, motion, time,
- 2. A Unit is one, a single thing, or a definite quantity.
3. A Number is a unit, or a collection of units.
4. 3Vn Abstract Number is a number used without ref-
erence to any particular thing or quantity ; as 3, ft, 756.
5~rk: Concrete Number is a number used with refer-
ence to some particular thing or quantity; as 21 ffturs, 4
cents, 230 miles.
45. A Simple Number is eithej: an abstract nuiriter, or a
concrete number of but one denomination; as 48, 52 pounds,
36 days.
7. A Compound Number is a concrete number expressed
in two or more denominations ; as 4 bushels 3 pecks, 8 rods
4 yards 2 feet 3 inches.
8. An Integral Number, or Integer, is a number which
expresses whole things; as 5, 12 dollars, 17 men.
9. A Fractional Number, or Fraction, is a number
which expresses equal parts of a whole thing or quantity;
as A, f of a pound, 7 5 y of a bushel.
1 0. Like Numbers have the same kind of unit, or ex-
press the same kind of quantity. Thus, 74 and 16 are like
numbers; so are 74 pounds, 16 pounds, and 12 pounds;
also, 4 weekb 3 days, and 16 minutes 20 seconds, both being
used to express units of time.
8 SIMPLE NUMBERS.
11. Unlike Numbers have different kinds of units, or
are used to express different kinds of quantity. Thus, 36
miles, and 15 daya ; 5 hours 36 minutes, and 7 bushels 3
pecks.
1. Arithmetic is the Science of numbers, and the Art
of computation.
13.. The Five Fundamental Operations of Arithmetic
are, Notation and Numeration, Addition, Subtraction,
Multiplication, and Division.
NOTATION AND NUMERATION.
14:J^Notation is a method of writing or expressing
numbers by characters ; and,
l5jNumeration is a method of leading numbers ex-
pressed by characters.
IGf^Two systems of Notation are in general use the
Roman and Arabic.
THE ROMAN NOTATION
M7. Employs seven capital letters to express numbers,
thus :
Letters, I V X L C D M
Values, one, five, ten, fifty, hu n n ^ed, hunted, thoSLnd.
%18. The Roman notation is founded upon the following
principles :
1st. Repeating a letter repeats its value. Thus, II rep-
resents two, XX twenty, CCC three hundred.
2d. If a letter of any value be placed after one of greater
value, its value is to be united to that of the greater. Thus,
XI represents eleven, LX sixty, DC six hundred.
3d. If a letter of any value be placed before one of greater
NOTATION AND NUMERATION. 9
value, its value is to be taken from that of the greater.
Thus, IX represents nine, XL forty, CD four hundred.
4th. If a letter of any value be placed between two letters,
each of greater value, its value is to be taken from the
yinited value of the other two. Thus, XIV represents four-
teen, XXIX twenty-nine, XCIV ninety-four.
TABLE OF ROMAN NOTATION.
1 is One. XVIII is Eighteen.
II " Two. XIX -' Nineteen.
III " Three. XX " Twenty.
IV " Four. XXI " Twenty-one.
V " Five. XXX " Thirty.
VI l< Six. XL " Forty.
VII " Seven. L " Fifty.
VIII " Eight. LX " Sixty.
IX " Nine.. LXX " Seventy.
X " Ten. LXXX " Eighty.
XI " Eleven. XO " Ninety.
XII " Twelve. C " One hundred.
XIII " Thirteen. CO " Two hundred.
XIV " Fourteen. D " Five hundred. ^
XV " Fifteen. DO ' Six hundred. -
XVI " Sixteen. M " One thousand.
XVII " Seventeen.
Express the following numbers by the Roman notation:
1. Fourteen. 6. Fifty-one.
2. Nineteen. 7. Eighty-eight.
3. Twenty-four. 8. Seventy- three.
4. Thirty-nine. 9. Ninety-five.
5. Forty-six. 10. One hundred one.
. Employs ten characters or figures to express numbers.
10 SIMPLE NUMBERS.
Tim?,
Figures, 01 234.56789
Names and ) naught, one, two, three, four, five, six, seven, eight, nine.
values, \ cip r)
30. The first character is called naught, because it has
no value of its own. . The other nine characters are called
significant figures, because each has a value of its own.
31. As we have no single character to represent ten, we
express it by writing the unit, 1, at the left of the cipher, 0,
thus, 10. In the same manner we represent
2 tens, 3 tens, 4 ten*, 5 tens 6 tens, 7 tens, 8 tens, 9 tens,
or or ur or or or or or
twenty, thirty, forty, fifty, sixty, seventy. eighty, ninety,
20; 30; 40; 50; 60; 70; 80; 90.
33. When a number is expressed by two figures, the right
hand figure is called units, and the left hand figure tcn&.
We express the numbers between 10 and 20, thus :
eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen.
11, 12, 13, 14, 15, 16, 17, 18, 19.
In like manner we express the numbers between 20 and
30, thus : 21, 22, 23, 24, 25, 26, 27, 28, 29, &c.
UThe greatest number that can be expressed by two figures
is 99.
33. We express one hundred by writing the unit, 1, at
the left hand of two ciphers ; thus, 100. In like manner
we write two hundred, three hundred, &c., to nine hundred.
Thus:
one two three four five six seven eight nine
hundred, hundred, hundred , hundred, hundred, hundred, hundrod,hundred, hundred.
100, 200, 300, 400, 500, 600, 700, 800, 900.
3J:. When a number is expressed by three figures, the
right hand figure is called ujiits, the second figure tens, and
tho left hand figure Jmin/rc</s.
NOTATION AND NUMERATION. 11
As the ciphers have, of themselves, no value, but are
always used to denote the absence of value in the places they
occupy, we express tens and units with hundreds, by writing,
in place of the ciphers, the numbers representing the tens
and units. To express one hundred fifty, we write 1 hun-
dred, 5 tens, and units ; thus, 150. To express seven
hundred ninety-two, we write 7 hundreds, 9 tens, and 2
units ; thus,
. 3
792
The greatest number that can be expressed by thne
figures is 999.
Express the following numbers by figures :
1. Write one hundred twenty-five.
2. Write four hundred eighty-three.
3. Write seven hundred sixteen.
4. Express by figures nine hundred. W
5. Express by figures two hundred ninety.
6. Write eight hundred nine.
7. Write five hundred five.
8. Write five hundred fifty-seven.
95. We express one thousand by writing the unit, 1, at
the left hand of three ciphers ; thus, 1000. In the same
manner we write two thousand, three thousand, &c., to nine
thousand; thus, ^
one two three four five six seven eight nine
thousand, thousand, thousand, thousand, thousand.thousand, thousand, thousand, thousand.
1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000.
9G. When a number is expressed by four figures, the
places, commencing at the right hand, are units, tens, hun-
dreds, thousands.
12 SIMPLE XUMBE11S.
To express hundreds, tens, and units with thousands, we
write in each place the figure indicating the number we
wish to express in that place. To write four thousand two
hundred sixty-nine, we write 4 in the place of thousands, 2
in the place of hundreds, 6 in the place of tens, 9 in the
place of units ; thus,
4269
The greatest number that can be expressed by four figures
is 9999.
Express the following numbers by figures :
^ 1. One thousand two hundred.
2. Five thousand one hundred sixty.
3. Three thousand seven hundred forty-one.
4. Eight thousand fifty six.
r 5. Two thousand ninety.
6. Seven thousand nine.
7. One thousand one.
8. Nine thousand four hundred twenty-seven.
9. Four thousand thirty-five.
10. One thousand nine hundred four.
Read the following numbers :
11. 76; 128; 405; 910; 116; 3414; 1025.
^12. 2100 ; 5047 ; 7009 ; 4670 ; 3997 ; 1001.
27. Next to thousands come tens of thousands, and next
to these come hundreds of thousands, as tens and hundreds
come in their order after units.
Ten thousand is expressed by removing the unit, 1, oni
place to the left of the place of thousands, or by writing it
NOTATION AND NUMERATION. 13
at the left hand of four ciphers ; thus, 10000 ; and one
hundred thousand is expressed by removing the unit, 1,
still one place further to the left, or by writing it at the left
hand of five ciphers ; thus, 100000. We can express thou-
sands, tens of thousands, and hundreds of thousands in one
number, in the same manner as we express units, tens, and
hundreds in one number. To express five hundred twenty-
one thousand eight hundred three, we write five in the sixth
place, counting from units, 2 in the fifth place, 1 in the
fourth place, 8 in the third place, in the second place,
(because there are no tens), and 3 in the place of units ;
thus,
* tn
II *J I I 1 I
5 21803
The greatest number that can be expressed by Jive figures
is 99999 ; and by six figures, 999999. 9
Write the following numbers in figures :
%1. Twenty thousand.
^2. Forty-seven thousand.
^3. Eighteen thousand one hundred.
Twelve thousand three hundred fifty.
Thirty-nine thousand five hundred twenty-two.
^6. tS^een thousand two hundred six.
Jk7. Eleven thousand twenty-four.
^8. Forty thousand ten.
^ 9. Sixty thousand six hundred.
Two hundred twenty thousand.
One hundred fifty-six thousand.
Eiglit hundred forty thousand three hundred.
14 SIMPLE NUMBERS.
Read the following numbers:
13. 5^06; 1^304; 96071; &470 ; 20^410.
14. 36.741; 40(\560; 13,061; 4^300; lOOplO.
*- For convenience in reading large numbers, we may point
them off, by commas, into periods of three figures each,
counting from the right hand or unit figure. This point-
ing enables us to read the hundreds, tens, and units in each
period with facility as seen in the following
NUMERATION TABLE.
2 2 | ;
ggs 'S-S 3 "23 'gas
Sis 5S 5ls 5jJ 2S
876,556,789,012,345
fifth fourth third second first
period, period, period, period, period.
_ I. ]ngures occupying different places in a number, as
units, tens, hundreds, &c., are said to express different or-
ders of units. 9
29. In numerating, or expressing numbers verbally,^he
various orders of units have the following names : ^
ORDERS.
NAMES.
1st order is called
Units.
2d order "
Tens.
3d order "
Hundreds.
4th order " "
Thousands.
5th order "
Tens of thousands.
6th order " "
Hundreds of thousands.
7th order " "
Millions.
8th order " "
Tens of millions.
9th order "
Hundreds of millions.
&c., &c.
<&c., &c.
NOTATION AND NUMERATION. 15
"Write and read the following numbers :
1. One unit of the third order, two of the second, five of
the first. Ans. 125 ; read, one. hundred twenty-Jive.
2. Two units of the 5th order, four of the 4th ; five of the
2d, six of the 1st.
Ans. 24056; read, twenty-four thousand fifty-six.
3. Seven units of the 4th order, five of the third ; three
of the 2d, eight of the 1st.
4. Two units of the 7th order, nine of the 6th ; four of
the 3d, one of the 1st, seven of the 2d.
5. Three units of the 6th order, four of the 2d.
6. Nine units of the 8th order, six of the 7th, three of
the 5th, seven of the 4th, nine of the 1st.
7. Four units of the 10th order, six of the 8th, four of
the 7th, two of the 6th ; one of the 3d, five of the 2d.
8. Eight units of the 12th order, four of the llth, six of
the 10th, nine of the 7th 7 three of the 6th, five of the 5th,
two of the 3d, eight of the 1st.
30. Since the number expressed by any figure depends
upon the place it occupies, it follows that figures have two
values, Simple and Local.
31. The Simple Value of a figure is its value when ta-
ken alone ; thus, 4, 7, 2.
33. The Local Value of a figure is its value when used
with another figure or figures in the same number. Thus,
in 325, the local value of the 3 is 300, of the 2 is 20, and
of the 5 is 5 units.
NOTE. When a figure occupies units' place, its simple and local values
are the same.
33. The leading principles upon which the Arabic nota-
tion is founded are embraced in the following
16 SIMPLE NUMBERS.
GENERAL LAWS.
I. AU numbers are expressed ~by applying the ten figure*
to the different orders of units.
II. The different orders of units increase from right to
left, and decrease from left to right, in a tenfold ratio.
III. Evert/ removal of a figure one place to the left, in-
creases its local value tenfold ; and every removal of a fig-
ure one place to the right, diminishes its local value tenfold.
From this analysis of the principles of Notation and
Numeration, we derive the following rules :
RULE FOR NOTATION.
I. Beginning at the left hand, write the figures belonging
to the highest period.
II. Write the hundreds, tens, and units, of each success-
ive period in their order, placing a cipher wherever an order
of units 'is omitted. .
RULE FOR NUMERATION.
I. Separate the number into periods of three figures each,
commencing at the right hand.
II. Beginning at the left hand, read each period sepa-
rately, and give the name to each period, except the last, or
period of units.
34. Until the pupil can write numbers readily, it may
be well for him to write several periods of ciphers, point
them off, over each period write its name, thus.
Trillions, Billions, Millions, Thousands, Units.
000, 000, 000, 000, 000
NOTATION AND NUMERATION. 17
and then write the given numbers underneath, in. their ap-
propriate places.
EXERCISES IN NOTATION AND NUMERATION.
* 4
Express the following numbers by figures :
1. Four hundred thirty-six.
2. Seven thousand one hundred sixty-four.
3. Twenty-six thousand twenty-six.
4. Fourteen thousand two hundred eighty.
5. One hundred seventy-six thousand.
6. Four hundred fifty thousand thirty-nine.
7. Ninety-five million.
8. Four hundred eighty-three million eight hundred
sixteen thousand one hundred forty-nine.
9. Nine hundred thousand ninety.
10. Ten million ten thousand ten hundred ten.
Point off, numerate, and read the 'following numbers :
11. 8240.
12. 400900.
13. 308.
14. 60720.
%15. 111111.
16. 57468139.
17. 5628.
18. 11111111.
19. . 370005.
20. 9400706342.
21. 38429526.
22. 11111111111.
23. Write seven million thirty-six.
24. Write five hundred sixty-three thousand four.
25. Write one million ninety-six thousand.
26. A certain number contains . 3 units of the seventh
order, 6 of the fifth, 4 of the fourth, 1 of the third, 5 of the
second, and 2 of the first ; what is the number ?
27. What orders of units are contained in the number
290648 ?
18
SIMPLE NUMBERS.
ADDITION.
35. Addition is the process of uniting several numbers
of the same kind into one equivalent number.
SO. TJie Sum, or Amount, is the result obtained.
ADDITION TABLE.
2 and 1 are 3
3 and 1 are 4
4 and 1 are 5
5 and 1 are 6
2 and 2 are 4
3 and 2 are 5
4 and 2 are 6
5 and 2 are 7
2 and 3 are 5
3 and 3 are 6
4 and 3 are 7
5 and 3 are 8
2 and 4 are 6
Sand 4 are 7
4 and 4 are 8
5 and 4 are 9
2 and 5 are 7
3 and 5 are 8
4 and 5 are 9
5 and 5 are 10
2 and 6 are 8
3 and 6 are 9
4 and 6 are 10
5 and 6 are 11
2 and 7 are 9
Sand 7 are 10
4 and 7 are 11
5 and 7 are 12
2 and 8 are 10
Sand 8 are 11
4 and 8 are 12
5 and 8 are 13
2 and 9 are 11
3 and 9 are 12
4 and 9 are 13
5 and 9 are 14
2 and 10 are 12
3 and 10 are 13
4 and 10 are 14
5 and 10 are 15
2 and 11 are 13
3 and 11 are 14 j
4 and 11 are 15
5 aud 11 are 1 6
2 and 12 are 14
3 and 12 are 15 '
, 4 and f 2 are 16
5 and 12 ar^l"
6 and 1 are 7
1 7 and-^1 are 8
8 and .1 are 9
;9nd 1 are 10
6 and 2 are 8
7 and 2*|re 9
8 and 2 are 10 ""
9 and 2 arc 11
6 and 3 are 9
7 and 3 are 10
. 8 and 3 are 11
9 and 3 are 12
6 and 4 are 10
7 and 4 are 11
'i8,a,nd 4 are 12
9 and 4 are 13
6 and 6 are 11
7 and 5 are 12
Sand 5 are 13
9 and 5 are 14 '
6 and 6 are 12
7 and 6 are 13
8 and 6 are 14
9 and 6 are 15
G and 7 are 13
7 and 7 are 14
Sand 7*el5;
9 and 7 are 16
6 and 8 are 14
7 and 8 are 15
8 and 8 are 16
9 and 8 are 17
6 and 9 are 15
7 and 9 are 16
Sand 9 are 17
9 and 9 are 18
6 and 10 are 16
7 and 10 are 17
8 and 10 are 18
9 and 10 are 19
6 and 11 are 17
7 and 11 are 18
8 and 11 are 19
9 and 11 are 20
6 anil 12 are 18
7 and 12 are 19
8 and 12 are 20
9 and 12 are 21
10 and 1 are 11
Hand 1 are 12
12 and 1 are 13
13 and 1 are 14
10 and 2 are 12
11 and 2 are 13
12 and 2 are 14
13 and 2 are 15
10 and 3 are 13
Hand 3 are 14
12 and 3 are 15
13 and 3 are 16
10 and 4 are 14
Hand 4 are 15
12 aud 4 are 6
13 aud 4 are 17
10 and 5 are 15
11 and 5 are 16
12 and 5 are 17
13 and 5 are 18
10 and G are 16
Hand 6 are 17
12 and 6 are 18
13 and 6 are 19
10 and 7 arc 17
11 and 7 are 18
12 and 7 are 19
13 and 7 are 20
10 and 8 are 18
11 and 8 are 19
12 and 8 are 20
13 and 8 are 21
Wand 9 are 19
11 and 9 are 20
12 and 9 are 21
13 and 9 are 22
10 and 10 are 20
11 and 10 are 21
12 and 10 are 22
13 and 10 are 23
10 an'l 11 are 21
11 "and 1 1 are 22
12 and 11 are 23
13 and 11 are 24
10 and 12 are 22
11 and 12 are 23
12 and 12 are 24
13 and 12 are 25
ADDITION. 19
MENTAL EXERCISES.
1. A farmer paid 6 dollars for a straw-cutter, and 9 dol-
lars for a plow ; how much did he pay for both ?
ANALYSIS. He paid the sum of 6 dollars and 9 dollars, which
:s 15 dollars. Therefore, he paid 15 dollars for both.
2. John gave 4 apples to James, 8 to Henry, and 9 to
Asa ; how many did he give to all ?
8. Gave 7 dollars for a barrel of flour, 9 dollars for a
hundred weight of sugar, and 6 dollars for a tub of butter ;
how much did I give for the whole ?
4. I have two pear trees ; one tree produced 12 bushels
of pears, aud the other 11 bushels ; how many bushels did
both produce ?
5. A man bought 4 cords of wo^d for 12 dollars, and 7
bushels of corn for 5 dollars ; how much did he pay for
both?
6. James gave 11 cents for a slate, and had 8 cents left ;
how many cents had he at first ?
7. A lady paid 5 dollars for a bonnet, 10 dollars for a
shawl, and had 7 dollars left ; how much money had she at
first ?
8. In a shop are 8 men, 9 boys, and 6 girls, at work ;
how many persons are at work in the shop ?
9. Rollin bought a quire of paper for 12 cents, a slate
for 13 cents, and gave 10 cents to a beggar j how much
money did he pay out in all ?
10. A man bought 4 bushels of wheat for 7 dollars, 18
bushels of corn for 11 dollars, and 2 cords of wood for 5
dollars ; how much did he pay for the whole ?
11. A farmer has 6 cows in one yard, 9 in another, and
as many in the third yard as in both the others ; how
many cows has he ?
20
SIMPLE NUMBERS.
PROMISCUOUS A
2 and 5 are how many ?
6 and 2 are how many ?
2 and 4 are how many ?
8 and 9 are how many ?
9 and 4 are how many ?
4 and 7 are how many ?
8 and 6 are how many ?
6 and 8 are how many ?
7 and 2 are how many ?
DDITION TABLE.
7 and 9 are how many ?
6 and 5 are how many ?
3 and 6 are how many ?
4 and 4 are how many ?
7 and 8 are how many ?
9 and 3 are how many ?
5 and 4 are how many ?
3 and 8 are how many ?
5 and 6 are how many ?
3 and 9 are how many ?
4 and 5 are how many ?
9 and 8 are how many ?
8 and 5 are how many ?
4 and 9 are how many ?
5 and 4 are how many ?
2 and 7 are how many ?
7 and 5 are how many ?
5 and 2 are how many ?
5 and 8 are how many ?
3 and 7 are how many ?
6 and 4 are how many ?
7 and 6 are how many ?
6 and 8 are how many ?
9 and 5 are how many ?
8 and 3 are how many ?
9 and 6 are how many ?
5 and 7 are how many ?
6 and 9 are how many ?
7 and 7 are how many ?
8 and 4 are how many ?
8 and 7 are how many ?
4 and 8 are how many ?
9 and 2 are how many ?
5 and 3 are how many ?
6 and 6 are how many ?
7 and 4 are how many ?
4 and 6 are how many ?
7 and 3 are how many ?
2 and 8 are how many ?
5 and 9 are how many ?
8 and 8 are how many ?
6 and 7 are how many ?
5 and 5 are how many ?
9 and 7 are how many *?
9 and 9 are how many 1
37. The Sign of Addition is the perpendicular cross, -f,
called plus. It indicates that the numbers connected by it
are to be added ; as 3 -f- 5 -j- 7, read 3 plus 5 plus 7.
38. The Sign of Equality is two short, parallel, hori-
zontal lines, =. It indicates that the 'numbers, or combi-
nation of numbers, connected by it are equal j as 4 -J- 8 =
9 -f 3 ; read the sum of 4 plus 8 is equal to the sum of 9
plus 3
ADDITION. 21
CASE I.
39. "When the amount of each column is less
than 10.
I. A drover bought three flocks of sheep. The first
contained 232, the second 422, and the third 245; how
many did he buy in all ?
OPERATION. ANALYSIS. We arrange the numbers so
. j that units of like order shall stand in the
.gj'l same column. We then add the columns
232 separately, for convenience commencing at
422 the right hand, and write each result under
245 *h e c l un:) n added. Thus, we have 5 and 2
and 2 are 9, the sum of the units ; 4 and 2
Amount, 899 and 3 are 9, the sum of the tens ; 2 and 4
and 2 are 8, the sum of the hundreds.
Hence, the entire amount is 8 hundreds 9 tens and 9 units, or
899, the Answer.
EXAMPLES FOR PRACTICE.
(2.) (3.) (4.) (5.)
403 164 510; 234
271 321 176 324
124 510 203 140
Ans. 798
(6.) (7.) (8.) (9.)
1234 2041 3102 4100
2405 3216 2253 1523
5140 1500 4014 2041
Ans. 8779
10. What is the sum of 421, 305 and 5162 ?
II. What is the sum of 3121, 436 and 2002 ?
22 SIMPLE NUMBERS.
CASE II.
4O. "When the amount of any column equals or
exceeds 10.
1. A merchant pays 397 dollars for freights, 476 dollars
for a clerk, and 873 for rent of a store ; what is the amount
of his expenses ?
OPERATION. ANALYSIS. We arrange the numbers so
that units of like order shall stand in the same
. column. We then add the first, or right hand
column, and find the sum to be*16 units, or 1
_ ten and 6 units ; writing the 6 units under
1746 the column of units, we add the 1 ten to the
column of tens, and find the sum to be 24
tens, or 2 hundreds and 4 tens; writing the 4 tens under tUe
column of tens, we add the 2 hundreds to the column of hundreds,
and find the sum to be 17 hundreds, or 1 thousand and 7 hun-
dreds ; writing the 7 hundreds under the column of hundreds,
and the 1 in thousands' place, we have the entire sum, 1746.
NOTES. 1. In adding, learn to pronounce the partial results without naming the
figures separately. Thus, in the operation given for illustration, say 3, 9, 16 ; 8,
15, 24 ; 10, 14, 17.
2. When 1 he sum of any column is greater than 9, the process of adding the
tens to the next column is called carrying.
41. From the proceeding examples and illustrations we
deduce the following
RULE. I. Write the numbers to be added so tliat all the
wiits of the same order shall stand in the same column ;
that is, units under units, tens under tens, &c.
II. Commencing at units, add each column separately,
and write the sum underneath, if it be less than ten.
III. If the sum of any column be ten or more than ten,
write the unit figure only, and add tlie ten or tens to the
next column,
IV. Write the entire sum of the last column.
ADDITION. 23
PROOF. .Begin with the right hand or unit column, and
add the figures in each column in an opposite direction from
that in which they were first added ; if the two results agree,
the work is supposed to be right.
EXAMPLES FOR PRACTICE.
(1.) (2.) (3.) (4.) (5.)
inches, feet. pounds. yards. miles.
142 325 75 407 1270
325 46 276 96 342
476 674 508 2584 79
943 1045 859 3087 1691
(6.) (7.) (8.) (9.) (10.)
842 376 426 713 4761
396 407 397 86 374
472 862 450 345 83
205 94 294 60 19
11. What is the sum of 912 -f 342 -f 187 -j- 46 ?
Ans. 1487.
12. What is the sum of 214 -f 425 +90 -f 37 ?
Ans. 766.
13. What is the sum of 56 feet, 450 feet, and 680 feet ?
Ans. 1186 feet.
14. What is the sum of 1942 dollars, and 685 dollars ?
15. A man paid 375 dollars for a span of horses, 160
dollars for a carriage, and 87 dollars for a harness ; how
much did he pay for all ? Ans. 622 dollars.
16. A man traveled 476 miles by railroad, 390 miles by
steamboat, and 120 miles by stage ; how many miles in all,
did he travel ? Ans. 986.
17. A carpenter built a house for 2464 dollars, a barn for
496 dollars, and outhouses for 309 dollars ; how much did
he receive for building all ?
SIMPLE NUMBEKS.
18. A merchant bought at public auction 520 yards of
broadcloth, 386 yards of muslin, 92 yards of flannel, and
156 yards of silk ; how many yards in all ?
19. A father divided his estate among his four sons, giv-
ing each 2087 dollars ; what was the amount of his estate ?
20. Three persons deposited money in a bank ; the first
4780 dollars, the second 3042 dollars, and the third 407
dollars ; how much did they all deposit ?
21. Five men engage in business as partners, and each
puts in 2375 dollars ; what is the whole amount of capital
invested ? ^ Ans. 11875 dollars.
(22.) (23.)
765 347
381 192
976 763
315 410
169 507
Am. 2606
(26.) (27.)
767346 374205
432761 108497
386109 643024
508763 879638
Am. 2094979
29. 3720 -f 647
(24.)
630
815
456
307
960
(25.)
4603
7106
972
385
64
(28.)
4076315
5632870
8219634
3827692
190 -f 82 = how many ?
Ans. 4639.
30. 962 -|- 2161 -f 500 + 75 = how many ?
Ans. 3698,
31. 4170 -j- 1009 -f 642 + 120 -f 18 = how many ?
32. 3000 + 47602 + 805 + 1266 + 76 = how many ?
33. 69 + 4030 -f 349 -f 1384 + 72 + 400 = how many?
'ADDITION. 20
34. What is the um of two thousand eight hundred fif-
ty-six, twelve thousand eighty-four, seven hundred forty-
two, and sixty-nine ? Arts. 14751.
35. What is the amount of twenty thousand five hundred
ten, six thousand nine hundred forty-four, and three thou-
sand two hundred ? Ans. 30654.
36. What is the sum of forty-seven thousand fifty, nine
thousand one hundred six, fourteen hundred ninety-two, and
five hundred twelve ? Ans. 58160.
37. What is the sum of one hundred forty thousand three
hundred thirty-four, seventy nine-thousand six hundred five,
twenty-five hundred twenty-five, and three thousand sixty-
nine? Ans. 225533.
38. What is the amount of five hundred thousand five
hundred five, eighty-four thousand two hundred, fifteen
thousand six hundred twenty, and seventeen hundred sev-
enteen? ' Ans. 602042.
89. How many men in an army consisting of 26840
infantry, 6370 cavalry, 3250 dragoons, 750 artillery, and
\ 320 miners 1 Ans. 37530.
M:0. A merchant deposited 125 dollars in bank on Mon-
day, 91 on Tuesday, 164 on Wednesday^ 200 on Thursday,
196 on Friday, and 73 on Saturday; how much did he de-
posit during the week 1
*41. By selling a farm for 8586 dollars, 684 dollars were
lost ; how much did the farm cost ?
-*42. If I were born in 1840, when will I be 63 years old?
. 43.. A man willed his estate to his wife, two sons and
three daughters; to his daughters he gave 1565 dollars
apiece, to his sons 3560 dollars each, and to his wife 4720
* dollars ; . how much was his estate 1 Ans. 16535 dollars.
44. A man engaging in trade, gained 450 dollars the first
2
26 SIMPLE NUMBERS.
year, 684 dollars the second, and as much the third as he
gained during the first and second ; how much was his whole
gain 1 Ans. 2268 dollar?.
45. I bought three village lots for 12570 dollars, and
sold them so as to gain 745 dollars on each lot ; for how
much did I sell them 1 Avis. 14805 dollars.
46. A has 3240 dollars, B has 5672 dollars, and C has
1000 more than A and B together ; how many dollars have
all 1 Ans. 18824 dollars.
47. A man was 32 years old when his son was born ; how
old will he be when his son is 3G years old 1 Ans. 68 years.
48. The Old Testament contains 39 books, 929 chapters,
23214 verses, 592439 words, and 2728100 letters; the New
Testament contains 37 books, 269 chapters, 7959 verses,
181153 words, and 838380 letters; what is the total num-
ber of each in the Bible 1 ^
Ans. 76 books, 1198 chapters, 31173 verses, 773592
words, and 3566480 letters.
49. The number of immigrants landed in New York in
1858 was 78589, in 1859, 79322, and in 1860, 103621 ;
what was the total number landed in the three years ?
Ans. 261532.
50. In 1860, the population of New York was 814277,
of Philadelphia 568034, of Boston 177902, of New Orleans
170766, of St. Louis 162179, of Chicago 109429, and 3t
Cincinnati 160000 ; what was the total population of these
cities 1 Ans. 2162587 ^
51. In the year 1856, the United States exported molasses
to the value of 154630 dollars; in 1857,108003 dollars;
in 1858, 115893 dollars; what was the value of the molas-
ses exported in those three years'? Ans. 378526 dollar*.
52. During the same years, respectively, the United States
ADDITION. 27
exported tobacco to the value of 1829207 dollars, 1458553
dollars, and 2410224 dollars ; what was the total value of the
tobacco exported in those years 1 Ans. 5697984 dollars.
53. How many miles from the southern extremity of Lake
Michigan to the Gulf of St. Lawrence, .passing through
Lake Michigan, 330 miles ; Lake Huron, 260 miles ; River
St. Clair, 24 miles; Lake St. Clair, 20 miles; Detroit
River, 23 miles; Lake Erie, 260 miles; Niagara River, 34
miles ; Lake Ontario, 180 miles ; and the River St. Law-
rence, 750 miles? Ans. 1881 miles.
54. At the commencement of the year 1858 there were
in operation in the New England States, 3751 miles of rail-
road ; in New York, 2590 miles ; in Pennsylvania, 2546 ;
in Ohio, 2946; in Virginia, 1233 ; in Illinois, 2678 ; and
in Georgia, 1233 ; what was the aggregate number of miles
in operation in all these States 1 Ans. 16977.
55. The number of pieces of silver coin made at the Uni-
ted States Mint at Philadelphia in the year 1858, were as
follows: 4628000 half dollars, 10600000 quarter dollars,
690000 dimes, 4000000 half dimes, and 1266000 three-
cent pieces ; what was the total number of pieces coined 1
Ans. 21184000.
(56.) (57.) (58.) (59.)
344 843 1186 81988
388 738 513 380167
613 237 740. 108424
803 218 1820 193686
825 347 955 144225
412 288 736 112558
322 483 810 107481
886 753 511 176826
620 834 1179 145851
5213 8450 1451206
28
SIMPLE NUMBERS.
(60.)
35938
49172
56546
82564
69789
47321
77563
83563
54973
38137
54246
95864
48135
37975
48467
(61.)
47197
63956
85678
35495
16457
94667
76463
34698
17179
93965
81367
29787
79826
31275
59689
(62.)
12380
98795
23442
87639
91758
19347
81731
29342
75659
35446
98237
12845
87677
23444
39878
(63.)
456568
754712
567346
543678
3427G6
768345
563875
547427
945956
165675
756431
354747
543864
567456
621367
(64.)
768856
674387
978874
567678
568594
639678
669657
594886
695756
789568
689689
638786
675968
958789
769896
153674
331767
355989
(65.)
576654
678456
754543
786567
964432
699678
978321
678789
564673
895437
569128
678982
869771
668339
956234
195876
957412
573375
(66.)
987654
123456
876864
234246
765183
345927
654678
456432
345719
765391
673123
437987
566789
544321
891389
219721
625247
431321
(67.)
9873785
1239564
7591074
3517569
8598674
2513756
8454210
7656754
5467856
5645781
7893344
8216677
4569911
6543344
9576677
1539900
6662233
4235566
11522492
13046667
9945448
99796675
SUBTRACTION.
29
SUBTRACTION.
42. Subtraction is the process of determining the
difference, between two numbers of the same unit value.
43. The Difference or Remainder is the result obtained.
SUBTRACTION TABLE.
1 from 2 leaves 1
2 from 3 leaves 1
3 from 4 leaves 1
4 from 5 leaves 1
1 from 3 leaves 2
2 from 4 leaves 2
3 from 5 leaves 2
4 from 6 leaves 2
1 from 4 leaves 3
2 from 5 leaves 3
3 from 6 leaves 3
4 from 7 leaves 3
1 from 5 leaves 4
2 from 6 leaves 4
3 from 7 leaves 4
4 from 8 leaves 4
1 from 6 leaves 5
2 from 7 leaves 6
3 from 8 leaves 5
4 from 9 leaves 5
1 from 7 leaves 6
2 from 8 leaves 6
3 from 9 leaves 6
4 from 10 leaves 6
1 from 8 leaves 7
2 from 9 leaves 7
3 from 10 leaves 7
4 from 11 leaves 7
1 from 9 leaves 8
2 from 10 leaves 8
3 from 11 leaves 8
4 from 12 leaves 8
1 from 10 leaves 9
2 from 11 leaves 9
3 from 12 leaves 9
4 from 13 leaves 9
1 from 11 leaves 10
2 from 12 leaves 10
3 from 13 leaves 10
4 from 14 leaves 10
5 from 6 leaves 1
6 from 7 leaves 1
7 from 8 leaves 1
8 from 9 leaves 1
5 from 7 leaves 2
6 from 8 leaves 2
7 from 9 leaves 2
8 from 10 leaves 2
5 from 8 leaves 3
6 from 9 leaves 3
7 from 10 leaves 3
8 from 11 leaves 3
5 from 9 leaves 4
6 from 10 leaves 4
7 from 11 leaves 4
8 from 12 levves 4
5 from 10 leaves 5
6 from 11 leaves 5
7 frm 12 leaves 5
8 from 13 leaves 5
5 from 11 leaves 6
6 from 12 leaves 6
7 from 13 leaves 6
8 from 14 leaves 6
5 from 12 leaves 7
6 from 13 leaves 7
7 from 14 leaves 7
8 from 15 leaves 7
5 from 13 leaves 8
6 from 14 leaves 8
7 from 15 leaves 8
8 from 16 leaves 8
5 from 14 leaves 9
6 from 15 leaves 9
7 from 16 leaves 9
8 from 17 leaves 9
5 from 15 leaves 10
6 from 16 leaves 10
7 from 17 leaves 10
8 from 18 leaves 10
9 from 10 leaves 1
10 from 11 leaves 1
11 from 12 leaves 1
12 from 13 leaves 1
9 from 11 leaves 2
10 from 12 leaves 2
11 from 13 leaves 2
12 from 14 leaves 2
9 from 12 leaves 3
10 from 13 leaves 3
11 from 14 leaves 3
12 from 15 leaves 3
9 from 13 leaves 4
10 from 14 leaves 4
11 from 15 leaves 4
12 from 16 leaves 4
9 from 14 leaves 5
10 from 15 leaves 5
11 from 16 leaves 5
12 from 17 leaves 6
9 from 15 leaves 6
10 from 1(3 leaves 6
11 from 17 leaves 6
12 from 18 leaves 6
9 from 16 leaves 7
10 from 17 leaves 7
11 from 18 leaves 7
12 from 19 leaves 7
9 from 17 leaves 8
10 from 18 leaves 8
11 from 19 leaves 8
12 from 20 leaves 8
9 from 18 leaves
10 from 19 leaves 9
11 from 20 leaves 9
12 from 21 leaves 9
9 from 19 leaves 10
10 from 20 leaves 10
11 from 21 leaves 10
12 from 22 leaves 10
80 SIMPLE NUMBERS.
MENTAL EXERCISES.
1. A grocer having 20 boxes of lemons, sold 12 boxes \
how many boxes had he left]
ANALYSIS. He had left the difference between 20 boxes and
12 boxes, which is 8 boxes. Therefore, he had 8 boxes left.
2. If a man earn 12 dollars a week, and spend 7 for pro-
visions, how many dollars has he left f (
3. If I borrow 15 dollars, and pay 9 dollars, how many
dollars remain unpaid ?
4. John had 11 marbles, and lost 5 of them; how many
had he left?
5. From a cistern containing 22 barrels of water, 9 barrels
leaked out ; how many barrels remained 1
6. In a school are 24 boys and Ii2 girls ; hoW many more
boys than girls ?
7. From a piece of cloth containing 17 yards, 8 yards
were cut; how many yards remained ?
8. Grin paid 15 dollars for a coat, and 9 dollars for a
pair of pantaloons ; how niuch more did he pay for the coat
than for the pantaloons ? '
9. Cora is 23 years old, and her brother is 10 years
younger ; how old is her brother ?
10. A jeweler bought a watch for 11 dollars, and sold it
for 1 8 dollars ; how much did he gain ?
11. A boy gave 21 cents for some pictures, which were
worth no more than 17 cents; how much more than their
value did he give for them 1
12. A grocer bought a barrel of sugar for 16 dollars,
but it not proving as good as he expected, he sold it for 11
dollars ; how much did he lose on it ?
SUBTRACTION.
PROMISCUOUS SUBTRACTION TABLE.
5 from 1 i how many ?
5 from 9 how many ?
9 from 10 how many ?
6 from 7 how many ?
7 from 12 how many ?
9 from 12 how many ?
5 from 10 how many ?
6 from 11 how many ?
6 from 14 how many?
8 from 15 how many?
5 from 11 how many?
7 from 10 how many ?
3 from 13 how many ?
9 from 11 how many ?
6 from 12 how many ?
8 from 10 how many ?
8 from 9 how many ?
7 from 16 how many ?
2 from 11 how many?
5 from 8 how many ?
9 from 14 how many ?
9 from 13 how many ?
7 from 9 how many ?
2 from 10 how many ? .
4 from 11 how many ?
3 from 10 how many ?
5 from 12 how many ?
7 from 13 how many ?
8 from 12 how many ?
9 from 16 how many ?
6' from 13 how many ?
4 from 12 how many?
8 from 16 how many ?
9 from 1 5 how many ?
7 from 11 how many?
3 from 12 how many ?
6 from 15 how many ?
9 from 18 how many ?
6 from 10 how many ?
4 from 13 how many ?
7 from 15 how many ?
8 from 17 how many ?
4 from 10 how many ?
7 from 14 how many ?
3 from 11 how many ?
5 from 13 how many ?
9 from 17 how many ?
8 from 14 how many?
44. The Minuend is the number to be subtracted from.
45. The Subtrahend is the number to be subtracted.
46. The Sign of Subtraction is a short horizontal
line, , called minus. When placed between two numbers,
it denotes that the one after it is to be taken from the
one before it. Thus, 8 6=2, .is read 8 minus 6 equals
2, and denotes that 6, the subtrahend, taken from 8, the
minuend) equals 2, the remainder.
32 SIMPLE NUMBERS.
CASE i.
47. When no figure in the subtrahend is greater
than the corresponding figure in the minuend.
1. From 574 take 323.
OPERATION. ANALYSIS. We write the less num
Minuend, 574 her under the greater, with units un-
Subtrahend, 823 d
777 draw a line underneath. Then, be-
der ' ginning at the right hand, we sub-
tract separately each figure of the subtrahend from the figure
above it in the minuend. Thus, 3 from 4 leaves 1, which is the
difference of the units; 2 from 7 leaves 5, .the difference of the
tens ; 3 from 5 leaves 2, the difference of the hundreds. Hence,
we have for the whole difference, 2 hundreds 5 tens and 1 unit,
or 251.
EXAMPLES FOR PRACTICE.
(1.) (2.) (3.) (40
Minuend, 876 349 637 508
Subtrahend, 435 212 431 104
Remainder, 441 137 206 404
(5.) (6.) (7.) (8.)
987 753 438 695
647 502 421 535
340 251 17 160
(9.) (10.) (11.) (12.)
From 7642 8730 2369 9786
Take 3211 6430 2104 3126
4431 2300 265 6660
SUBTRACTION. 33
Remainders.
13. From 4376 take 1254. 3122.
14. From 70342 take 5*0130. 20212.
15. From 137647 take 16215. 121432.
16. Subtract 32014 from 86325. 54311.
17. Subtract 217356 from 719568. 502212.
18. 437615 213502 = how many ? 224113. ,
19. 732740 11520 = how many ? 721220
20. 2042674 32142 = how many ?
21. 8461203 7161003 = how many ?
22. From three thousand two hundred seventy-six, take
two thousand one hundred forty-three.
23. From one hundred eighty-three thousand four hun-
dred sixty, take fifty-two thousand one hundred fifty.
Am. 131310.
24. A man bought a piece of property for 7634 dollars,
and sold the same for 3132 dollars ; how much did he lose ?
Ans. 4502 dollars.
25. A merchant sold goods to the amount of 41763 dol-
lars, and by so doing gained 11521 dollars ; how much did
the goods cost him 1 Ans. 30242 dollars.
26. A drover bought 3245 sheep, and sold 1249 of them
how many sheep had he left ?
27. A general before commencing a battle had 18765 men
in his army ; after the battle he had only 8530 ; how many
men did he lose 1 Ans. 10235.
28. Two persons bought a block of buildings for 69524
dollars ; one paid 47321 dollars ; how much did the other
pay 1 Ans. 22203 dollars.
29. If a man's annual income is 13460 dollars, and hig
expenses are 3340 dollars, how much does he save ?
Ans. 10120 dollars.
34 SIMPLE NUMBEKS.
CASE II.
48. When any figure in the subtrahend is greater
than the corresponding figure in the minuend.
I. From 846 take 359.
OPERATION. ANALYSIS. Since we cannot take 9 units from
| ^ _jo 6 units, we add 10 units to 6 units, making 16
J2 units; and 9 units from 16 units leave 7 units.
But as we have added 10 units, or 1 ten to the
minuend, we shall have a remainder 1 ten too
'. Q I large, to avoid which, we add 1 ten to the 5 tens
in the subtrahend, making 6 tens. We can not
take 6 tens from 4 tens; so we add 10 tens to 4, making 14
tens ; 6 tens from 14 tens leave 8 tens. Now, having added 10
tens, or 1 hundred, to the minuend, we shall have a remainder 1
hundred too large, unless we add 1 hundred to the 3 hundreds
in the subtrahend, making 4 hundreds ; 4 hundreds from 8 hun-
dreds leave 4 hundreds, and we have for the total remainder, 487.
NOTE. The process of adding 10 to the minuend is sometimes called borrowing
10; and that of adding 1 to the next figure of the subtrahend, carrying one .
4tO. From the preceding example and illustration we
have the following general
RULE. I. Write the less number under the greater, plac-
ing units of the same order in the same column.
II. Beginning at the right hand, take each figure of the
subtrahend from the figure above it, and write the result un
dern&ath.
III. If any figure in the subtrahend be greater than the
corresponding figure above it, add 10 to that upper figure
before subtracting, and then add 1 to the next left hand fig-
ure of the subtrahend.
PROOF. 1st. Ad'd the remainder to the subtrahend; the
sum will be equal to the minuend. Or,
2d. Subtract the remainder from the minuend; the dif-
ference will be equal to the subtrahend.
SUBTRACTION.
36
EXAMPLES EOR PRACTICE.
Minuend,
00
753
(20
6731
(3.)
3248
(4.)
90361
Subtrahend,
Remainder,
469
2452
1863
6284
284
4279
1385
84077
(5.)
miles.
(6.)
bushels.
(70
dollars.
(8.)
feet.
3146
19472
45268
24760
2529
14681
24873
3478
617
4791
20395
23282
(9.)
rods.
40307
(10.)
days.
14605
(11.)
acres.
23617
(12.)
gallons.
980076
38421
1886
8341
14309
94087
6264
9308
885989
(13.)
men.
17380
(14.)
sheep.
282731
(15.)
barrels.
80014
(16.)
tons.
941000
3417
90756
43190
5007
13963
191975
36824
935993
(17.)
8077097
(18.)
3000001
(19.)
1970000
1829164
2199077
1361111
1247933
800924
608889
36
SIMPLE NUMBERS.
(20.)
6000000
999999
500Q001
(21.)
8000800
457776
7543024
(22.)
103810040
91300397
12509643
Ans. 224130.
23. 234100 9970 == how many ?
24. 3749001 349623=how many ?
25. 4000320 20142 = how many ?
26. 14601896 764059 = how many ?
27. From 4716359 take 2740714. Ans. 1975645.
28. From 7867564 take 2948675. Ans. 4918889.
29. From 7788996 take 849842. Ans. 6939154.
30. From 1073563 take 182000. Ans. 891563.
31. From 1111111 take 111112. Ans. 999999.
82. Subtract 1234509 from 8643587. Ans. 7409078.
33. Subtract 1000 from 1100000. Ans. 1099000.
34. Subtract 100701 from 846587.
35. Subtract 432986702100 from 539864298670.
36. Subtract 29176807982 from 86543298765.
37. A speculator boughf wild lands for 10580 dollars,and
sold them for 7642 dollars; how much did he lose ?
Ans. 2938 dollars.
38. Napoleon the Great was born in 1769, and died in
1821 ; how old was he at his death ? Ans. 52 years.
39. Gunpowder was invented in 1330 ; and printing in
1440 ; how many years between the two ? Ans. 110.
40. George Washington was born in 1732, and died in
1799 ; how old was he at his death ? Ans. 67 years.
41. The first newspaper published in America was issued
at Boston in 1704 ; how long Was that before the death of
Benjamin Franklin, which occurred in 1790 ?
Ans. 86 years.
PROMISCUOUS EXAMPLES. 37
42. The first steamboat in the United States, built by
Robert Fulton, in 1807, made a trip from New York
to Albany in 33 hours ; how many years from that time to
the visit of the Great Eastern to this country in 1860 ?
Ans. 53 years.
43. Queen Victoria was born in 1819 ; what will be her
age in 1862 ? Ans. 43 years.
44. The United States contain 2983153 square miles,
and the British North American Provinces 3125401 square
miles. How many square miles does the latter country ex-
ceed the former ? Ans. 142248.
EXAMPLES COMBINING ADDITION AND SUBTRACTION.
1. A farmer having 450 sheep, sold 124 at one time, and
96 at another ; how many had he left ? Ans. 230.
2. If a man's income is 175 dollars a month, and he pays
25 dollars for rent, 44 dollars for provisions, and 18 dollars
for other expenses, how much will he have left ?
Ans. 88 dollars.
3. A man gave his note for 3245 dollars. He paid at
one time 780 dollars, and at another 484 dollars ; how much
remained unpaid ? Ans. 1981 dollars?
4. A man paid 140 dollars for a horse and 165 dollars for a
carriage. He afterward sold them both for 300 dollars ] did
he gain or lose, and how much ? Ans. Lost 5 dollars.
5. A flour merchant having 700 barrels of flour on hand,
sold 278 barrels to one man, and 142 to another ; how many
barrels had he left ? ~~ Ans. 280 barrels.
6. Three men bought a farm for 9840 dollars. The first
paid 2672 dollars, the second paid 3089 dollars, and the
third the remainder ; how much did the third pay ?
Ans 4079 dollars.
88 SIMPLE NUMBERS.
7. A man bought a house for 1500 dollars, and having
expended 315 dollars for repairs, sold it for 2000 dollars ;
how much was his gain ? Ans. 185 dollars.
8. Henry Jones owns property to the amount of 36748 dol-
lars, of which he has invested in real estate 12850 dollars, in
personal property 9086 dollars, and the remainder he has in
bank ; how much has he in bank ? Ans. 14812 dollars.
9. A grocer bought 275 pounds of butter of one farmer,
and 318 pounds of another; he afterward sold 210 pounds
to one customer, and 97 to another ; how many pounds had
he left ? Ans. 286 pounds.
10. A man deposited in bank 10476 dollars ; he drew
out at one time .2356 dollars, at another 1242, and at anoth-
er 737 dollars ; how much had he remaining in bank ?
Ans. 6141 dollars.
11. Borrowed of my neighbor at one time 680 dollars, at
another time 910 dollars, and at another time 218 dollars.
Having paid him 1309 dollars, how much do I still owe
him ? Ans. 499 dollars.
12. A man bought 3 lots ; for the first he paid 2480 dol-
lars, for the second 3137 dollars, and for the third as much
as for the other two ; he afterward sold them for 15000
dollars ; how much was his gain ? Ans. 3766 dollars.
13. A farmer raised 1864 bushels of wheat, and 1129
bushels of corn. Having sold 1340 bushels of wheat, and
1000 bushels of corn, how many bushels of each has he re-
mainiDg ? Ans. 524 bushels, and 129 bushels.
14. A gentleman worth 25800 dollars, bequeathed his es-
tate so that each of his two sons should have 9400 dollars,
and his daughter the remainder. How much was the
daughter's portion ?
MULTIPLICATION.
39
MULTIPLICATION.
50. Multiplication is the process of taking one of two
given numbers as many times as there are un*its in the other.
51. The Product is the result obtained.
MULTIPLICATION TABLE.
Once 1 is 1
2 times 1 are 2
3 times 1 are 3
4 times 1 are 4 i
Once 2 is 2
2 times 2 are 4
3 times 2 are 6
4 times 2 are 8
Once 3 is 3
2 times 3 are 6
8 times 8 are 9
4 times 3 are 12
Once 4 is 4
2 times 4 are 8
3 times 4 are 12
4 times 4 are 16
Once 5 is 6
2 times 5 are 10
3 times 5 are 15
4 times 5 are 20
Once 6 is 6
2 times 6 are 12
3 times 6 are 18
4 times 6 are 24
Once 7 ia 7
2 times 7 are 14
3 times 7 are 21
4 times 7 are 28
Once 8 is 8
2 times 8 are 16
3 times 8 are 24
4 times 8 are 32
Once 9 is 9
2 times 9 are 18
3 times 9 are 27
4 times 9 are 36
Once 10 ia 10
2 times 10 are 20
3 times 10 are 30
4 times 10 are 40
Once 11 is 11
2 times 11 are 22
3 times 11 are 33
4 times 11 are 44
. Once 12 is 12
2 times 12 are 24
8 times 12 are 36
4 times 12 are 48
5 times 1 are 5
6 times 1 are 6
7 limes 1 are 7 .
8 times 1 are 8
5 times 2 are 10
6 times 2 are 12
7 times 2 are 14
8 times 2 are 16
5 times 3 are 15
6 times 3 are 18
7 times 3 are 21
8 times 3 are 24
5 times 4 are 20
6 times 4 are 24
7 times 4 are 28
8 times 4 are 32
5 times 5 are 25
6 times 5 are 30
7 times 6 are 35
8 times 5 are 40
5 times 6 are 30
6 times 6 are 36
7 times 6 are 42
8 times 6 are 48
5 times 7 are 35
6 times 7 are 42
7 times 7 are 49
8 times 7 are 56
5 times 8 are 40
6 times 8 are 48
7 times 8 are 56
8 times 8 are 64
5 times 9 are 45
6 times 9 are 54
7 times 9 are 63
8 times 9 are 72
5 times 10 are 50
6 times 10 are 60
7 times 10 are 70
8 times 10 are 80
5 times 11 are 55
6 times 11 are 66
7 times 11 are 77
8 times 11 are 88
5 times 12 are 60
6 times 12 are 72
7 times 12 are 84
8 times 12 are 96
9 times 1 are 9
10 times 1 are 10
11 times 1 are 11
12 times 1 are 12
9 times 2 are 18
10 times 2 are 20
11 times 2 are 22
12 times 2 are 24
9 times 3 are 27
10 times 3 are 30
11 times 3 are 33
12 times 3 are 36
9 times 4 are 36
10 times 4 are 40
lltimea 4 are 44
12 times 4 are 48
9 times 5 are 45
10 times 5 are 50
11 times 5 are 55
12 times 5 are 66
9 times 6 are 54
10 times 6 are 60
11 times 6 are 66
12 times 6 are 72
9 times 7 are 63
10 times 7 are 70
11 times 7 are 77
12 times 7 are 84
9 times 8 are 72
10 times 8 are 80
11 times 8 are 88
12 times 8 ane 96
9 times 9 are 81
10 times 9 are 90
11 times 9 are 99
12 times 9 are 108
9 times 10 are 90
10 times 10 are 100
11 times 10 are 110
12 times 10 are 120
9 times 11 are 99
10 times 11 are 110
11 times 11 are 121
12 times 11 are 132
9 times 12 are 10S
10 times 12 are 120
11 times 12 are 132
12 times 12 are 144
4:0 SIMrLE NUMBERS.
MENTAL EXERCISES.
1. At 9 cents a pound, what will 7 pounds of sugar cost/
ANALYSIS. Since one pound costs 9 cents, 7 pounds will cost
7 times 9 cents, or 63 cents. Therefore, at 9 cents a pound, 7
pounds of sugar will cost 63 cents.
2. At 6 dollars ? ^eek, what will 8 weeks 7 board cost ?
3. When flour is ( dollars a barrel, what will 11 barrels
cost?
4. If Rollin can earn 10 dollars in one month, how much
can he earn in 4 months ? in 9 months ? in 11 months ?
5. What will be the cost of 12 pounds of coffee, at 9
cents a pound ?
6. At 5 dollars a ton, what will 9 tons of coal cost ?
7. At 4 dollars a yard, what will 8 yards of cloth cost ?
8. If a pair of boots cost 5 dollars, what will be the cost
of 3 pairs ? of 6 pairs ? of 7 pairs ? of 11 pairs 1
9. Since 12 inches make a foot, how many inches in 3 feet ?
in 5 feet ? in 7 feet ? in 9 feet ? in 12 feet ?
10. At five cents a quart, what will 6 quarts of milk
cost ? 10 quarts ? 11 quarts ?
11. If a man earn 8 dollars in a week, how much can he
earn in 6 w;eeks ? in 7 weeks ? in 8 weeks ? in 9 weeks ?
12. If 9 bushels of apples buy one barrel of flour, how
many bushels will be required to buy 3 barrels ? 5 barrels ?
7 barrels *? 9 barrels ?
13. If 4 men can do a piece of work in 8 days, how
many days will it take one man to do it ?
14. If 7 men can build a wall in 3 days, how long will it
take one man to build it ?
15. If a barrel of potatoes last 6 persons 3 weeks, how
many weeks will it last one person ?
MULTIPLICATION.
PROMISCUOUS MULTIPLICATION TABLE.
2 times 8 are how many 1
3 times 9 are how many 1
4 times 8 are how many 1
7 times 5 are how many ?
9 times 4 are how many ?
6 times 3 are how many ?
4 times 9 are how many 1
5 times 9 are how many ?
7 times 6 are how many 1
2 times 9 are how many ?
6 times 5 are how many ?
4 times 7 are how many ]
9 times 3 are how many 1
5 times 7 are how many ?
5 times 8 are how many 1
9 times 5 are how many 1
6 times 4 are how many 1
8 times 3 are how many ?
3 times 7 are how many ?
8 times 9 are how many 1
6 times 8 are how many 1
5 times 6 are how many ]
7 times 3 are how many 1
6 times 6 are how many ?
9 times 7 are how many 1
3 times 8 are how many 1
7 times 4 are how many ?
7 times 7 are how many ?
4 times 2 are how many ?
9 times -9 are how many ?
4 times 3 are how many ?
6 times 9 are how many ?
2 times 6 are how many 1
8 times 5 are how many ?
4 times 4 are how many ?
9 times 8 are how many 1
8 times 7 are how many ?
5 times 4 are how many 1
3 times 5 are how many ?
3 times 4 are how many ?
8 times 6 are how many ?
7 times 8 are how many ?
5 times 3 are how many 1
3 times 6 are how many 1
8 times 8 are how many ?
2 times 4 are how many ?
5 times 9 are how many ?
9 times 8 are how many ?
3 times 3 are how many ?
2 times 3 are how many ?
7 times 4 are how many ?
times 8 are how many ?
3 times 6 are how many ?
6 times 10 are how many ?
The Multiplicand is the number to be taken.
The Multiplier is the number which shows how
many times the multiplicand is to be taken.
54. The Factors are the multiplicand and multiplier.
55. The Sign of Multiplication is the oblique cross,
X It indicates that the numbers connected by it are to
be multiplied together ; thus 9x6 = 54, is read 9 times
equals 54.
42 SIMPLE NUMBERS.
NOTES. 1. Factors axe producers, and the multiplicand and multiplier are called
factors because they produce the product.
2. Multiplication is a short method of performing addition when the numbers to
be added are equal.
CASE I.
56. When the multiplier consists of one figure.
1. Multiply 374 by 6.
OPERATION. ANALYSIS. In this example it is re-
quired to take 374 six times. If we
take the units of each order 6 times,
Multiplicand, 374 we shall take the entire number 6
times. Therefore, writing the multi-
f the mul "
Product, 2244
tiplicand, we proceed as follows: 6
times 4 units are 24 units, which is 2 tens and 4 units ; write the
4 units in the product in units' place, and reserve the 2 tens to
add to the next product; 6 times 7 tens are 42 tens, and the
two tens reserved in the last product added, are 44 tens, which
is 4 hundreds and 4 tens ; write the 4 tens in the product in
tens' place, and reserve the 4 hundreds to add to the next prod-
uct; 6 times 3 hundreds are 18 hundreds, and 4 hundreds add-
ed are 22 hundreds, which, being written in the product in the
places of hundreds and thousands, gives, for the entire product,
2244.
57. The unit value of a number is not changed by re-
peating the number. As the multiplier always expresses
times, the product must have the same unit value as the mul
tiplicand. But, since the product of any two numbers will
be the same, whichever factor is taken as a multiplier, either
factor may be taken for the multiplier or multiplicand.
NOTE. In multiplying, learn to pronounce the partial results, as in addition,
without, naming tin; numbers separately. Thus, in the lasl example, instead of say-
ing ti times 4 are -4, fi times 7 are 42 and 2 to carry an.- 44. ti limes 3 are 18 and 4
to carry are 22; prtmouncfr only the results, 24, 44, 22, performing the operation*
qaentally. Tina will greatly facilitate the process of multiplying.
MULTIPLICATION.
EXAMPLES FOR PRACTICE.
Multiplicand,
(2.)
842
(3.)
625
(40
718
Multiplier,
Product,
4
6
7
3368
3750
5026
(6.)
4328
(7.)
5073
(8.)
1869
8
5
25365
4
34624
7476
(10.)
7186
(11.)
9010
(12.)
4079
3
7
6
21558
63070
24474
(14.)
340071
(15.)
760892
2
4
680142
3043568
* (6.)
937
fc
2811
29385
(13.)
6394
51152
9881150
17. Multiply 473126 by 9. Ans.
18. Multiply 30789167 by 7. Ans.
19. Multiply 87231420 by 8. Ans.
20. What will be the cost of 9380 bushels of wheat, at 9
shillings a bushel ? Ans. 84420 shillings.
21. What will be the cost of 4738 tons of coal, at 4 dol-
lars a ton ? Ans. 18952 dollars.
22. In one mile are 5280 feet; how many feet in 8
miles ? Ans. 42240 feet.
44 SIMPLE NUMBERS.
CASE II.
58. When the multiplier consists of two or more
figures.
1. Multiply 746 by 23.
OPERATION. ANALYSIS. Writ-
Multiplicand, 746 ing the multipli-
cand and multipli-
f times the mul- er as in Case I, we
figure in the mul-
multiplier, precisely as in Case I. We then multiply by the 2 tens.
2 tens times 6 units, or 6 times 2 tens, are 12 tens, equal to 1
hundred, and 2 tens ; we place the 2 tens under the tens figure
in the product already obtained, and add the 1 hundred to the
next hundreds produced. 2 tens times 4 tens are 8 hundreds,
and the 1 hundred of the last product added are 9 hundreds ;
we write the the 9 in hundreds' place in the product. 2 tens
times 7 hundreds are 14 thousands, equal to 1 ten thousand and 4
thousands, which we write in their appropriate places in the
the product. Then adding the two products, we have the en
tire product, 17158.
Hence we deduce the following general
II OLE. I. Write the multiplier under the multiplicand,
placing units of the same order under each other.
II. Multiply the multiplicand by each figure of the multi-
plier successively, beginning with the unit figure, and write
the first figure of each partial product under the figure of
the multiplier used, writing down and carrying as in addi-
tion.
III. If there are partial products, add them, and their
mm will be the product required.
MULTIPLICATION.
45
PROOF. Multiply the multiplier by the multiplicand, and
if the product is the same as the first result, the work is
correct.
NOTH. When the multiplier contains two or more figures, the several results ob-
tained by multiplying by each figure are called partial products.
EXAMPLES FOB PRACTICE.
(1.) (20
(3.)
34732 56784
34075
14 24
36
138928 227136
204450
34732 113568
102225
486248 1362816
1226700
4. Multiply 177242 by 19.
Am. 3367598.
5. Multiply 1429689 by 55.
6. Multiply 364111 by 56.
Ant. 20390216.
7. Multiply 78540 by 95.
An*. 7461300.
8. Multiply 6555 by 39.
An*. 255645.
9. Multiply 76419 by 17.
Ans. 1299123.
10. Multiply 26517 by 45.
Ans. 1193265.
11. Multiply 108336 by 58.
Ans. 6283488.
12. Multiply 209402 by 72.
Ans. 15076944.
13. Multiply 342516 by 56.
Ans. 19180896.
14. Multiply 764131 by 48.
Ans. 36678288.
15. There are 52 weeks in a year ; how many weeks in
1861 years? Ans. $6772 weeks.
16. An army of 5746 men having plundered a city, took
so much money that each man received 37 dollars ; how
much money was taken ? Ans. 212602 dollars.
17. If it cost 47346 dollars to build one mile of railroad,
how much will it cost to build 76 miles ?
Ans. 3598296 dollars.
SIMPLE NUMBERS.
190784
190784
47696
6868224
(19.)
560341
304
2241364
1681023
170343664
(20.)
243042
265
1215210
1458252
486084
64406130
21. Multiply 45678 by 333. Ans. 15210774.
22. Multiply 202842 by 342. Ans. 69371964.'
23. Multiply 9636799 by 489. Ans. 4712394711.
24. Multiply 3064125 by 807. Ans. 2472748875.
25. Multiply 5610327 by 2034.
26. Multiply 1900731 by 4006. Ans. 7614328386.
27. A gentleman bought 307 horses for shipping, at the
rate of 105 dollars each ; how much did he pay for the
whole ?
28. What would be the value of 976 shares of railroad
stock, at 98 dollars a share ? Ans. 95648 dollars.
29. A man bought 48 building lots, at 1236 dollars each;
how much did they all cost him 1 Ans. 59328 dollars.
30. How many yards of broadcloth in 487 pieces, each
piece containg 37 yards ? Ans. 18019 yards.
31. If it require 135 tons of iron for one mile of railroad,
how many tons will be required for 196 miles ?
Ans. 26460.
32. How many oranges in 356 boxes, each box contain-
ing 264 oranges ? Ans. 93984.
33. If it require 6894 shingles for the roof of a house,
how many shingles will be required for 19 such houses ?
MULTIPLICATION. 47
34. 37896X149 =how many ? A-ns. 5646504.
35. 8567X462 =how many 1 ? Ans. 3957954.
36. 6793X842 =how many ? Ans. 5719706.
37. 674200 X 2104 =how many? Ans. 1418516800.
38. 15607X3094 =how many? Ans. 48288058.
39. 83209X4004 =how many ?
40. Multiply 31416 by 175.
41. Multiply 40930 by 779. Ans. 31884470.
42. Multiply 4567 by 9009. Ans. 41144103.
43. Multiply 7071 by 556. Ans. 3931476.
44. Multiply 291042 by 125. Ans. 36380250.
45. Multiply 54001 by 5009.
46. Multiply twelve thousand thirteen, by twelve hun-
dred four. Ans. 14463652.
47. Multiply thirty-seven thousand seven hundred ninety-
six, by four hundred eight.
48. Multiply one million two hundred forty-six thousand
eight hnndred fifty-three, by nine thousand seven.
-4ns. 11230404971.
49. What will be the cost of building 128 miles of rail-
road, at 6375 dollars per mile 1 Ans. 816000 dollars.
50. A crop of cotton was put up in 126 bales, each bale
containing 572 pounds ; what was the weight of the entire
crop ? Ans. 72072 pounds.
51. Two towns, 243 miles apart, are to be connected by a
railroad, at a cost of 39760 dollars a mile ; how much will
be the entire cost of ^he road ? Ans. 9661680 dollars.
52. Allowing an acre of land to produce 105 bushels, how
much would 246 acres produce ? Ans. 25830 bushels.
53. If a garrison of soldiers consume 5789 pounds of
bread a day, how much will they consume in 287 days 1
Ans. 1661443 pounds
48 SIMPLE NUMBERS.
CONTRACTIONS.
CASE I.
59. When the multiplier is a composite num-
ber.
A Composite Number is one that may be produced by
multiplying together two or more numbers ; thus, 18 is a
composite number, since 6x3=18 ; or ; 9x2=18 ; or, 3X
3X2=18.
@. The Component Factors of a number are the sev-
eral numbers which, multiplied together, produce the given
number ; thus, the component factors of 20 are 10 and 2,
(10X2=20); or, 4 and 5, (4x5=20) ; or, 2 and 2 and
5, (2X2X5=20).
XOTE. The pupil must not confound the factors with the parts of a number.
Thus, the/acto?-s of which twelve is composed, are 4 and 3, (4X3=12) ; while the
parts of which 12 is composed are 8 and 4, (8+4=12), or 10 and 2, (10+2=12)
The factors are multiplied, while the parts are added, to produce the number.
I. What will 32 horses cost, at 174 dollars apiece ?
OPERATION. ANALYSIS. The fac-
Muitipiicand 174 cost of 1 horse. tors of 32 are 4 and
1st factor, 4 8. If we multiply the
cost of 1 horse by 4,
696 cost of 4 horses. we obta in the cost of
2d factor, 8 4 horses; and by mul-
tiplying the cost of 4
product, 5568 cost of 32 horses. horses by 8j we obtain
the cost of 8 times 4 horses, or 32 horses, the number bought
61. Hence we have the following
RULE. I. Separate the composite number into two or
more factors.
II. Multiply the multiplicand "by one of these factors, and
that product l)y. another, and so on until all the factors have
Lecn used) the last product will be the product required.
NOTE. The product of any nupiber of factors will be the same in whatever order
they arc multiplied. Thus, 4X-';X5 60, and 5X4X3-CO.
{
MULTIPLICATION. 49
EXAMPLES TOR PRACTICE.
1. Multiply 521 by 16=4x4. ! Am. 8336.
2. Multiply 4350 by 25 = 5x5. Ans. 108750.
3. Multiply 10709 by 36=6x6. Ans. 385524.
4. Multiply 21700 by 27=3x9. Am. 585900.
5. Multiply 783473 by 42=6x7. jAns. 32905866.
6. Multiply 764131, by 48=6X8. lAns. 36678288.
7. Multiply 40567 by 96=8x12. | Ans. 3894432.
8. Multiply 182642 by 120=4x5Xf
\Ans. 21917040.
9. Multiply 20704 by 84=3X4X7.; Ans. 1739136.
10. Multiply 564120 by 140=4X5X7.
I Ans. 78976800.
11. What will 56 acres of land cost, at 147 dollars an
acre? Ans. 8232 dollars.
12. What will 75 yoke of cattle cf>st, at 184 dollars a
yoke ? Ans. 13800 dollars.
13. If a ship sail 380 miles a day, how far will she sail
in 45 days 1 Ans. 17100 miles.
14. What is the value of 3426 pounds of butter, at 18
cents a pound ? tAns. 61668 cents.
15. What would be the cost of 125 horses, at 208 dollar
each ? Ans. 26000 dollars.
16. What would be the value of 1^42 acres of land, at
28 dollars an acre ?
17. What will be the cost of 28 pieces of broadcloth, each
piece containing 42 yards, at 4 dollar
Ans. 4704 dollars.
18. What will be the cost of 16 sa<fks of coffee, each sack
containing 7 r > pounds, at 9 cents a pound ?
' Ans. 10800 cents.
a yard
50 SIMPLE NUMBERS.
CASE II.
62. When the multiplier is 10, *100, 1000, &c.
If we annex a cipher to the multiplicand, each figure is
removed one place toward the left, and consequently the
value of the whole numher is increased ten fold. If two
ciphers are annexed, each figure is removed two places to-
ward the left, and the value of the numher is increased one
hundred fold j and every additional cipher increases the
value tenfold.
Hence the following
RULE. Annex as many ciphers to the multiplicand as
there are ciphers in the multiplier ; the number so formed
will be the product required.
EXAMPLES FOR PRACTICE.
1. Multiply 246 by 10. Ans. 2460.
2. Multiply 97 hy 100. Ans. 9700.
3. Multiply 1476 hy 1000. Ans. 1476000.
4. Multiply 7361 by 10000. Ans. 73610000.
5. At 47 dollars an acre, what will 10 acres of land cost ?
Ans. 470 dollars.
6 What will be the cost of 100 horses, at 95 dollars a
head ? Ans. 9500 dollars.
7. What will be the cost of 1000 fruit trees, at 18 cents
apiece? Ans. 18000 cents.
8. If one acre of land produce 28 bushels of wheat,
how many bushels will 100 acres produce ? Ans. 2800.
9. If a man save 386 dollars a year, how much will he
save in 10 years ? Ans. 3860 dollars.
10. If the freight on a barrel of flour from Chicago to
New York be 47 cents, how much will it be on 100000 bar-
rels ? Ans. 4700000 cents.
MULTIPLICATION. 61
CASE III.
63. When there are ciphers at the right hand of
one or both of the factors.
1. Multiply 7200 by 40.
OPERATION. ANALYSIS. The multiplicand, fac-
Muibpiicand, 7200 tored, is equal to 72 X 100 ; the mul-
Moltipiier, 40 tiplier, factored, is equal to 4 x 10
and as these factors taken in any
order will give the same product,
we first multiply 72 by 4, then this product by 100 by annex-
ing two ciphers, and this product by 10 by annexing one a
pher. Hence, the following
RULE. Multiply the significant figures of the multipli-
cand by those of the multiplier, and to the product annex as
many ciphers as there are ciphers on the right of either or
loth factors.
EXAMPLES FOR PRACTICE.
(1.) (2.) (3.)
Multiply 3900 1760 37200
By 8000 3500 730000
31200000 880 1116
528 2604
6160000 271560000CO
4. Multiply 7030 by 164000. Ans. 1152920000.
5. Multiply 27600 by 48000. Ans. 1324800000.
6. Multiply 403700 by 30200. Ans. 12191740000.
7. At 150 dollars an acre, what will be the cost of 500
acres ol land 1 Ans. 75000 dollars.
8. What will be the freight on 4000 barrels of flour, at
50 cents a barrel 1 Ans. 200000 cents.
9. If there are 560 shingles in a bunch, how many shin-
gles in 26ITO bunches ? Ans. 14560000.
52 SIMPLE NUMBERS.
EXAMPLES COMBINING ADDITION, SUBTRACTION, AND
MULTIPLICATION.
1. Bought 9 cords of wood at 3 dollars a cord, and 15
tons of coal at 5 dollars a ton ; what was the cost of the
wood and coal ? Ans. 102 dollars.
2. A grocer bought 6 tubs of butter, each containing 64
pounds, at 14 cents a pound; and 4 cheeses, each weighing
42 pounds, at 8 cents a pound ; how much was the cost of
the butter and cheese ? Ans. 6720 cents.
3. If a clerk receive 540 dollars a year salary, and pay
180 dollars for board, 116 dollars for clothing, 58 dollars
for books, and 75 dollars for other expenses, how much will
he have left at the close of the year ? Ans. Ill dollars.
4. A farmer having 2150 dollars, bought 536 sheep at
2 dollars a head, and 26 cows at 23 dollars a head ; how
much money had he left ? Ans. 480 dollars.
5. A man sold three horses ; for the first he received 275
dollars, for the second 87 dollars less than for the first, and
for the third as much as for the other two ; how much did
he receive for the third ? Ans. 463 dollars.
6. Bought 76 hogs, each weighing 416 pounds, at 7
cents a pound, and sold the same at 9 cents a pound ; how
much was gained ? Ans. 63232 cents.
7. A man bought 14 cows at 26 dollars each, 4 horses
at 112 dollars each, and 125 sheep at 3 dollars each ] he sold
the whole for 1237 dollars ; did he gain or lose, and how
much? Ans. Gained 50 dollars.
8. B has 174 sheep, C has three times as many lacking
98, and D has as many as B and C together ; how many
eheep has D 1 Ans. 598.
9. There are 36 tubs of butter, each weighing 108
pounds ; the tubs which contain the butter, each weigh 19
PKOMISCUOUS EXAMPLES. 63
pounds j how much is the weight of the butter without the
tubs ? Ans. 3204 pounds.
10 A man paid for building a house 2376 dollars, and for
his farm 4 times as much lacking 970 dollars ; how much
did he pay for both ?
11. A merchant bought 9 hogsheads of sugar at 32 dol-
lars a hogshead, and sold it for 40 dollars a hogshead ; how
much did he gain ? Ans. 72 dollars.
12. Bought 360 barrels of flour for 2340 dollars, and sold
the same at 8 dollars a barrel ; how much was gained by
the bargain ? Ans. 540 dollars.
13. A farmer sold 462 bushels of wheat at 2 dollars a
bushel, for which he received 75 barrels of flour at 9 dol-
lars a barrel, and the balance in money ; how much money
did he receive ? Ans. 249 dollars.
14. Two persons start from the same point, and travel in
opposite directions ; one travels at the rate of 28 miles a
day, the other at the rate of 37 miles a day ; how far apart
will they be in 6 days ? Ans. 390 miles.
15. If a man buy 40 acres of land at 35 dollars an acre,
and 56 acres at 29 dollars an acre, and sell the whole for
32 dollars an acre, how much does he gain or lose ?
Ans. Gains 48 dollars.
16. In an orchard, 76 apple trees yield 18 bushels of ap-
ples each, and 27 others yield 21 bushels each ; how much
would the apples be worth, at 30 cents a bushel ?
Ans. 58050 cents.
17. A man bought two farms, one of 136 acres at 28
dollars an acre, and another of 140 acres at 33 dollars an
acre ; he paid at one time 4000 dollars, and at another time
1875 dollars ; how much remained unpaid ?
Ans. 2553 dollars.
64: SIMPLE NUMBERS.
DIVISION.
G4:. Division is the process of finding how many times
one number is contained in another.
G5. The Quotient is the result obtained, and shows how
many times the divisor is contained in the dividend.
DIVISION TABLE.
1 in 2 2 times
2 in 4 2 times
8 in 6 2 times
1 in 3 3 times
2 in 63 times
3 in 9 3 times
1 in 4 4 times
2 in 8 4 times
8 in 12 4 times
1 in 5 5 times
2 in 10 5 times
3 in 15 5 times
1 in 6 6 times
2 in 12 6 times
3 in 18 6 times
1 in 7 7 times
2 in 14 7 times
,3 in 21 7 times
1 in 8 8 times
2 in 16 8 times
8 in 24 8 times
1 in 9 9 times
2 in 18 9 times
8 in 27 9 times
4 in 8 2 times
5 in 10 2 times
6 in 12 2 times
4 in 12 3 times
5 in 15 3- times
6 in 18 3 times
4 in 16 4 times
B in 20 4 times
6 in 24 4 timea
4 in 20 5 times
6 in 25 5 times
6 in 30 5 times
4 in 24 6 times
5 in 30 6 times
6 in 36 6 times
4 in 28 7 times
5 in 35 7 times
6 in 42 7 times
4 in 32 8 times
5 in 40 8 times
6 in 48 8 times
4 in 36 9 times
5 in 45 9 times
6 in 54 9 times
7 in 14 2 times
8 in 16 2 times
9 in 18 2 times
7 in 21 3 times
8 in 24 3 times
9 in 27 3 times
7 in 28 4 times
8 in 32 4 times
9 in 36 4 times
7 in 35 6 times
8 in 40 5 times
9 in 45 5 times
7 in 42 6 times
8 in 48 6 times
9 in 54 6 times
7 in 49 7 times
8 in 56 7 times
9 in 63 7 times
7 in 66 8 times
8 in 64 8 times
9 in V2 8 times
7 in 63 9 times
8 in 72 9 times
9 in 81 9 times
10 in 20 2 times
11 in 22 2 times
12 in 24 2 times
10 in 30 3 times
11 in S3 3 times
12 in 36 3 times
10 in 40 4 times
11 in 44 4 times
12 in 48 4 times
10 in 60 5 times
11 in 55 5 times
12 in 60 5 times
10 in 60 6 times
11 in 66 6 times
12 in 72 6 times
10 In 70 7 times
11 in 77 7 times
12 in 84 7 times
10 in 80 8 times
11 in 88 8 times
12 in 96 8 times
10 in 90 9 times
11 in 99 9 times
12 in 108 9 times
DIVISION. 56
MENTAL EXERCISES.
1. How many barrels of flour ; at 6 dollars a barrel can be
bought for 30 dollars ?
ANALYSIS. Since 6 dollars will buy one barrel of flour, 30 dol-
lars will buy as many barrels as 6 dollars, the price of one barrel,
Is contained times in 30 dollars, which is 5 times. Therefore, at
dollars a barrel, 5 barrels of flour can be bought for 30 dollars.
2. How many oranges, at 4 cents apiece, can be bought
for 28 cents ?
3. How many tons of coal ; at 5 dollars a ton, can be
bought for 35 dollars ?
4. When lard is 7 cents a pound, how many pounds can
be bought for 49 cents ? for 63 cents ? for 84 cents ?
5. If a man travel 48 miles in 6 hou^how far does he
travel in one hour ?
6. At 3 cents apiece, how many lemons can be bought
for 24 cents ? for 30 cents ? for 36 cents ?
7. If you give 55 cents to 5 beggars, how many cents do
you give to each ?
8. If a man build 42 rods of wall in 7 days, how many
rods can he build in 1 day ?
9. At 4 dollars a cord, how many cords of wood can be
bought for 20 dollars ? for 28 dollars ? for 32 dollars ?
10. A farmer paid 33 dollars for some sheep, at 3 dollars
apiece ; how many did he buy ?
11. At 7 cents a pound, how many pounds of sugar can
be bought for 63 cents ? for 84 cents ?
12. If a man spend 5 cents a day for cigars, how many
days will 50 cents last him ? 60 cents 1
13. At 12 cents a pound, how many pounds of coffee can
be bought for 48 cents? for 72 cents? for 96 cents? for
120 cents ?
56
SIMPLE NUMBERS.
PROMISCUOUS D
6 in 36, how many times ?
7 in 42, how many times 1
9 in 81, how many times 1
5 in 35, how many times ?
8 in 72, how many times 1
9 in 27, how many times 1
4 in 20, how many times ?
6 in 54, how many times ?
I VISION TABLE.
9 in 63, how many times ?
6 in 12, how many times 1
7 in 28, how many times ?
4 in 16, how many times ?
7 in 49, how many times ?
4 in 36, how many times ?
8 in 64, how many times ?
8 in 40, how many times ?
8 in 32, how many times ?
5 in 45, how many times ?
6 in 42, how many times ?
8 in 56, how many times 1
7 in 63, how many times ?
3 in 27, how many times ?
7 in 21, how JM^J times 1
8 in 16, how many times 1
4 in 28, how many times ?
8 in 32, how many times ?
6 in 48, how many times ?
9 in 45, how many times ?
8 in 48, how many times ?
7 in 56, how many times ?
3 in 21, how many times ?
6 in 54, how many times ?
4 in 12, how many times ?
7 in 35, how many times ?
5 in 10, how many times ?
7 in 14, how many times ?
4 in 24, how many times 1
5 in 30, how many times ?
9 in 36, how many times ?
6 in 30, how many times ?
2 in 16, how many times ?
4 in 32, how many times ?
6 in 24, how many times ?
9 in 72, how many times ?
5 in 10, how many times ?
4 in 8, how many times ?
5 in 20, how many times ?
2 in 10, how many times ?
66. The Dividend is the number to be divided.
67. The Divisor is the number to divide by.
68. The Sign of Division is a short horizontal line,
with a point above and one below, -+-. It indicates that the
number before it is to be divided by the number after it.
Thus, 20 .-*- 4 = 5, is read, 20 divided by 4 is equal to 5.
Division is also expressed by writing the dividend above f
and the divisor below, a short horizontal line ;
12
Thus, ~-= 4, shows that 12 divided by 3 equals 4.
DIVISION. 57
CASE I.
69. When the divisor consists of one figure.
1. How many times is 4 contained in 848 ?
OPERATION. ANALYSIS. After writing the divisor
Dividend, O n the left of the dividend, with a line
DiTisor > between them, we begin at the left hand
oi an( * Sa 7 : ^ is contained in 8, 2 times,
and as 8 in the dividend is hundreds,
the 2 in the quotient must be hundreds ; we therefore write 2
in hundreds' place under the figure divided. 4 is contained in
4, 1 time, and since 4 denotes tens, we write 1 under it :a tens'
place. 4 in 8, 2 times, and since 8 is units, we write 2 in units'
place under it, and we have 212 for the entire quotient.
EXAMPLES' FOR PRACTICE.
(2.) (3.) (4.)
Wrisor, 3)936 Dividend, 2)4862 4)48844
312 Quotient. 2431 12211
5. Divide 9963 by 3. Ans. 3321.
6. Divide 5555 by 5. Ans. 1111.
7. Divide 68242 by 2. Ans. 34121.
8. Divide 66666 by 6.
When the divisor is not contained in the first figure of
the dividend, we find how many times it is contained in the
first two figures.
9. How many times is 4 contained in 2884 ?
OPERATION. ANALYSIS. As we cannot divide 2 by 4,
4)2884 we say 4 is contained in 28, 7 times, and
write the 7 in hundreds' place; then 4 is
721 contained in 8, 2 times, which we write m
tens' place under the figure divided ; and 4 is contained in 4, 1
time, which we write in units' place in the quotient, and we
have the entire quotient, 721.
58 SIMPLE NUMBERS.
EXAMPLES FOR PRACTICE.
(10.) (11.) (12.)
3)2469 5)3055 2)148624
823 611 ^ 74312
13. Divide 4266 by 6. Ans. 711.
14. Divide 36488 by 4. Ans. 9122.
15. Divide 72999 by 9. Ans. 8111.
16. Divide 21777 by 7.
After obtaining the first figure of the quotient, if the di-
visor is not contained in any figure of the dividend, place a
cipher in the quotient, and prefix this figure to the next
one of the dividend.
NOTE. To prefix means to place before, or at the left hand.
17. How many times is 6 contained in 1824 ?
OPERATION. ANALYSIS. Beginning as in the last ex-
6 ) 1824 amples, we say, 6 is contained in 18, 3 times
which we write in hundreds' place in the
quotient ; then 6 is contained in 2 no times,
BO we write a cipher (0) in tens' place in the quotient, and pre-
fixing the 2 to the 4, we say 6 is contained in 24, 4 times, which
we write in units' place, and we have 304 for the entire quo-
tient.
EXAMPLES FOR PRACTICE.
(18.) (19.) (20.)
4)3228 7)28357 3)912246
807 4051 304082
21. Divide 40525 bj 5. Ans. 8105.
22. Divide 36426 by 6. Ans. 6071.
23. Divide 184210 by 2. Ans. 92105.
24. Divide 85688 by 8. Ans. 10711.
25. Divide 273615 by 3. Ans. 91205.
DIVISION. 69
After dividing any figure of the dividend, if there be a
remainder, prefix it mentally, to the next figure of the divi-
dend, and then divide this number as before.
31. How many times is 4 contained in 943 ?
OPERATION. ANALYSIS. Here 4 is contained in
4 ) 943 9, 2 time?, and there is 1 remainder,
which we prefix mentally to the next
235 ... 3 Rem. figure, 4, and say 4 is contained in 14,
3 times, and a remainder of 2, which we prefix to 3, and say, 4
is contained in 23, 5 times, and a remainder of 3. This 3 which
is left after performing the last division should be divided by the
divisor 4 ; but the method of doing it cannot be explained here,
and so we merely indicate the division by placing the divisor
under it ; thus, f. The entire quotient is written 235 1, which
may be read, two hundred thirty-five and three divided, ly four,
or, two hundred thirty-five and a remainder oj three.
NOTE. When the process of dividing is performed mentally, and the results only
are written, as in the preceeding examples, the operation is termed Sliort Division.
From the foregoing examples and illustrations, we deduce
the following
RULE. I. Write the divisor at the left of the dividend,
with a line between them.
II. Beginning at the left hand, divide each figure of the
dividend by the divisor , and write the result under the divi-
dend.
III. If there be a remainder after dividing any figure,
regard it as prefixed to the figure of the next lower order in
the dividend, and divide as before.
IV. Should any figure or part of the dividend be less
than the divisor, write a cipher in the quotient, and prefix
the number to the figure of the next lower order in the divi-
dend > and divide as before.
V. If there be a remainder after dividing the last figure,
place it over the divisor at the right hand of the quotient.
60
SIMPLE NUMBEKS.
PROOF. Multiply the divisor and quotient together, and
to the product add the remainder, if any ; if the result be
equal to the dividend, the work is correct.
NOTES. 1. This method of proof depends on the fact that division is the revers*
of multiplication. The dividend answers to the product, the divisor to one of tha
factors, and the quotient to the other factor.
2. In multiplication the two factors arc given, to find the product : in division,
the product and one of the factors are given, to find the other factor.
EXAMPLES FOR PRACTICE.
1. Divide 8430 by 6.
OPERATION. PROOF
Divisor. 6)8430 Dividend. 1405 Quotient.
1405 Quotient
(20
5)730490
146098
5. Divide
6. Divide
7. Divide
8. Divide
9. Divide
10. Divide
11. Divide
12. Divide
13. Divide
14. Divide
15. Divide
16. Divide
17. Divide
(3.)
7)510384
72912
87647 by 7.
94328 by 8.
43272 by 9.
377424 by 6.
975216 by 8.
46375028 by 7.
4763025 by 9.
42005607 by 7.
72000450 by 9.
97440643 by 8.
65706313 by 9.
3627089 by 6.
4704091 by 7.
6 Divisor.
8430 Dividend.
(40
8)6003424
750428
Quotients.
12521.
11791.
4808.
62904.
121902.
6625004.
529225.
6000801.
8000050.
12180801.
7300701J.
604514JJ.
672013.
DIVISION. 61
18. Divide 16344 dollars equally among 6 men; how
much will each man receive ? Ans. 2724 dollars.
19. How many barrels of flour, at 7 dollars a barrel, can
be bought for 87605 dollars ? Ans. 12515 barrels.
20. In one week there are 7 days ; how many weeks in
23044 days ? Ans. 3292 weeks.
21. If 5 bushels of wheat make 1 barrel of flour, how
many barrels of flour can be made from 314670 bushels ?
Ans. 62934 barrels.
22. By reading 9 pages a day, how many days will be re-
quired to read a book through which contains 1161 pages?
Ans. 129 days.
23. At 4 dollars a yard, how many yards of broadcloth
can be bought for 1372 dollars 1 Ans. 343 yards.
24. If a stage goes at the rate of 8 miles an hour, how
long will it be in going 1560 miles ? Ans, 195 hours.
25. There are 3 feet in 1 yard; how many yards in
206175 feet 1 ? Ans. 68725 yards.
26. Five partners share equally the loss of a ship and car-
go, valued at 760315 dollars ; how much is each one's
share of the loss ? Ans. 152063 dollars.
27. If a township of 64000 acres be divided equally
among 8 persons, how many acres will each receive ?
Ans. 8000 acres.
28. A miller wishes to put 36312 bushels of grain into 6
bins of equal size ; how many bushels must each bin con-
tain ? Ans. 6052 bushels.
29. How many steps of 3 feet each would a man take in
walking a mile, or 5280 feet ? Ans. 1760 steps.
30. A gentleman left his estate, worth 36105 dollars, to
be shared equally by his wife and 4 children ; how much
did each receive ? Ans. 7221 dollars.
62 SIMPLE NUMBERS.
CASE II.
7O. "When the divisor consists of two or more
figures.
NOTE. To illustrate more clearly the method of operation, we mil first take ao
example usually performed by Short Division.
1. How many times is 4 contained in 1504 ?
OPERATION. ANALYSIS. First. We find how many times
4)1504(376 ^ e Divisor ^> * s contained in 15, the first par-
- 2 tial dividend, which we find to be 3 times*
and a remainder. We place this quotient
30 figure at the right hand of the dividend, with
28 a line between them. Second. To find the
remainder, we multiply the divisor 4, by this
quotient figure 3, and place the -product 12,
24 under the figures divided. We subtract tho
product from the figures divided, and have
a remainder of 3. Third. Bringing down the next figure of
the dividend to the right hand of the remainder, we have 30,
the second partial dividend. Then 4 is contained in 30, 7 times
and a remainder. Placing the 7 at the right hand of the last
quotient figure, and multiplying the divisor by it, we place the
product 28, under the figures last divided, and subtract as
before. To the remainder 2, bring down the next figure 4 of
the given dividend, and we have 24 for the third partial divi-
dend. Then 4 is contained in 24, 6 times. Multiplying and
subtracting as before, we find that nothing remains, and we
have for the entire quotient 376.
NOTE. When the whole process of division is written out as above, the operation
is termed Long Division. The principle however is the name as Short Division.
Solve the following examples, by Long Division.
2. Divide 4672 by 8. . Ans. 584.
3. Divide 97636 by 7. Ans. 13948.
4. Divide 37863 by 9. Ans. 4207.
5 Divide 394064 by 11. Am. 35824.
DIVISION. 63
6. How many times is 23 contained in 17158 ?
OPERATION. ANALYSTS. As 28 is not contained in the
23)17158(746 first two figures of the dividend, we find how
IQl many times it is contained in 171, as the first
partial dividend* 23 is contained in 171, 7
105 times, which we place in the quotient on the
92 right of the dividend. We then multiply
the divisor 23, by the quotient figure 7, and
138 subtract the product 161, from- the part of
138 the dividend used, and we have a remainder
of 10. To, this remainder we bring down the
next figure of the dividend, making 105 for the second partial
dividend. Then, 23 is contained in 105, 4 times, which we
place in the quotient. Multiplying and subtracting as before,
we have a remainder of 13, to which we bring down the next
figure of the dividend, making 138 for the third partial divi-
dend. 23 is contained in 138, 6 times; multiplying and sub-
tracting as before, nothing remains, and we have for the entire
quotient, 746.
From the preceding illustrations we derive the following
general
RULE. I. Write the divisor at the left of the dividend,
as in short division.
II. Divide the least number of the left hand figures in
the dividend that will contain the divisor one or more times,
and place the quotient at the right of the dividend, with a
line between them.
III. Multiply the divisor by this quotient figure, subtract
the product from the partial dividend used, and to the re-
mainder bring down the next figure of the dividend.
IV. Divide as before, until all the figures of the dividend
have been brought down and divided.
V. If any partial dividend will not contain the divisor,
64 SIMPLE NUMBERS.
place a cipher in the quotient, and bring down the next
figure of the dividend, and divide as before.
VI. If there be a remainder after dividing all the figures
of the dividend, it must be written in the quotient, with the
divisor underneath.
NOTES. 1. If any remainder be equal to, or greater than the divisor, the quotient
figure is too small, and must be increased,
2. If the product of the divisor by the quotient figure be greater than the partial
dividend, the quotient figure is too large, and must be diminished.
PROOF. The same as in short division.
7 1 . The operations in long division consist of five prin-
cipal steps, viz. :
1st. Write down the numbers.
2d. Find how many times.
3d. Multiply.
4th. Subtract.
5th. Bring down another figure.
EXAMPLES FOR PRACTICE.
7. Find how many times 18 is contained in 36838.
OPERATION. PROOF.
Dividend. Quotient.
DiTisor, 18)36S38(2046jg 2046 Quotient.
36 18 Divisor.
83 16368
72 2046
118 36828
108 10 Remainder.
10 Remainder 36838 Dividend.
DIVISION. 65
8. Divide 79638 by 36. 9. Divide 93975 by 84.
OPERATION. OPERATION.
86)79638(2212/j 84)93975(1118|j
72 84
76 99
72 84
43 157
36 84
~78 735
72 672
6 Rem. 63 Bern.
10. Divide 408722 by 136. 11. Divide 104762 by 109.
OPERATION. OPERATION.
136)408722(3005 109)104762(961
408 981
722
680 654
42 &. 122
109
12. Divide 178464 by.16. Am. 11154.
13. Divide 15341 by 29. Ans. 529.
14. Divide 463554 by 39. Ans. 11886.
15. Divide 1299123 by 17. Ans. 76419.
16. Divide 161700 by 15. An*. 10780.
17. Divide 47653 by 24.
18. Divide 765431 by 42.
SIMPLE NUMBERS.
19. Divide 6783 by 15.
20. Divide 7831 by 18.
21. Divide 9767 by 22.
22. Divide 7654 by 24.
23. Divide 767500 by 23.
24. Divide 250765 by 34.
25. Divide 5571489 by 43.
26. Divide 153598 by 29.
27. Divide 301147 by 63.
28. Divide 40231 by 75.
29. Divide 52761878 by 126.
30. Divide 92550 by 25.
31. Divide 7461300 by 95.
32. Divide 1193288 by 45.
33. Divide 5973467 -by 243.
34. Divide 69372168 by 342.
35. Divide 863256 by 736.
36. Divide 1893312 by 912.
37. Divide 833382 by 207.
38. Divide 52847241 by 607.
39. Divide 13699840 by 342.
40. Divide 946656 by 1038.
41. Divide 46447786 by 1234.
42. Divide 28101418481
by 1107.
43. Divide 48288058. by 3094.
44. Divide 47254149 by 4674.
45. A man bought 114 acres of land for 4104 dollars ,
what was the average price per acre ? Ans. 36 dollars.
46. Nine thousand dollars was paid to 75 operatives:
how much did each receive? Ans. 120 dollars.
Quotients,
Rem,
452
3.
435
1.
443
21.
318
22.
33369
13.
7375
15.
129669
22.
5296
14.
4780
7.
536
31.
418745
8.'
3702
78540
26517
23.
24582
41.
202842
204.
1172
664.
2076
4026
87063
40058
4.
912
37640
26.
25385201
15607
10110
974.
9.
DIVISION. 67
47. There are 24 hours in a day ; how many days in
11424 hours ? Ans. 476.
48. In one hogshead are 63 gallons ; how many hogs-
heads in 6615 gallons ? Ans. 105.
49. If a man travel 48 miles a day, how long will it take
him to travel 1296 miles'? Ans. 27 days.
50. If a person can count 8677 in an hour, how long
will it take him to count 38369694 ? Ans. 4422 hours.
51. If it cost 5987520 dollars to construct a railroad 576
miles long, what will be the average cost per mile ?
Ans. 10395 dollars.
52. The Memphis and Charleston railroad is 287 miles
in length, and cost 5572470 dollars; what was the average
cost per mile ? Ans. 19416rr 7 5 8 7 dollars.
53. A garrison consumed 1712 barrels of flour in 107
days ; how much was that per day ? Ans. 16 barrels.
54. How long would it take a vessel to sail from New
York to China, allowing the distance to be 9072 miles, and
the ship to sail 144 miles a day 1 Ans. 63 days.
55. How long could 27 men subsist on a stock of provis-
ion, that would last 1 man 3456 days ? Ans. 128 days.
56. A drover received 10362 dollars, for 314 head of cat-
tle ; how much was their average value per head 1
Ans. 33 dollars.
57. If 42864 pounds of cotton be packed in 94Jmles, what
is the average weight of each bale 1 Ans. 456 pounds.
58. If a field containing 42 acres produce 1659 bushels
of wheat, what will be the numbor of bushels per acre ?
Ans. 39f 4 bushels.
59. In what time will a reservoir containing 109440 gal-
lons, be emptied by a pump discharging 608 gallons per
hour 1 Ans. 180 hours.
68 SIMPLE NUMBERS.
CONTRACTIONS.
CASE I.
72. When the divisor is 10, 100, 1000, &c.
1. Divide 374 by 10.
OPERATION. ANALYSIS. Since we have shown,
1!0^37'4 th&t to remove a figure one place to-
ward the left by annexing a cipher
Quotient, 37---4Rem. i ncrea ses its value tenfold, or multi-
or, 37 T 4 Q, Ans. p\[ cs it by 10, so, on the contrary, by
cutting ofi or taking away the right
hand figure of a number, each of the other figures is removed
one place toward the right, and, consequently, the value of each
is diminished tenfold, or divided by 10.
For similar reasons, if we cut off two figures, we divide by
100, if three, we divide by 1000, and so on. Hence the
RULE. From the right hand of the dividend cut of as
many figures as there are ciphers in the divisor. Under the
figures so cut off, place the divisor, and the Mchole will form
the quotient.
EXAMPLES FOR PRACTICE.
Quotients. Kern's.
2. Divide 13705 by 100. 137 5.
3. Divide 50670 by 100. 506 70.
4. Divide 320762 by 1000. 320 762.
5. Divi<Jp 14030731 by 10000. 1403 731.
6. Divide 9021300640 by 100000. 90213 640.
7. A man sold 100 acres of land for 3725 dollars ; how
much did he receive an acre ? Ans. 37f\fty dollars.
8. Bought 1000 barrels of flour for 6080 dollars ; how
much did it cost me a barrel ? Ans. 6 T H$jj dollars.
9. Paid 12560 dollars for 10000 bushels of wheat; how
much was the cost per bushel ? Ans. 1^^^ dollars.
DIVISION. 69
CASE III.
73. When there are ciphers on the right hand
of the divisor.
1. Divide 437661 by 800.
OPERATION. ANALYSIS. In this example we
, 8'|00)4376i61 resojve 800 into the factors 8 and
~547 61 Rem ^ an( * Divide ^ rst ky 100, by cut-
ting off two right hand figures of the
dividend, and we have a quotient of 4376, and a remainder of 61.
We next divide by 8, and obtain 547 for a quotient ; and the
entire quotient
2. Divide 34716 by 900.
OPERATION. ANALYSIS. Dividing as in the last
9jOO)347jl6 example, we have a quotient of 38,
Quotient, 38-- 516 Rem. and a remainder of 5 after dividing
or 38 3 l & Ans. by 9 ' which we prefix to the fi S ures
cut off from the dividend, making a
true remainder of 516, and the entire quotient 38|^. Hence
RULE. I. Cut off the ciphers from the right of the divi-
sor, and the same number of figures from the right of the
dividend.
II. Divide the remaining figures of the dividend by the
remaining figures of the divisior, and the result will be the
quotient. If there be a remainder after this division, pre-
fix it to the figures cut off from the dividend, and this will
form the true remainder.
EXAMPLES FOR PRACTICE.
Quotients. Bern's.
3. Divide 46820 by 400. 117 20.
4. Divide 130627 by 800. 163 227.
5. Divide 76173 by 320. 238 13.
6 Divide 378000 by 1200. 315
70 SIMPLE NUMBERS.
7. Divide 674321 by 11200. 60 2321.
8. Divide 64613214 by 4000. 16153 1214.
9. Divide 146200 by 430. 340
10. Divide 7380964 by 23000. 320 20964.
11. Divide 58677000 by 1800. 32598 600.
EXAMPLES IN THE PRECEDING RULES.
1. A speculator bought at different times 320 acres, 175
acres, 87 acres, and 32 acres of land, and afterward sold 467
acres ; how many acres had he left 1 Ans. 147 acres.
2. Two men travel in opposite directions ; one travels 31
miles a day, the other 43 miles a day ; how far apart will
they be in 12 days ? Ans. 888 miles.
3. A tobacconist has 6324 pounds of tobacco, which he
wishes to pack in boxes containing 62 pounds each ; bow
many boxes must he procure to contain it ? Ans. 102.
4. A farmer sold 15 tons of hay at 9 dollars a ton, and
25 cords of wood at 4 dollars a cord, and wished to divide
the amount equally among 5 creditors ; how much would
each receive ? Ans. 47 dollars.
5. If you deposit 216 cents each week in a savings bank,
and take out 89 cents a week, how many cents will you
have in bank at the end of 36 weeks ? Ans. 4572 cents.
6. The product of two numbers is 8928, and one of the
numbers is 72 ; what is the other number ? Ans. 124.
7. The dividend is 7280, and the quotient is 208 ; what is
the divisor 1 Ans. 35.
8. What is the remainder after dividing 876437 by
16900 1 Ans. 14537.
9. A man sold 6 horses at 125 dollars each, 25 head of
cattle at 30 dollars each, and with the proceeds bought
land at 25 dollars an acre ; how many acres did he buy 1
Ans. 60 acres.
PROMISCUOUS EXAMPLES. 71
10. If a Mechanic receives 784 dollars a year for labor,
and his expenses are 426 dollars a year, how much can he
gaspe in 6 years ? Ans. 2148 dollars.
. 11. A farmer sold 40 bushels of wheat at 2 dollars a
bushel, and 16 cords of wood at 3 dollars a cord. He re-
ceived 15 yards of cloth at 4 dollars a yard, and the re-
mainder in money ; how much money did he receive ?
Ans. 68 dollars.
12. How many pounds of cheese worth 10 cents a pound,
can be bought for 22 pounds of butter worth 15 cents a
pound ? Ans. 33 pounds.
13. If 56 yards of cloth cost 336 dollars, how much will
12 yards cost, at the same rate ? Ans. 72 dollars.
14. If 100 barrels of flour cost 600 dollars, what will
350 barrels cost, at the same rate ? Ans. 2100 dollars.
15. How long can 60 men subsist on an amount of food
that will last 1 man 7620 days ? Ans. 127 days.
16. If I buy 225 barrels of flour for 1125 dollars, and
sell the same for 1800 dollars, how much do I gain on each
barrel? Ans. 8 dollars.
17. A man sold his house and lot for 5670 dollars, and
took his pay in bank stock at 90 dollars a share ; how many
shares did he receive ? Ans. 68 shares.
18. How many pounds of tea worth 75 cents a pound,
ought a man to receive in exchange for 27 bushels of oats,
worth 50 cents a bushel ? Ans. 18 pounds.
19. The quotient of one number divided by another is
40, the divisor is 364, and the remainder 120 ; what is the
dividend? Ans. 14680.
20. How many tons of hay at 12 dollars a ton, must be
given for 21 cows at 24 dollars apiece ? Ans. 42 tons.
72 SIMPLE NUMBEKS.
21. Bought 150 barrels of flour for 1050 dollars, and sold
107 barrels of it at 9 dollars a barrel, and the remainder at
7 dollars a barrel ; did I gain or lose, and how much ?
Ans. gained 214 dollars.
22. A mechanic earns 45 dollars a month, and his neces-
sary expenses are 27 dollars a month. How long will it
take him to pay for a farm of 50 acres, at 27 dollars an
acre ? Ans. 75 months.
23. How many barrels of flour at 7 dollars a barrel, will
pay for 30 tons of coal, at 4 dollars a ton, and 44 cords of
wood, at 3 dollars a cord ? Ans. 36 barrels.
PROBLEMS IN SIMPLE INTEGRAL NUMBERS.
T4. The four operations that have now been considered,
viz., Addition, Subtraction, Multiplication, and Division, are
all the operations that can be performed upon numbers, and
hence they are called the Fundamental Rules*
Tt>. In all cases, the numbers operated upon and the re-
sults obtained, sustain to each other the relation of a whole
to its parts. Thus,
I. In Addition, the numbers added are the parts, and the
sum or amount is the whole.
II. In Subtraction, the subtrahend and remainder are the
parts, and the minuend is the whole.
III. In Multiplication, the multiplicand denotes the val-
ue of one part, the multiplier the number of parts, and the
product the total value of the whole number of parts.
IV. In Division, the dividend denotes the total value of
the whole number of parts, the divisor the value of one part,
and the quotient the number of parts ; or the divisor the
number of parts, and the quotient the' value of one part.
PROBLEMS. 3
76. Let the pupil be required to illustrate the following
problems by original examples.
Problem 1. Given, several numbers, to find their sum.
Prob. 2. Given, the sum of several numbers and all of
them but one, to find that one.
Pi*ob. 3. Given, two numbers, to find their difference.
Prob. 4. Given, the minuend and subtrahend, to find the
remainder.
Prob. 5. Given, the minuend and remainder, to find tha
subtrahend.
Prob. 6. Given, the subtrahend and remainder, to find
the minuend.
Prob. 7. Given, two or more numbers, to find their prod-
uct.
Prob. 8. Given, the multiplicand and multiplier, to find
the product.
Prob. 9. Given, the product and multiplicand, to find the
multiplier.
Prob. 10. Given, the product and multiplier, to find the
multiplicand.
Prob. 11. Given, two numbers, to find their quotitots.
Prob. 12. Given, the divisor and dividend, to find the
quotient.
Prob. 13. Given, the divisor and quotient, to find the
dividend.
Prob. 14. Given, the dividend and quotient, to find the
divisor.
Prob. 15. Given, the divisor, quotient, and remainder, to
find the dividend.
Prob. 16. Given, the dividend, quotient, and remainder
to find the divisor.
74 FRACTIONS.
FRACTIONS.
DEFINITIONS, NOTATION, AND NUMERATION.
7 7. If a unit be divided into 2 equal parts, one of the
parts is called one half.
If a unit be divided into 3 equal parts, one of the parts is
called one third, two of the parts two thirds.
If a unit be divided into 4 equal parts, one of the parts is
called one fourth, two of the parts two fourths, three of the
parts three fourths.
If a unit be divided into 5 equal parts, one of the parts is
called one fifth, two of the parts two fifths, three of the
parts three fifths, &c.
And since one half, one third, one fourth, and all other
equal parts of an integer or whole, thing, are each in them-
selves entire and complete, the parts of a unit thus used
are called fractional units ; and the numbers formed from
them, fractional numbers. Hence
7l A Fractional Unit is one of the equal parts of an
integral unit.
TO. A Fraction is a fractional unit, or a collection of
fractional units.
8O. Fractional units take their name, and their value,
from the number of parts into which the integral unit is
divided. Thus, if we divide an orange into 2 equal parts,
the parts are called halves; if in to 3 equal parts, thirds;
if into 4 equal parts, fourths, &c. ; and each third is less in
value than each half, and e&c\i fourth less than each third}
and the greater the number of parts, the leas their value.
DEFINITIONS, NOTATION, AND NUMERATION. 75
The parts of a fraction are expressed by figures ; thus,
One half is written A
One third " ^
Two thirds " f
One fourth " \
Two fourths f
Three fourths f
"
One fifth is written
Two fifths
One seventh " ^
Three eighths " f
Five ninths " f
Eiht tenths " -
To write a fraction, therefore, two integers are required,
one written above the other with a line between them.
8 1 . The Denominator of a fraction is the number below
the line. It shows into how many parts the integer or unit
is divided, and determines the value of the fractional unit.
82. The Numerator is the number above the line. It
numbers the fractional units, and shows how many are
taken.
83. Thus, if one dollar be divided into 4 equal parts,
the parts are called fourths, the fractional unit being one
fourth, and three of these parts are called three fourths of a
dollar, and may be written
3 the number of parts or fractional units taken.
4 the number of parts or fractional units into which the dollar is divided.
84. The Terms of a fraction are the numerator and
denominator, taken together.
80. Fractions indicate division, the numerator answer-
ing to the dividend, and the denominator to the divisor.
Hence,
86. The Value of a fraction is the quotient of the nu-
merator divided by the denominator.
Thus ; the quotient of 4 divided by 5 is |, or J expresses
<he QUOtie/lt of Which { 1 is the dividend.
\ 5 is the divisor.
76 FK ACTIONS.
1. What is 1 half of 8?
ANALYSIS. It is the quotient of 8 divided by 2, which is 4;
or, it is a number, which taken 2 times, will make 8, which is 4.
Therefore, 4 is 1 half of 8.
2. What is 2 thirds of 9 ?
ANALYSIS. Since 1 third of 9 is 3, 2 thirds of 9 is 2 times 3,
which is 6. Therefore, 2 thirds of 9 is 6.
Hence, to obtain one half, one third, one fourth, or any
fractional part of a number, we divide that number by the
denominator of the fraction expressing the parts ; and to
obtain any given number of such parts, we multiply that
part by the number of parts expressed by the numerator
of the same fraction.
8. What is 1 fourth of 12 ? 3 fourths of 12 ?
4. What is 1 fifth of 20 ? 3 fifths ? 4 fifths ?
5. What is 1 eight of 40 ? 3 eighths ? 5 eighths ?
6. What is 2 sevenths of 2^1 ? 5 sevenths of 35 ? 6 sev-
enths of 49?
7. What is 1 ninth of 63 ? 2 ninths of 27 ? 4 ninths of
36 ? 5 ninths of 45 ? 7 ninths of 81 ?
8. What is 1 twelfth of 48 ? 5 twelfths ? 7 twelfths ?
9. If a pound of coffee cost 15 cents, how much will 1
third of a pound cost ? 2 thirds ?
10. A farmer having 60 sheep, sold 1 fifth of them to
one man, and 3 fifths to another ; how many did he sell to
both?
11. A boy having 48 cents, spent 5 eighths of them ;
how many had he left ?
12. Paid 108 dollars for a horse, and 9 twelfths as much
for a carriage ; how much did the carriage cost ?
13. William had 120 pennies, and James had 7 tenths
as many ; how many had James ?
NOTATION AND NUMERATION. 77
87. It is often required to express by a fraction, what
part one number is of another number.
1. What part of 5 is 3 ?
ANALYSIS. Since 1 is 1 fifth of 5, 3 must be 3 times 1 fifth of
5, or 3 fifths of 5. Therefore, 8 is 8 fifths of 5.
NOTK. The number preceded by the word of is generally made the denominator
or divisor, and the other number called the part, the numerator or dividend.
2. What part of 6 is 3? 4? 5? 1?
3. What part of 9 is 2? 3? 5? 6? 1? 4?
4. What part of 10 is 7? 6? 3? 1? 9? 8? 4?
6. What part ot 12 is a? 5? 6? 8? 9? 11 10? 11?
6. What part of 14 is 5? 7? 9? 3? 6? 11? 8? 15?
7. What part of 15 bushels, is 3 bushels ? 7 bushels ? 9
bushels 11 bushels?
8. What part of 18 dollars, is 7 dollars? 5 dollars? 9
dollars ? 17 dollars ?
9. If 6 oranges cost 30 cents, what part of 30 cents will
1 orange cost ? 2 oranges ? 3 oranges ? 5 oranges ?
EXAMPLES IN WRITING AND READING FRACTIONS.
JSxpress the following fractions by figures :
1. Nine twelfths. Ans. T %.
2. Eleven fifteenths. Ans. jj.
8. Twenty-four for ty-ninths. Ans. ||.
4. Forty-four sixty-ninths. Ans. ||.
5. One hundred twenty, four hundred fiftieths.
Read the the following fractions :
6- ft, ii, ii, flfc, M,M, W. If?-
7. If the fractional unit is 28, express 9 fractional units,
16, 17; 22; 27.
8* If the fractional unit is 96, express 27 fractional
units ; 42 ; 75
78 FRACTIONS.
88. Fractions are distinguished as Proper and Improper.
A Proper Fraction is one whose numerator is less than
its denominator ; its value is less than the unit, 1 . Thus,
T 7 2> 7 5 g> To> II are proper fractions.
An Improper Fraction is one whose numerator equals
or exceeds its denominator ; its value is never less than the
unit, 1. Thus, ^, |, -Lo. ? jyi, |o ? J^Q are improper fractions.
89. A Mixed Number is a number expressed by an
integer and a fraction ; thus, 4j, 17Jf , 9 T % are mixed num-
bers.
REDUCTION.
....~ 9O. The Reduction of a fraction is the process of chang-
ing its terms, or its form, without altering its value.
CASE I.
91. To reduce fractions to their lowest terms.
A fraction is in its lowest terms when no number greater
than 1 will exactly divide both numerator and denominator
without a remainder.
1. Reduce f to its lowest terms. ^
ANALYSIS. It is plain, that the numerator 2, and the denom-
inator 4, are both divisible by 2, without remainders; hence
2-j-2_l
4-f-2~2
The terms thus obtained, viz., 1, the numerator, and 2, the de-
nominator, are not divisible by any number larger than 1, and
therefore are the smallest terms by which the value of can be
expressed.
2. Reduce | to its lowest terms.
3. Reduce -f^ to its lowest terms. _
4. Reduce | to its lowest fe rnis. . '/.
5. Reduce - to its lowest terms.
6. Reduce ' jj to it* lowest forms.
REDUCTION. 79
7. Reduce || to its lowest terms.
OPERATION. ANALYSIS. Dividing both terms
9V8 24 . 9Y24 i_2 of a fraction by the same number
^/gO 3l)> TVSO 15'
o\i2 4 A does not alter the value of the frac-
J " g 4 , tion or quotient; hence, we divide
both terms of j by 2 , and obtain
|J; dividing both terms of this fraction by 2, we have || as
the result ; finally, dividing the terms of this fraction by 3, we
have |, and as no number greater than 1 will divide the terms
of this fraction without a remainder, | are the lowest terms
in which the value of |J can be expressed. We may obtain
the final result more readily, by dividing the terms of this frac-
tion by the largest number that will -divide both without a re-
mainder, as in the above example ; if we divide by 12, we obtain
, the answer. Hence the
RULE. Divide the terms of the fraction by any numbei
greater than 1, that will divide both without a remainder,
and the quotients obtained in the same manner, until no num-
ber greater than 1 will so divide them ; the last quotients
will be the lowest terms of the given fraction.
it?
EXAMPLES FOR PRACTICE.
8. Reduce ^J to its lowest terms. Ans. |.
9. Reduce y 7 ^ to its lowest terms. Ans. f .
^0. Reduce T 9 T 8 2 to its lowest terms. Ans. g.
11. Deduce jff to its lowest terms. Ans. J.
12. Reduce 7 || to its lowest terms. **Ans. l|.
13. Reduce |4| to its lowest terms. Ans. 7 |
14. Reduce HI to its lowest terms. Ans. f,
15. Reduce -ff^ to its lowest terms. Ans. ^.
16. Reduce ||g to its lowest terms. ^4ns. ||.
-17. Reduce 3 ^ ff to its lowest terms. Ans. i.
18. Reduce ||J to its lowest terms. Ans. ^%.
19. Reduce $g-J-jj to its lowest terms. ^4n*. j^f J.
80 FRACTIONS.
CASE II.
93. To change an improper fraction to a whole
or mixed number.
1. In if how many times 1 ?
ANALYSIS. Since 1 equals J, L 2 equal as many times 1, as
| are contained times in \f, which are 3 times. Therefore,
L 2 are 3 times 1, or 3.
2. How many times 1 in y ? in J g 8 ? in \ ?
3. How many times 1 in 2 ^ ? in 2 g 4 ? in 3 / ?
4. How many times 1 in fi g 4 ? in f g ? in 4 g 8 ?
5. How many times 1 in 7 ^ 2 ? in ff ? in f f ?
Nora. When the denominator is not an exact divisor of the numerator, the re-
sult will be a mixed number.
6. In \f how many times 1 1
OPERATION. ANALYSIS. Since 1 equals ^, \f equal
as many times 1 as 7 is contained times
in 16, which is 2S times. Hence the
,
2 1 Ans.
Divide the numerator by the denominator.
EXAMPLES FOR PRACTICE.
7. In J-ffi- how many times 1 ? u4ns. 244.
8. In 2 T 2 2 8 of a year how many years ? Ans. 19.
9. In 12 -| 4 of a pound how many pounds? Ans. 107."
10. In m of a mile how many miles 1 Ans. 6.
11. In 7 JC 7 of a rod how many rods? Ans. 21J.
12. In 2 f$ 5 of a dollar how many dollars?
13. Reduce f g to a whole number. Ans. 6.
14. Reduce y/ to a mixed number. Ans. 5|.
15. Reduce 8 7 2 5 4 to a whole number. -4ns. 18.
Reduce ^f 6 to a mixed number. * Ans. 60f.
17. Change 3 || 6 to a mixed number. 67|.
18. Change 2 j| 4 to a whole number. An*. 52.
REDUCTION. 81
CASE m.
93. To reduce a whole or mixed number to an im-
proper fraction.
1. How many thirds in 4 ?
ANALYSIS. Since in 1 there are 3 thirds, in 4 there are 4
times 3 thirds, or 12 thirds. Therefore, there are 1* in 4.
2. How many fourths in 2 1 in 3 ? in 5 ?
3. How many halves in 5 ? in 7 ? in 8 ? in 9 ?
4. How many sixths in 3 ? in 5 ? in 7 ? in 10 ?
5. How many tenths in 4 ? in 8 1 in 9 ? in 6 ?
6. How many fifths in 2 whole oranges ? in 4 1 in 5 ?
7. How many eighths in 4 whole dollars ? in 5 ? in 6 ?
8. In 3| dollars how many eighths of a dollar ?
OPERATION.
35 ANALYSIS. Since in 1 dollar there are 8
g H eighths, in 3 dollars there are 3 times 8
eighths, or 24 eighths, and 5 eighths added,
24-5 = \ 9 make- 2 /-.'
BULE. Multiply the whole number by the denominate*
of the fraction ; to the product add the numerator, and un-
der the result write the denominator.
EXAMPLES FOR PRACTICE.
9. Reduce 6| to an improper fraction. Ans. 2 ? 7 .
10. Reduce 7f to an improper fraction. Ans. 6 ^ 8 .
11. Reduce 15 to a fraction whose denominator is 7.
Ans. ij}.
12. Reduce 120 to twelfths. Ans. J f J.
13. In 242| of an acre how many thirds of an acre ?
14. In 75| bushels how many eighths f Ans. 6 g 7 .
^ 15. In 24 pounds how many sixteenths? Ans. s T 8 g 4 .
16. In 52 weeks how many sevenths? Ans. 3 4 .
17. Change 14^ to an improper fraction. Ans. \\ 6 t
62 ' ITKACTION8.
CASE IV.
94. T 3 reduce two or more fractious to a com-
mon denominator.
A Common Denominator is a denominator common to
two or more fractions.
NOTK. Any number that can be divided by each of the denominators of the
given fractions, may be taken for the common denominator.
1. Reduce \ an,d f to fractions having a common de-
nominator.
ANALYSIS. 12 is exactly divisible by 4 and 3, and may there-
fore be taken for a common denominator. Since in 1 there are
12, in -1 of 1 there must be 1 of If or J^ and in | of 1 there
must be | of ||, or ^. Therefore 1 and | are equal to ^
** '
2. Reduce | and | to a common denominator. *
3. Reduce | and | to a common denominator.
4. Reduce -J and | to a common denominator.' .
5. Reduce J- and | to a common denominator! *
OPERATION. ANALYSIS. We multiply the terms of the
__25 first fraction |, by the denominator 5 of the
SQ second, and the terms of the fraction |, by
the denominator 6 of the first. This must re-
_ duce each fraction to the same denominator
gQ^ f or gg^jj new d enomma tor will be the pro
duct of the given denominators. Hence the
RULE. Multiply both terms of each fraction by the d&
nominators of all the other fractions.
NOIK. Mixed numbers must first be reduced to improper fractions. .
EXAMPLES FOR PRACTICE.
6. Reduce % and | to a common denominator.
An,. -
ADDITION. 88
7. .Reduce j and | to a common denominator.
/. H, U-
8. Reduce | and | to a common denominator.
AM. jf, Jf.
9. Reduce | and T 7 ^ to a common denominator.
Ans. J j, I?.
10. Reduce | and T 5 2 to a common denominator.
Am. if, ti-
11. Reduce , f , and to a common denominator.
Am. if, jj, if.
12. Reduce j, |, and ^ to a common denominator.
13. Reduce |, J, and | to a common denominator.
14. Reduce 1^, f , and | to a common denominator.
Ans. J^ 8 -, f4, f
15. Reduce T 7 ^, 2|, and f to a common denominator.
16. Reduce -f^, 3^, |, and | to a common denominator.
I, ,
ADDITION.
95. The denominator of a fraction determines the value
of the fractional unit; hence,
I. If two or more fractions have the same denominator,
their numerators express fractional units of the same value.
II. If two or more fractions have different denominators,
their numerators express fractional units of different values.
And since units of the same value only can be united into
one sum, it follows,
III. That fractions can be added only when they have
the same fractional unit or common denominator.
84 FRACTIONS.
1. What is the sum of i, 1,1,1?
ANALYSIS. When fractions have a coinmor denominator,
their sum is found by adding their numerators, and placing the
sum over the common denominator. Thus, 1+34-4 + 2=10,
the sum of the numerators ; placing this sum over the common
denominator 5, we have L 2, the required sum.
2. What is the sum of T 3 , T 4 y and T ^ ?
3. What is the sum of f , f , 4 and ?
4. What is the sum of J, f , , f and f ?
5. A boy paid | of a dollar for a pair of gloves, of a
dollar for a knife, and J of a dollar for a slate ; how much
did he pay for all 1
6. A father distributed some money among his children,
as follows : to the first he gave -f^ of a dollar, to the second
T 3 2 , to the third T 7 3 , to the fourth T 9 2 , and to the fifth T 4 2 ;
how much did he give to all ?
7. What is the sum of f and f ?
OPERATION. ANALYSIS. As the giv-
|-f |=||+ 5 8 ff =f | Ans. en fractions have not a
common denominator, we reduce them to the same fractional
unit, (94) and then add their numerators, 27+835,' placing
the sum over the common denominator 36, we obtain ||-
hence the following
RULE. I. When the given fractions have the same de-
nominator, add the numerators, and under the sum write the
common denominator.
II. When they have not the same denominator, reduce
them to a common denominator, and then add as before.
NOTE. If the amount be an improper fraction, reduce it to A whole or a mixed
number.
EXAMPLES FOR PRACTICE.
8. What is the sum of f and | ? Ans. 1 T 7 5 .
9. What is the sum of J and f ? Ans. \\.
2 / ADDITION. 85
10. What is the sum of f and ? Ans. |J.
11. Add |, | and | together. Ans. 1J.
12. Add -f , and f together. Ans. l-J^.
13. Add T %, |, | and \ together. Ans. 2f .
14. Add 3, jtnd |^ togetlier.
15. Add |/4, | and f together.
16. What is the sum of |, f and 1 Ans.
17. What is the sum of f , | and | ? ^4ras. If 1.
18. What is the sum of f , f and 1 1 Ans. 2 fSL 9
To add mixed numbers, add the fractions and integers
separately, and then add their sums.
NOTE. If the mixed numbers are small, they may be reduced to improper
fractions, and then added after the usual method.
19. What is the sum of 14|, 21 and 9| ?
OPERATION. ANALYSIS. By reducing the frac-
141 = 14^ tions to a common denominator, and
214 =21A$ adding them, we obtain || or 1^.,
9 3 = 944 which added to the sum of the inte-
45P Ans. ral numbers > S ives 45 il' the Ans.
20. What is the sum of 3|, 12| and 25f ? Ans. 41|.
21. What is the sum of |, 15, 42-J and 50 ?
22. What is the sum of 30|, 1J, 16^ and ||?
23. Bought 3 pieces of cloth containing 45^, 881, and
35| yards ; how many yards in the 3 pieces ?
Ans. 119^2 yards.
24. Three men bought a horse. A paid 31| dollars, B
paid 43 T 5 3 dollars, and C paid 47 1 dollars ; what was the
cost of the horse 1 Ans. 122 dollars.
25. If it take 5^ yards of cloth for an overcoat, 4| yards
for a dress coat, 2| yards for a pair of pantaloons, and | of
a yard for a vest, how many yards of cloth will it take for
the whole suit? Ans. 12| yards.
.
86 FRACTIONS. ^ *
SUBTRACTION.
96. The process of subtracting one fi action from anoth-
er is based upon the following principles :
I. One number can be subtracted from another only
when the two numbers have the same unit valifc. Hence,
II. ?h subtraction of fractions, the minuend and subtra-
hend must have a common denominator,
1. From T 9 2 subtract T 6 3 .
ANALYSIS. Since the fractions have a common denominator,
the difference is obtained, by subtracting the less numerator 5,
from the greater 9, and writing the result over the common der
nominator 12 ; we thus obtain J^ the required difference.
2. From | subtract f .
3. From jj subtract T \.
4. Subtract 4J from f|.
5. James had J of a bushel of walnuts, and sold | of
them ] how many had he left ?
6. Harvey had jf of a dollar, and gave T 5 ff of a dollar to
a beggar ; how much had he left ?
7. Subtract | from f .
OPERATION. ANALYSIS. As the given frac-
| f = 2i izj^/r <4**f. tions have not a common de-
nominator, we first reduce them to the same fractional unit,
(94) and then subtract the less numerator 9, from the greater
14, and write the result over the common denominator 21. We
thus obtain 5 5 T the required difference. Hence the following
RULE. I. When the fractions have the same denomina-
tor, subtract the less numerator from the greater, and place
the result over the common denominator.
II. When they have not a common denominator, reduce
tliem to a common denominator before subtracting.
*
t K % V
SUBTRACTION". 87
EXAMPLES FOR PRACTICE.
8. From J take f . Ans. j.
9. From | take f Ans. .
10. From f take f. ,4ns. -JJ.
11. From Jjj take . ^TW. -H-
12. Subtract f from f . ^ns. /,.
13. Subtract ^ from f Ans. fc
14. Subtract f from ij.
15. Subtract-^ from jj.
16. Subtract ^ from |. <4ns. 2 \.
17. Subtract | J from J. ^Ins. ^.
18. From 9| take 2|.
OPERATION. ANALYSTS. We first reduce the frac-
9|=9 T 4 2 tional parts, | and j, to a common de-
2|=:2- 9 7y nominator 12. Since we cannot take
^ from T 4 2 , we add 1 i| to T 4 -j, which
6 T5 J.TIS. makes I, and y 9 ^ from 1^ leaves T "^.
We now add 1 to the 2 in the subtrahend, and say, 3 from
9 leaves 6. We thus obtain 6^, the difference required.
Hence, to subtract mixed numbers, we may reduce the
fractional parts to a common denominator, and then subtract
the fractional and integral parts separately.
19. From 24| take 174. Ans, 7f
20. From 147| take 49}. Ans. 98 T 5 3 .
21. From 75^ take 40|. Ans. 3411.
22. From 63 T % take 22|. Ans. 40f.
23. Bought flour at 6| dollars a barrel, and sold it at 7|
dollars a barrel ; what was the gain per barrel ?
Ans. T 9 Q of a dollar.
24. From a cask of wine containing 38| gallons, 15| gal-
lons were drawn ; how many gallons remained ?
Am. 22 | gallons
88 FRACTIONS.
MULTIPLICATION".
CASE I.
97* To multiply a fraction by an integer.
1. If 1 pound of sugar cost $ of a dollar, how much will
3 pounds cost ?
ANALYSIS. If 1 pound cost i of a dollar, 3 pounds, which
are 3 times 1 pound, will cost 3 times ^ or | of a dollar. There-
fore, 3 pounds of sugar, at ^ of a dollar a pound, will cost j* of
a dollar.
2. If 1 horse eat | of a ton of hay in 1 month, how much
will 4 horses eat in the same time ?
3. At | of a dollar a bushel, what will be the cost of 2
bushels of pears ? of 3 bushels ? of 5 bushels ?
4. How many are 3 times f ? 5 times | ? 4 times J ?
6 times ? 9 times y\j ? 8 times f ?
5. If one yard of cloth cost | of a dollar, how much will
3 yards cost?
FIRST OPERATION. ANALYSIS. In the first operation we
|X3=-g 5 -=2^. multiply the fraction by 3, by multi-
SECOND OPERAT!ON. P 1 ?^ its numerator b 7 3 obtaining
_ 5 _ 01 ~ 5 " == ^ ^ ^ n * n * s case ^ ne ' Da ^ ue of the
fractional unit remains the same, but
we multiply the number taken, 8 times. In the second opera-
tion, we multiply the fraction by 3, by dividing its denominator
by 3, obtaining | = 2J. In this case, the value of the fractional
unit is multiplied, 8 times, but the number taken, is the same.
Hence,
Multiplying a fraction consists in multiplying it* nu-
merator, or dividing its denominator.
NOTK. Always divide the denominator when ft is exactly divisible by iue multi-
plier.
MULTIPLICATION. 89
EXAMPLES FOR PRACTICE.
6. Multiply by 5. Ans. 4f .
7. Multiply J by 4. Ans. 3 J.
8. Multiply T * by 6. ^TW. 5f .
9. Multiply 4f by 9. Ans. 4.
10. Multiply } ? by 3. ^ns. 1J.
11. Multiply |f by 14. Ans. 10.
12. Multiply 4| by 5.
OPERATION. ANALYSIS. In multiplying a
4j mixed number, we first multiply
5 . the fractional part, then the inte-
Or, ger, and then add the two pro-
If 4| = Y ducts. Thus, 5 X i = -I = If ;
20 ^X5==21f and 5x4 = 20, which added to
ITT If, gives 21|, the required re-
sult. Or, we may reduce the
mixed number to an improper fraction, and then multiply it.
13. Multiply 6| by 8. Ans. 54.
14. Multiply 9| by 7. Ans. 68f .
.15. If a man earn 1| in 1 day, how much will he earn in
10 days ? Ans. 18 f dollars.
16. What will 14 yards of cloth cost, at f of a dollar a
yard 1 Ans. 10 dollars.
17. At 3 dollars a cord, what will be the cost of 20
cords of wood ? Ans. 65 dollars.
18. If one man can mow Ij 9 ^ acres of grass in a day, how
many acres can 5 men mow? Ans. 9 acres.
19. What will 9 dozen eggs cost, at 14 cents a dozen ?
Ans. 130 J cents.
20. At 64 dollars a barrel, what will 30 barrels of flour
'
cost? Am. 204 dollars.
90 FRACTIONS.
CASE II.
98. To multiply an integer by a fraction.
1. At 9 dollars a barrel, what will | of a barrel of flour
cost?
ANALYSIS. Since 1 barrel of flour cost 9 dollars, f of a barrel
will cost 2 times of 9 dollars. of 9 dollars is 3 dollars, and
| of 9 dollars is 2 times 8 dollars, or 6 dollars. Therefore of
a barrel will cost 6 dollars.
2. If a yard of cloth be worth 8 dollars, what is | of a
yard worth 1
3. If an acre of land produce 25 bushels of wheat, how
much will \ of an acre produce ? f of an acre 1 | of an
acre ?
4. If a man earn 20 dollars in a month, how much can he
earn in of a month ? in f 1 in -^ ? in | ?
5. If a ton of hay cost 12 dollars, how much will ^ of a
ton cost ? | of a ton ? f of a ton ? | of a ton ?
6. At 60 dollars an acre, what will | of an acre of land
cost?
FIRST OPERATION. ANALYSIS. 4 fifths of an acre
5)60 P rice of 1 acre - will cost 4 times as much as 1 fifth
T2 cost of | of an acre. of an acre, or 4 times ^ of 60 dol-
4 lars. \ of 60 dollars is 12 dollars,
and 4 is 4 times 12, or 48 dollars,
4o cost of * of an acre. 6
the oost of | of an acre. In the
BECOND OPERATION. gecond operation> we multi pl y the
60 price of 1 acre. price of j acre by ^ afid obtain
240 dollars, the cost of 4 acres ;
5)240 cost of 4 acres.
L but as I of 1 acre is the same as
48 cost of 4 of an acre, t '
\ of 4 acres, we divide 240 dol-
lars, the cost of 4 acres, by 5, and obtain 48 dollars, the cost of
of of acre. Hence,
MULTIPLICATION. 9J
RULE. Multiplying an integer ly a fraction, consists in
multiplying by the numerator, and dividing the product by
the denominator.
7. Multiply 45 by |. Ans. 33f .
8. Multiply 68 by . Ans. 54f .
9. Multiply 105 by T 7 6 . Ans. 49.
10. Multiply 480 by f . Ans. 300.
11. At 16 dollars a ton, what will be the cost of j of a
ton of hay ? Ans. 12 dollars.
12. If a village lot is worth 340 dollars, what is f of it
worth ? Ans. 255 dollars.
13. If a hogshead of sugar is worth 75 dollars, what is
l of it worth ? Ans. 68| dollars.
14. If an acre of land produce 114 bushels of oats, how
many bushels will T 9 g of an acre produce ?
Ans. 64| bushels.
15. If a man travel 47 miles in a day, how far does he
travel in f of a day ? Ans. 29| miles.
CASE III.
99. To multiply a fraction by a fraction.
1. If a bushel of apples is worth | of a dollar, what is \
of a bushel worth ?
ANALYSIS. Since 1 bushel is worth \ of a dollar, \ of a bush-
el is worth \ times \ of a dollar ; equals f , and a \ of f is .
Therefore \ of a bushel is worth \ of a dollar.
2. If a yard of cloth cost A a dollar, how much will \ of
a yard cost ?
3. When oats are worth J of a dollar a bushel, what is |
of a bushel worth.
4. If a man own 4 of a vessel, and he sells \ of his share
what part of the vessel does he sell ?
92 FRACTIONS.
5. At | of a dollar a bushel, what will of a bushel of
corn cost ?
OPERATION. ANALYSIS. Since 1 bushel cost
|Xf= 1 4T=2 Ans. | of a dollar, | of a bushel will
cost | times | of a dollar. By multiplying the numerators 2
and 3 together, we obtain the numerator 6 of the product ; and
by multiplying the denominators 8 and 4 together, we obtain
the denominator 12 of the product, and thus we have -^=^ for
the required product. Hence we have the following
RULE. Multiply together the numerators for a new nu-
merator ', and the denominators for a new denominator, and
reduce the result to its lowest terms.
EXAMPLES FOR PRACTICE.
6. Multiply 4 by f . Ans. T V
7. Multiply | by f . Ans. 2 9 5 .
8. Multiply | by f . Ans. f .
- 9. Multiply | by f . Ans. 2 \.
10. Multiply -? 2 by f . Ans. T 6 2 .
11. What is the product of f , | and f ? Ans. ^.
12. What is the product of f , | and f ? Ans. ? %.
13. What is the product of J, f and ^ ? Ans. -|.
14. What is the product of f and ? ^Ins. j.
15. What is the product of f , 1|, 5 and j ?
OPERATION. When integers or m?'#-
|XlAX^X|= ec ^ numbers occur among
|X jXf Xf^V^S ^4ns. the given factors, they
may be reduced to improper fractions before multiplying ;
and an integer may be reduced to the form of a fraction by
writing 1 for its denominator ; thus 5=f .
16. What is the product of f , f and 2f ? Ans. Jf .
17. What is the product of 3, T 9 a and | ? Ans. 2f
MULTIPLICATION. 93
18. What is the product of |, T 5 T and f f ?
19. Find the value of f of f multiplied by f o
OPERATION.
NOTES. 1. Fractions with the word of between them are sometimes called com,'
pound fractions. The word of is simply an equivalent for the sign of multiplica-
tion, and signifies that the numbers between which it is placed are to be multiplied
together.
2. When the same factors occur in both numerator and denominator of fractions
to be multiplied together, they may be omitted and the remaining factors only
used; thus, 5 and 3 being found in both the numerators and denominators of the
above example may be omitted in multiplying.
20. Multiply | of f by | of . Ans. Ji-
21. Multiply | of 3 by | of 2^. Ans. 5|.
22. What is the product of T %, ^ of f and \ 1
Ans. ^.
^ 23. What is the product of f of T 7 T by 54 1 Am. 3.
24. What is the value of f times of f of 10 ?
Ans. |
25. What is the value of T 5 2 of f times \ of 3 f ?
Ans. f.
26. At | of a dollar a bushel, what will | of a bushel of
corn cost 1 . Ans. of a dollar.
27. When peaches are worth T 9 ^ of a dollar a .bushel,
what, is | of a bushel worth? Ans. ^ dollar.
28. Jane having | of a yard of silk gave | of it to her
sister ; what part of a yard did she give her sister ?
Ans. | of a yard.
29. When pears are worth J of a dollar a basket, what is
^ of | of a basket worth ? Ans. | of a dollar.
30. A man owning ^ of a ship, sold | of his share;
what part of the whole ship did he sell ? 'Ans. -||.
31. A grocer having ^f of a hogshead of molasses sold
$ of it ; what part of a hogshead remained 1
32. At of a dollar a yard, what will be the cost of -i of
8 yards of cloth 1 Ans. U dollars.
94 FRACTIONS.
DIVISION.
CASE I.
IOO. To divide a fraction by an integer.
1. If 3 pounds of raisins cost 5 of a dollar, how much
will 1 pound cost ?
ANALYSIS. If 3 pounds cost of a dollar, 1 pound which is
of 3 pounds, will cost of , or of a dollar. Therefore, 1
pound will cost of a dollar.
2. If 4 pounds of coffee cost ^ of a dollar, how much will
1 pound cost ?
3. If 5 marbles cost | of a dollar, how much will 1 mar-
ble cost ?
4. If J of a barrel of flour be equally divided among 6
persons, what part of a barrel will each have ?
5. If 4 of a box of tea be equally distributed among 8
persons, what part of a box will each have ?
6. Paid f of a dollar for 4 pounds of butter ; what was
the cost per pound ?
FIRST OPERATION. ANALYSIS. In the first operation
!-j-4= Ans. we divide the fraction by 4, by divid-
ing its numerator by 4, obtaining |.
SECOND OPERATION. In this case the value of the frartional
|-j-4=3 8 g = Ans. unit is unchanged, but we diminish
the number taken^ 4 times. Ii the
second operation we divide the fraction by 4, by multiplying
the denominator by 4, obtaining ^ 8 ff== |. In this case the val-
ue of the fractional unit is diminished 4 times, but the number
taken is the same. Hence,
Dividing a fraction consists in dividing its numerator, or
multiplying its denominator.
NOTB. We divide the numerator vrhen it is exactly divisible by the divisor^ oth-
erwise we multiply the denominator
DIVISION. 95
EXAMPLES JFOB PRACTICE.
7. Divide Jj by 3. Ans. .
8. Divide | by 4. Ans. J.
9. Divide j j by 5. ^4/is. T 2 5 .
10. Divide i| by 5. Ans. T 3 e .
11. Divide | by 9. ^dws. 6 a .
12. Divide | by 21. Ans. fo
13. Divide | of f by 12. ^s. T ' ff .
14. Divide | of f by 6. Ans. -fa.
15. Divide 4| by 7.
OPERATION.
4^= 2 ^ NOTE. We reduce the mixed num-
2gi-i-7=f -<4.?is. ber to an improper fraction and then
divide as before.
16. Divide 3| by 4. AM. JJ.
17. Divide 6j by 9. Ans. %%.
18. Divide 4 of 2^ by 3. Ans. j.
19. Divide 8^ by 12. ^Ins. f j.
20. Divide 13 j by 10. Ans. If.
21. Divide | of 8 by 20.- Ans. J.
22. If 6 persons agree to share equally | of a bushel of
grapes, what part of a bushel will each have ? Ans. |.
23. If 5 yards of sheeting cost T 9 <j of a dollar, what will
JL yard cost ? Ans. -f^ of a dollar.
24. If 8 bushels of apples cost 5| dollars, what will 1
Dushel cost ? Ans. | of a dollar.
25. If J of 10 pounds of butter cost l\ Collars, what
will 1 pound cost ? Ans. | of a dollar.
26. A man distributed J$ of a dollar equally among 6
beggars ; what part of a dollar did he give to each ?
27. If f of 9 cords of wood cost 12| dollars, what will
1 oord cost]
96 FRACTIONS.
CASE II.
1O1. To divide an integer by a fraction.
1. At j of a dollar a yard, how many yards of ribbon
can be bought for 2 dollars?
ANALYSIS. As many yards as | of a dollar, the price of 1
yard, is contained times in 2 dollars. Since in 1 dollar there
are 3 thirds of a dollar, in two dollars, there are 2 times 3 thirds,
or G thirds ; and 1 third is contained in 6 thirds, 6 times.
Therefore 6 yards of ribbon can be bought for 2 dollars.
2. When potatoes are | of a dollar a bushel, how n.any
bushels can be bought for 2 dollars ? for 4 dollars ? for 6
dollars ?
3. If a man spend | of a dollar a day for cigars, how long
will it take him to spend 3 dollars ? 5 dollars ? 6 dollars 9
4. At | of a dollar a bushel, how many bushels of corn
can be bought for 16 dollars 1
JIRST OPERATION. ANALYSIS. As many bushels as J of
a dollar, the price of 1 bushel, is con-
tained tiras in 16 dollars. But we can-
not divide integers by fifths, because
they are not of the same denomination.
20 bushels. Reducing 16 dollars to fifths by multi
SECOND ^OPERATION, plying by 5, we have 80 fifths, and 4
4)14 fifths is contained in SO fifths, 20 times,
> r the required number of bushels. In
the second operation, we divide the in-
20 bushels. teger by the numerator of the fraction,
and multiply the quotient by the denominator, which produces
the same result as in the first operation. Hence
Dividing "by a fraction consists in multiplying by the
denominator, and dividing the product by the numerator
of the divisor.
DIVISION.
97
EXAMPLES FOR PRACTICE.
6. Divide 18 by f .
6. Divide 14 by f .
7. Divide 11 by f .
8. Divide 75 by $.
9. Divide 120 by T 6 T .
10. Divide 96 by {?.
11. Divide 226 by &.
12. Divide 28 by 4|.
OPERATION.
28X3=84
84-4-14=6 Ans.
Ans. 27.
Am. 49.
AMS. 19f.
Ans. 83|.
Ans. 220.
Ans. 186.
Ans, 627J.
13. Divide 16 by
14. Divide 42 by
15. Divide 112 by
16. Divide 180 by
17. Divide 425 by
18. Divide 318 by
ANALYSIS. We reduce the mixed
number to an improper fraction, and
then divide the integer in the same
manner as by a proper fraction.
21. Ans. 7.
3A. Ans. 12.
6|. Ans. 17^.
7|. Ans. 25 -fy.
f Ans. 595.
2 V Ans. 1219.
19. When potatoes are ^ of a dollar a bushel, how many
bushels can be bought for 10 dollars ? Ans. 12 bush.
20. Divide 9 bushels of corn among some persons, giving
them T 3 g of a bushel each ; how many persons will receive
a share? Ans. 48.
21. At 2 1 dollars a cord, how many cords of wood can
be bought for 27 dollars ? Ans. 9 T 9 ? cords.
22. If a horse eat | of a bushel of oats in a day, in
how many days will he eat 20 bushels ? Ans. 36 days.
23. If a man walk 2 T 9 <j miles an hour, how many hours
will he require to walk 48 miles ? Ans. 16^| hours.
24. At Jg of a dollar a pound, how many pounds of rice
can be bought for 3 dollars 7 Ans. 48 pounds.
98 FRACTIONS.
CASE III.
1O2. To divide a fraction by a fraction.
1. At | of a dollar a pound, how many pounds of tea can
be bought for J of a dollar ?
ANALYSIS. As many pounds as | of a dollar, the price of 1
pound, is contained times in | of a dollar ; 2 fifths are contain-
ed in 4 fifths, 2 times. Therefore 2 pounds can be bought for
J of a dollar.
Hence we see, that when fractions have a common denomina-
tor, division may be performed by dividing the numerator of
the dividend by the numerator of the divisor.
2. How many pine-apples at T 3 of a dollar each, can be
bought for T 6 jj of a dollar? for y^? for }|?
3. If a horse eat ^ of a bushel of oats in 1 day, in how
many days will he eat | of a bushel If? y> ? \* ?
4. At ^ of a dollar a bushel, how many bushels of ap-
ples can be bought for f of a dollar ? for ^ ? for f ?
5. At | of a dollar a pound, how many pounds of tea can
be bought for | of a dollar ?
FIRST OPERATION. ANALYSIS. As many pounds
= 3 %; |=2U- as I of a dollar, the price of
io'^slj IB Ans. 1 pound, is contained times
SECOND OPERATION in | of a dollar. f equal
-*-f=:|Xf = g 5=1 B Ans - 2 8 <J i e( l ual if' and 8 twenti-
eths are contained in 15 twen-
tieths 1| times. Or, as in the second operation, we have multi-
plied the dividend |, by the denominator 5, of the divisor, and di-
vided the result by the numerator 2, of the divisor. Iience,by
inverting the terms of the divisor the two fractions will stand in
uch relation to each other, that we can multiply together the
two upper numbers for the numerator of the quotient, and the
two lower numbers for the denominator, as shown in the second
operation. Hence *
DIVISION. 99
*
KULE. I. Reduce integers and mixed numbers to im-
proper fractions.
II. Invert the terms of the divisor, and proceed as in mul-
tiplication.
EXAMPLES FOR PRACTICE.
6. Divide T % by T %. Ans. 3.
7. Divide ^ by \. Ans. 2.
8. Divide f by f . Ans. jf .
9. Divide J by f . Ans. 2 T 3 g .
10. How many times is T 7 ^ contained in -} J ? Ans. 2|.
11. How many times is f contained in | ? Ans. -fa.
12. How many times is A contained in j| ? ^4ns. 1|.
13. Divide | oi'| by f. Ans. f
14. Divide f of f by &. Ans. If
15. Divide ji by 4 of f. . Ans. 7 T V
16. Divide | of by f of J. ^ns. 1 T ^.
17. At ^ of a dollar a pound, how many pounds ot su-
gar can be bought for | of a dollar ? Ans. 5| pounds.
18. At T 7 ^ of a dollar a pint, how much wine can be
bought for \ of a dollar ? Ans. f of a ^pint.
19. At -| of I of a dollar a yard, how many yards of rib-
bon can be bought for T 7 ^ of a dollar ? Ans. 2| yards.
20. At ^ of a dollar a yard, how many yards of silk can
be bought for | of a dollar ? Ans. 2| yards.
21. A man owning f of a copper mine, divided his share
equally among his sons, giving them T 6 ff each ; how many
sons had he ? Ans. 2.
22. If ^ of a bushel of pears cost | of a dollar, how much
will 1 bushel cost 1 Ans. ^ of a dollar.
23. How much corn at $ of a dollar a bushel, can be
bought for | of a dollar. Ans. { of a bushel.
100 FRACTIONS.
^
PROMISCUOUS EXAMPLES.
1. In 25 T 9 ff pounds how many IGtlis of a pound]
2. Reduce ^V to a mixed number. Ans. 11||.
3 o o u
3. Reduce ^{jj to its lowest terms. Ans. j.
4. In 7 8 5 5 9 of a day how many days 1
5. Change 42 pounds to sevenths of a pound.
6. Reduce 21 i to an improper fraction, ylns. J | 9 .
7. Reduce 126| to thirds. Ans. -f-&.
8. Reduce 1 1 to its lowest terms. Ans. |.
9. Reduce 4 and | to a common denominator.
10. Reduce 36 to a fraction whose denominator is 12.
11. What is the sum of |, | and { 1 Ans. If.
12. Add together T 9 ^, ^ and 34. Ans. 4|.
13. What is the difference between f and | ?
14. Reduce T 9 ff , f and | to a common denominator.
15. Sold 9f cords of wood to one inan ; and 12 T 9 g to an-
other ; how much did I sell to both ]
16. Paid 87 T 9 o dollars for a horse, and 62^ dollars for a
wagon ; how much more was paid for the horse than the
wagon ? Ans. 25| dollars.
17. A farmer having 234{| acres of land, sells at one
time 42| acres, at another time 61|, and at another 70^-
asres ; how many acres has he left ? Ans. 60 T 6 g acres.
18. A speculator bought 120 bushels of wheat, for 136|
dollars, and sold it for 197 1 dollars; how much did he gain?
19. Bought 12 pounds of coffee at ^ of a dollar a pound,
and 9 pounds of tea at J of a dollar a pound; what was the
cost of the whole ? Ans. Sy 7 ^ dollars.
20. Bought 10 bushels of wheat, at 1^ dollars a bushel,
and 14 bushels of corn, at | of a dollar a bushel ; which
cost the more, and how much ?
Ans. the corn, 3 \ dollars.
PKOMISCUOUS EXAMPLES. 101
21. Paid 12 dollars for some cloth, at the rate of J of a
lollar a yard ; how many yards was purchased ?
22. If 8 oranges cost f of 14 dollars, what will 1 orange
3ost ? Ans. yL of a dollar.
23. A man bought J of a farm and sold f of his share ;
what part of the whole farm did he sell ? what part had he
eft 1 ? Am. Sold 44.
24. If a barrel of sugar is worth 22 dollars, what is T 7 ^ of
it worth ? Ans. 15 f dollars.
25. How many hours will it take a man to travel 136
miles, if he travel 3| miles an hour ? Ans. 41 2 9 g hours.
26. How many barrels of apples can be bought for 18
dollars, at ! T 3 g dollars a barrel 1 Ans. 15 T 3 ^ barrels.
27. If the smaller of two fractions be T 4 T , and the differ-
ence f , what is the greater ] Ans. ||.
28. If the sum of two fractions is 1|, and one of them is
5 3, what is the other 1 Ans. f J.
29. If the dividend be f f, and the quotient f , what is
the divisor? Ans. 1^.
30. If the divisor be T 9 g, and the quotient 3 J, what is the
dividend ? Ans. 2 T 4 5 .
31. How many bushels of oats worth | of a dollar a bush-
el, will pay for f of a barrel of flour worth 9 dollars a bar-
rel 1 Ans. 15 bushels.
32. At | of a dollar a rod, what will it cost to dig J of |
of 5^ rods of ditch ? Ans. T 9 T 9 3 dollars.
33. If a man has 24 J bushels of clover seed, and he sells
| of it, how much has he left ? Ans. Q^ bushels.
34. A man had 6 lots of land, each containing 37|
acres ; how many acres did they all contain ?
35. If | of a ton of hay can be bought for 15 dollarsu
what part of a ton can be bought for 1 dollar?
102 DECIMALS.
DECIMAL FRACTIONS.
NOTATION AND NUMERATION.
1O3. Decimal Fractions are fractions which have for
their denominator 10, 100, 1000, or 1 with any number of
ciphers annexed.
Decimal fractions are commonly called decimals.
Since T V= T V$y, Tiu=T<5oo> &c -> tlie denominators of
decimal fractions increase and decrease in a tenfold ratio,
the same as simple numbers.
1 04:. In the formation of Decimals a unit is divided in-
to 10 equal parts, called tenths ; each of these tenths is di-
vided into 10 other equal parts called hundredths ; each of
these hundredths into 10 other equal parts, called thou-
sandths j and soon. Since the denominators of decimal
fractions increase and decrease by the scale of 10, th-j same
as simple numbers, in writing decimals the denominators
may be omitted.
1O5. The Decimal sign (.) is always placed before deci-
mal figures to distinguish them from integers. It is com-
monly called the decimal point. Thus,
T 6 is expressed .6
" -279
.5 is 5 tenths, which = y 1 ^ of 5 units ;
.05 is 5 hundredths, " = tff ^ * tenths;
.005 is 5 thousandths, " = T \y of 5 hundredths.
And universally, the value of a figure in any decimal
place is T \j the value of the same figure in the next left
bond place.
NOTATION AND NUMERATION. 103
1OO. The relation of decimals and integers to each oth-
er is clearly shown by the following
DECIMAL NUMERATION TABLE.
5732754.573256
By examining this table we see that
Tenths are expressed by one figure.
Hundredths " " " two figures.
Thousandths " " " three "
1O7. Since the denominator of tenths is 10, of hun-
dredths 100, of thousands 1000, and so on, a decimal may
be expressed by writing the numerator only ; but in this
case the numerator or decimal must always contain as many
decimal places as are equal to the number of ciphers in the
denominator ; and the denominator of a decimal will al
ways be the unit, 1, with as many ciphers annexed as are
equal to the number of figures in the decimal or numerator,
The decimal point must never be omitted.
EXAMPLES FOR PRACTICE.
1. Express in figures seven-tenths. Ans. .7.
2. Write twenty-five hundredths. Ans. .25.
8. Write nine hundredths. Ans. .09.
4. Write one hundred twenty-five thousandths.
5. Write eighteen thousandths.
104 DECIMALS.
6. Write fifty-eight hundredths.
7. Write two hundred thirty-six thousandths.
8. Write one thousand three hundred twenty ten-thou-
sandths. Am. .1320.
9. Write seven hundred thirty-two ten-thousandths.
Read the following decimals :
.06 .143 .000 .479
.84 .037 .3240 .00341
.80 .472 .1026 .102367
1O8. A mixed number is a number consisting of inte-
gers and decimals ; thus, 71.406 consists of the integral
part, 71, and the decimal part, .406 ; it is read the same as
71 T 4 <j<&, 71 an( i 406 thousandths.
EXAMPLES FOE PRACTICE.
1. Write twenty-four, and four tenths. Ans. 24.4.
2. Write thirty-two, and five hundredths,
3. Write seventy-six, and forty-six thousandths.
4. Write one hundred twelve, and one hundred ninety
thousandths. Ans. 112.190.
5. Write sixty-three, and forty-four ten-thousandths.
6. Write seventy-five, and one hundred forty ten-thou-
sandths.
7. Write five, and 5 hundred thousanths.
8. Write sixteen, and 21 ten-thousan^tis.
9. Write eight, and 234 hundred thousai
10. Write forty, and 75 hundred thousandths.^
Ans. 40.00075.
11. Read the following numbers :
42.08 50.002 640.00010
81.110 161.0301 7.4230
120.0342 14.42000 3.01206
NOTATION AND NUMERATIO
1OO. From the foregoing explanations and illustrations
we derive the following important
PRINCIPLES OF DECIMAL NOTATION AND NUMERATION.
1. The value of any decimal figure depends upon its
place from the decimal point j thus .3 is ten times .03.
2. Prefixing a cipher to a decimal decreases its value the
same as dividing it by ten ; thus, .03 is ^ the value of .3.
3. Annexing a cipher to a decimal does not altar its val-
ue, since it does not change the place of the significant fig-
ures of the decimal ; thus, T 6 0, or, .6, is the same as y 6 ^, or
.60.
4. Decimals increase from right to left, and decrease from
left to right, in a tenfold ratio ; and therefore they may be
added, subtracted, multiplied, and divided the same as whole
numbers.
5. The denominator of a decimal, though never express-
ed, is always the unit 1, with as many ciphers annexed as
there are figures in the decimal.
6. To read decimals requires two numerations; first, from
units, to find the name of the denominator, and second, to-
wards units, to find the value of the numerator.
1 1 0. Having analyzed all the principles upon which
the writing and reading of decimals depend, we will now
present these principles in the form of rules.
RULE EOR DECIMAL NOTATION.
I. Write the decimal the same as a whole number, placing
ciplvers where necessary to give each significant figure its true
local value.
II. Place the decimal point before the first figure.
106 DECIMALS.
9
RULE FOR DECIMAL NUMERATION.
I. Numerate from the decimal point, to determine the de-
nominator.
II. Numerate towards the decimal point, to determine the
numerator.
III. Read the decimal as a whole number, giving it the
name of its lowest decimal unit, or right hand figure.
EXAMPLES FOR PRACTICE.
1. Write 325 ten-thousandths. Ans. .0325.
2. Write four hundred ten-thousandths.
3. Write 117 ten-thousandths.
4. Write ten ten-thousandths. Ans. .0010.
5. Write 250 millionths. Ans. .000250.
6. Write twelve hundred ten-thousandths.
7. Write 9 hundred-thousandths. Ans. .00009.
8. Read the following decimals.
.1236 .00061 .32760
.0080 .720000 040721
9. Write four hundred, and nine tenths.
Ans. 400.9.
10. Write twenty-seven, and fifty-six hundredths.
11. Write eighty-five, and one hundred fifty thousandths.
12. Write one thousand, and twelve millionths.
13. Write three hundred sixty-five, and one thousand
eight hundred seven hundred-thousandths.
Ans. 365.01807.
14. Write nine hundred ninety, and three thousand two
hundred fourteen millionths. Ans. 990.003214.
15. Read the following numbers :
71.03 11.0003 34.800000
126.326 240.01376 9.1263476
REDUCTION. > * 107
REDUCTION.
CASE I.
111. To reduce decimals to a common denomina-
tor.
1. Reduce .3, .09, .0426, .214 to a common denominator.
OPERATION. ANALYSIS. A common denominator must
.3000 contain as many decimal places as is equal to
.0900 the greatest number of decimal figures in any
of the given decimals. We find that the third
number contains four decimal places, and hence
10000 must be a common denominator. As annexing ciphers
to decimals does not alter their value, we give to each number
four decimal places, by annexing ciphers, and thus reduce the
given decimals to a common denominator. Hence,
RULE. Giv$ to qalh number the same number of deci-
mal places ^y annexing ciphers. tk *
EXAMPLES FOR PRACTICE.
2. Reduce .7, .073, .42, .0020 and .007 to a common de-
nominator.
3. Reduce .004, .00032, .6, .37 and .0314 to a common
denominator. *
4. Reduce 1 tenth, 46 hundredths, 15 thousandths, 462
ten-thousandths, and 28 hundred-thousandths, to a common
denominator.
5. Reduce 9 thousandths, 9 ten-thousandths, 9 hundred-
thousandths and 9 millionths to a common denominator.
6. Reduce 42.07, 102.006, 7.80, 400.01234 to a com.
mon denominator.
7. Reduce 300.3, 8.1003, 14.12614, 210.000009, and
1000.02 to a common denominator.
$' :
iafc
QECIMALS.
CASE II.
119. To reduce a decimal to a common fraction.
1. Reduce .125 to an equivalent common fraction.
OPERATION. ANALYSIS. Writing the decimal figures,
.125 = T Wo-==i .125, over the common denominator, 1000,
we have -Jfyjfo$=$.
RULE. Omit the decimal point, supply the proper de-
nominator, and then reduce the fraction toits lowest terms.
EXAMPLES FOR PRACTICE.
1. Reduce .08 to a common fraction. Ans. %.
2. Reduce .625 to a common fraction. Ans. f .
3. Reduce .375 to a common fraction. Ans. f,
4. Reduce .008 to a common fraction. Ans. T ^.
5. Reduce .4 to a common fraction. Ans. I.
5
6. Reduce .024 to a common fraction, Ans. T 4 T
CASE ill.
113. To rduce a common frffctJrai to a decimal
2. Reduce f to its equivalent decimal.
ANALYSIS. Since we can not di-
vide the numerator 3, by 4, we re-
duce it to tenths by annexing a ci-
pher, and then dividing we obtain 7
tenths, and a remainder of 2 tenths.
Reducing this remainder to 7mn-
dredths by annexing a cipher, and
dividing by 4, we obtain 5 hun-
dredths. The sum of the quotients
gives .75, the required answer.
OPERATION.
4)3.0(7 tenths.
9 Q
^J.O
4^20(5 hundredth*
20
Ans. .75.
or 4)3.00
.75 Ans.
RULE I. Annex ciphers to the numerator, and divide
~by the denominator.
II. Point off as many decimal places in the result as are
equal to the number of ciphers annexed*
ADDITION. 109
EXAMPLES FOR PRACTICE.
1. Reduce to a decimal. Ans. .5.
2. Reduce J to a decimal. Ans. .25.
3. Reduce f to a decimal. Ans. A.
4. Reduce | to a decimal. Ans. .8.
5. Reduce $ to a decimal. Ans. .125.
6. Reduce T 9 5 to a decimal. Ans. .9.
7. Reduce | to to a decimal. Ans. .625.
8. Reduce ^ to a decimal. Ans. .04.
9. Reduce T 5 g to a decimal. Ans. .3125.
10. What decimal is equivalent to J ? Ans. .85.
11. What decimal is equivalent to T 3 g ? ^4ns. .1875.
12. What decimal is equivalent to -^1 Ans. .016.
ADDITION.
1 ll. Since the same law of local value extends both to
the right and left of units' place; that is, since decimals and
simple integers increase and decrease uniformly hy the scale
of ten, it is evident that decimals may be added, subtracted,
multiplied and divided ^n the same manner as integers.
1. What is the sum of 4.314, 36.42, 120.0042, and
.4276]
OPERATION. ANALYSIS. We write the numbers so
4.314 that the figures of like orders of units shall
36.42 stand in the same columns ; that is, units
^ under units, tenths under tenths, hun-
dredths under hundredths, &c. This brings
161 1658 * ne Decimal Points directly under each^Dth-
er. Commencing at the right hand, we add
each column separately, and carry as in whole numbers, and in
the result we place a decimal point between units and tenths,
or directly under the decimal point in the numbers added
From this example we derive the following
110 DECIMALS.
RULE. I. Write the numbers so that the decimal pvint*
shall stand directly under each other.
II. Add as in ivlwle number s } and place the decimal point,
in the result, directly under the points in the numbers added.
EXAMPLES FOR PRACTICE.
2. What is the sum of 2.7, 30.84, 75.1, 126.414 and
3.06? Ans. 238.114.
3. What is the sum of 1.7, 4.45, 6.75, 1.705, .50 and
.05? Ans. 15.155.
4. Add 105.7, 19.4, 1119.05, 648.006 and 19.041.
Ans. 1911.197.
5. Add 48.1, .0481, 4.81, .00481, 481.
Ans. 533.96291.
6. Add 1.151, 13.29, 116.283, 9.0275 and .61.
Ans. 140.3^15.
7. Add .8, .087, .626, .8885 and .49628.
8. What is the sum of 91.003, 16.4691, 160.00471,
700.05, 900.0006, .03^5 ? Ans. 1867.55891.
9. What is the sum .of fifty-four, and thirty-four hun-
dredths; one, and\ine ten-thousanMths ; thre'e," and two
hundred seven milliomhs ; twenty-three thousandths; eight,
and nine tenths; four, and one hundred thirty-five thou-
sandths? Ans. 71.399107.
10. How many acres of land in four farms, containing
respectively, 61.843 acres, 120.75 acres, 142.4056 acres,
and 180.750 acres? Ans. 505.7486.
1L How many yards of cloth in 3 pieces, the first con-
taining 21^ yards, the second 36| yards, and the third
40.15 yards? Ans. 98.40.
12. A man owns 4 city lots, containing 32|, 36|, 40f,
42.73 rods of land respectively; how many rods in all?
Ans. 152.205 rods.
SUBTRACTION.
Ill
Ans. 76.9624
SUBTRACTION.
115. From 12 4.2750 take 47.3126.
OPERATION. ANALYSIS. Write the subtrahend un-
124.2750 der the minuend, placing units under
47.3126 units, tenths under tenths, &c. Com-
mencing at the right hand, we subtract
as in whole numbers, and in the remain-
tier we place the decimal point directly under those in the num-
bers above. If the number of decimal places in the minuend
and subtrahend are not equal, they may be reduced to the same
number of decimal places before subtracting, by annexing ci-
phers. Hence the
RULE 1. Write the numbers so that the decimal points
shall stand directly under each other.
II. Subtract gs in whole numbers, and place the decimal
point in the result directly under the points in the given
numbers.
EXAMPLES FOR PRACTICE.
(2) (3)
Minuend, 12.07 37.4562
Subtrahend, 4.3264 .97
Remainder, ^7.7436 36.4862 .628476
5. From 463.05 take 17.0613. Ans. 445.9887.
6. From 134.63 take 101.1409. Ans. 83.4891.
7. From 189.6145 take 10.151. Ans. 179.4635.
8. From 671.617 take 116.1. Ans 555.517.
9. From 480. take 245.0075. Ans. 234.9925.
10. Subtract .09684 from .145. Ans. .04816.
11. Subtract .2371 from .2754. Ans. .0383.
12. Subtract 215.7 from 271. Ans. 55.3.
13. Subtract .0007 from 107. Ans. 106.9993.
14. Subtract 1.51679 from 27.15. Ans. 25.63321.
112 DECIMALS.
15. Subtract 37i from 84.125. Ans. 46.625.
16. Subtract 3| from 9.3261. Ans. 5.5761.
17. Subtract 25.072 from 112|. Ans. 87.553.
18. A man owned fifty-four liundredths of a township
of land, and sold fifty-four thousandths of the same, how
much did he still own 1 Ans. .486.
19. From 10 take three millionths. Ans. 9.999997.
20. A man owning 475 acres of land, sold at different
times 80.75 acres, 100 J acres, and 125.625 acres; how
much land had he left ? \ Ans. 168.5 acres.
MULTIPLICATION.
116. 1. What is the product of .25 multiplied by .5.
OPERATION. ANALYSIS. We first multiply as in whole
.25 numbers ; then, since the multiplicand has 2
.5 decimal places and the multiplier 1, we point off
~~ 2 -{-1=3 decimal places in the product. The
ns >' reason for this will be evident, by considering
both factors common fractions, and then multiplying as in
(99), thus: .25= T ^an<1.5 = T 5 o; and ^XiV^Wi
which written decimally is .125 Ans. Hence the
RULE. Multiply as in whole numbers, and from the
riff Jit- hand of the product point off as many figures for dec-
imals as there are decimal places in Loth factors.
NOTES. 1. If there be not as many figures in the product as there are decimals
In both factors, supply the deficiency by prefixing ciphers.
2. To multiply a decimal by 10, 100, 1000, &c., remove the point as many place*
to the right as there are ciphers on the right of the multiplier.
EXAMPLES FOR PRACTICE.
(2) (3) (4)
.241 9.4263 .01346
.7 .5 .06
.1687 4.71315 .0008076
MULTIPLICATION. 113
5. Multiply 7.1 by 8.2. Ans. 58.22.
6. Multiply 15.5 by .08. Ans. 1.24.
7. Multiply 8.123 by .09. Ans. .73107.
8. Multiply 4.5 by .15. Ans. .675.
9. Multiply 450. by .02. Ans. 9.
10. Multiply 341.45 by .007. Ans. 2.39015.
11. Multiply 3020. by .015. Ans. 45.3ft?
12. Multiply .132 by .241. Ans. .031812.
13. Multiply .23 by .009. Ans. .00207.
14. Multiply 7.02 by 5.27. Ans. 36.9954.
*15. Multiply .004 by .04. Ans. .00016.
16. Multiply 2461. by .Q52&__- Ans. 130.1869.
17. Multiply .007853 by .035^1 J^s. .000274855.
18. Multiply 25.238 by 12.17. Ans. 307.14646.
19. Multiply .3272 by 10. Ans. 3.272.
20. Multiply .3272 by 100. Ans. 32.72.
21. Multiply .3272 by 1000. Ans. 327.2.
22. Find the value of .25X5Xl2. Ans. 1.5.
23. Find the value of .07x2.4 X-015. Ans. 00252.
24. Find the value of 6JX.8X3.16. Ans. 16.432.
25. If a man travel 3.75 miles an hour, how far will he
travel in 9.5 hours'? Ans. 35.626 miles.
26. If a sack of salt conialn 94.16 pounds, how many
pounds will 17 such sacks contain ?
Ans. 1600.72 pounds.
27. If a man spend .87 of a dollar in 1 day, how much
will he spend in 15.525 days ?
Ans. 13.50675 dollars.
28. One rod is equal to 16.5 feet; how many feet in
30.005 rods ? Ans. 495.0825.
29. How many gallons of molasses in .54 of a barrel,
there being 31.5 gallons in 1 barrel ? A ns. 17.01 gallons.
114 DECIMALS.
DIVISION.
117. 1. What is the quotient of .225 divided by .5 ?
OPERATION. ANALYSIS. We perform the division
,5).225 the same as in whole numbers, and the
only difficulty we meet with is in point-
.45 Ans. j n g og 1 the decimal places in the quotient.
To determine how many places to point off, we may reduce the
decimals to common fractions, thus; .225=-^^ and 5==-^,
performing the division as in (97), we have T 22_5 j _i__5^___2^5 j
X '- = -^Aj ; and this quotient expressed decimally, is .40.
Here we see that the dividend contains as many decimal places
as are contained in both divjsor and quotient. Hence the fol-
lowing
RULE. Divide as in whole numbers, and from the right
hand of the quotient point off as many places for decimals
as the decimal places in the dividend exceed those m the
divisor.
NOTES. 1. If the number of figures in the quotient be less than the excess of
the decimal places in the dividend over those hi the divisor, the deficiency must
be supplied by prefixing ciphers.
2. If there be a remainder after dividing the dividend, annex ciphers, and con-
tinue the division ; the ciphers annexed are decimals of the dividend.
3. The dividend must always contain at least as many decimal places as th
divisor, before commencing the division.
4. In most business transactions, the division is considered sufficiently exact
when the quotient is carried to 4 decimal places, unless great accuracy is required.
5. To divide by 10, 100, 1000, &c., remove the decimal point as many places to
the left as there are ciphers on the right hand of the divisor.
EXAMPLES FOR PRACTICE.
(2) (3) (4) (5)
.6).426 .8)3.7624 .05)81.60 .009).00207
.71. 4.703 1632. * ~S
DIVISION. 115
(6) (7) (8)
.075).9375(12.5 .288)18.0000(.0625 .0025)15.875(6350,
75 1728 150
187 720 87
150 576 75
375 1440 125
375 1440 125
9. Divide 44 by .4. Ans. 110.
10. Divide 15 by .25. Ans. 60.
11. Divide .3276 by .42. .Ans. .78.
12. Divide .00288 by .08. Ans. .036.
13. Divide .0992 by .32. Ans. .31.
14. Divide 17.6 by 44. Ans. .5.
15. Divide .0000021 by .0007. Ans. .003.
16. Divide .56 by 1.12. Ans. 5.
17. Divide 1496.04 by 10. Ans. 149.604.
18. Divide 1196.04 by 100. Ans. 14.9604.
19. Divide 1596.04 by 1000. Ans. 1.49604.
20. Divide 4.96 by 100. Ans. .0496.
21. Divide 10 by .1. Ans. 100.
22. Divide 100 by .2. Ans. 500.
23. If 2.5 acres produce 34.75 bushels of wheat, how
much does one acre produce ? Ans. 13.9 bushels.
24. If a man travels 21.4 miles a day, how many days
will he require to travel 461.03 miles?
25. If a man build 812.5 rods of fence in 100 days,
how many rods does he build each day?
26. Paid 131.15 for 61 sheep; how much was paid for
each ? Ans. 2.15 dollars.
116 DECIMALS.
PROMISCUOUS EXAMPLES.
1. Add twenty-five hundredths, six hundred fifty-four
thousandths, one hundred and ninety-nine thousandths, and
seven thousand five hundred sixty-nine ten-thousandths.
Ans. 1.8599.
2. From ten take ten thousandths. Ans. 9.99.
3. What is the difference between forty thousand, and
forty thousandths? Ans. 39999.960.
4. Multiply sixty-five hundredths, by nine hundredths.
Ans. .0585.
5. Divide 324 by 6400. Ans. .050625.
6. Reduce .125 to a common fraction. Ans. |.
7. Reduce J to a decimal fraction. Ans. .875.
8. Divide .016Q04 by .004. Ans. 4.001.
9. Reduce JX to a decimal fraction. Ans.. 68.
10. Reduce" .4, .007, .1142, .036, .00015, and .42, to a
common denominator.
11. At 13.9 dollars a ton, what will 2.5 tons of hay cost?
Ans. 34.75 dollars.
12. If a pound of sugar cost .09 dollars, how many
pounds can be bought for 5.85 dollars? Ans. 65 pounds.
13. If 40.02 bushels of potatoes are raised upon 1 acre
of land, how many acres would be required to raise 4580.64
bushels? Ans. 114.458 acres.
14. At 11 dollars a ton, how much hay can be bought
for 13.75 dollars? Ans. 1.25 tons.
15. If a man travel 32.445 miles in a day, how far can
he travel in .625 of a day? Ans. 20.278125 miles.
16. If 2 pounds of sugar cost .1875 dollars, what will
be the cost of 10 pounds? Ans. .9375 dollars.
17. If 3 barrels apples cost 19.125 dollars, what will be
the cost of 100 barrels ? Ans. 337.5 dollars.
UNITED STATES MONEY. 117
UNITED STATES MONEY.
118. United States Money is the legal currency
of the United States, and was established by act of Con-
gress August 8, 1786. Its denominations and their rela-
tive "values are shown in the following
TABLE.
10 mills (m.) make 1 cent, c.
10 cents " 1 dime, d.
* 10 dimes " 1 dollar, $.
10 dollars " 1 eagle, E.
NOTE. The currency of the United States is decimal currency, and is sometimes
called federal Money.
119. The character, $, before any number indicates
that it expresses United States money. Thus $75 expresses
75 dollars.
120. The dollar is the unit of United States
money; dimes, cents, and mills are fractions of a dollar,
and are separated from the dollar by the decimal point (.) ;
thus, two dollars one dime two cents five mills are written
$2.125.
121. By examining the above table we find
1st. That the dollar being the unit, dimes, cents and
mills are respectively tenths, hundredths and thousandths
of a dollar.
2d. That the denominations of United States money
increase and decrease the same as simple numbers and dec-
imals, and are expressed according to the decimal system of
notation.
Hence we conclude that
United States money may le added, subtracted, multi-
plied and divided in the same manner as decimals.
118 UNITED STATES MONEY.
Dimes are not read as dimes, but the two places of dimes
and cents are appropriated to cents; thus 1 dollar 3 dimes 2
cents, or $1.32, are read one dollar thirty-two cents; hence,
When the number of cents is less than 10, we write a
cipher before it in the place of dimes.
NOTE. The half cent is frequently written as 5 mills : thus, 24% cents, written
$.245.
EXAMPLES FOR PRACTICE.
1. Write five dollars twenty-five cents. Ans. 85.25.
2. Write four dollars eight cents. Ans. $4.08*^
3. Write twelve dollars thirty-six cents.
4. Write seven dollars sixteen cents.
6. Write ten dollars ten cents.
7. Write sixty-five cents four mills. $.654.
8. Write one dollar five cents eight mills. $1.058.
9. Write eighty-seven cents five mills. Ans. $.875.
10. Write one hundred dollars one cent one mill.
Ans. $100.011.
11. Read $4.07; $3.094; $10.50; $25.02.
KEDUCTION.
122. 1. How many cents are there in 75 dollars 7
OPERATION. ANALYSIS. Since in 1 dollar there are
75 100 cents, in 75 dollars there are 75 times
100 100 cents or 7500 cents. To multiply by
10, 100, &c., we annex as many ciphers to
7500 cents. the mu i t i p iicand as there are ciphers in the
multiplier, ( 62 ). Hence
To change dollars to cents, multiply by 100 ; that is, an-
nex TWO ciphers. And
To change dollars to mills, annex THREE ciphers.
To change cents to mills, annex ONE cipher.
REDUCTION. 119
EXAMPLES FOR PRACTICE.
2. Reduce $24 to cents. Ans. 2400 cents.
8. Reduce $42 to cents. Ans. 4200 cents.
4. Reduce $14 to mills. Ans. 14000 mills.
5. Reduce $102 to cents.
6. Change $35 to mills.
7. Change 66 cents to mills. Ans. 660 mills.
8. Change 73 cents to mills.
NOTE. To change dollars and cents, or dollars, cents, and mills to mills, remor*
the decimal point and sign, $.
9. Change $4.28 to cents. Ans. 428 cents.
10. Change $18.07 to cents. Ans. 1807 cents.
11. Change $6.325 to mills. Ans. 6325 mills.
12. In $7.01 how many cents?
13. In 94 cents how many mills ?
14. In $51 how many cents 1
1. In 3427 cents how many dollars?
OPERATION. ANALYSIS. Since 100 cents equal
1/00)34/27 1 dollar, 3427 cents equal as many
dollars as 100 is contained times
$34.27 Ans. m 3427, which is 34.27 times.
To divide by 10, 100, &c., cut off as many figtires from the right
of the dividend as there are ciphers in the divisor, ( 72 )
Hence
To change cents to dollars, divide by 100 ; that is, point
off TWO figures from the right. And
To change mills to dollars, point off, THREE figures.
To change mills to cents, point off ONE figure.
EXAMPLES FOR PRACTICE.
2. Change 972 cents to dollars. Ans. $9.72.
3. Change 1609 cents to dollars. Ans. $16.09.
4. Change 3476 mills to dollars Ans. $3.476
120 UNITED STATES MONEY.
5. In 34671 cents how many dollars ?
6. 10307 cents how many dollars 1
7. In 203062 mills how many dollars? Ans. $203.062.
8. Reduce 672 mills to cents. Ans. $.672.
9. Reduce 3104 mills to dollars.
10. Reduce 17826 cents to dollars.
ADDITION.
123. 1. What is the sum of $12.50, $8.125, $4.076,
$15.375 and $22?
OPBBATION.
$12.50
8.125 ANALYSIS. "Writing dollars under do' -
4.076 lars, cents under cents, &c., so that the
15.375 decimal points shall stand under each
22.000 other, we add and point off as in ad-
dition of decimals. Hence the following
$oZ.U7o Ans.
RULE. I. Write dollars under dollars, cents under cents, &c.
II. Add as in simple numbers, and place the point in the
amount as in addition of decimals.
EXAMPLES FOR PRACTICE.
(2) (3) (4) (5)
$ 42.64 $100.375 $750.00 $1042.875
126.085 13.09 140.07 427.035
304.127 65.82 35.178 50.50
14.42 400.00 6.004 7.08
6. What is the sum of 30 dollars 9 cents ; 200 dollars 63
cents ; 27 dollars 36 cents 4 mills, and 10 dollars 16 cents ?
Ans. $268.244.
7. Add 390 dollars 37 cents 5 mills, 187 dollars 50
cents, 90 dollars 5 cents 5 mills, and 400 dollars 40 cents.
Ans. $1068.33.
ADDITION. 121
, 8. A lady paid $45.40 for some furs, $12.375 for a dress,
$5 for a bonnet and $1.125 for a pair of gloves; how much
did she pay for all ?
9. A farmer sold a cow for $20, a horse for $96.50, a
yoke of oxen for $66.875, and a ton of hay for $9.40;
how much did he receive for all ? Ans. $192.775.
10. Bought a hat for $4.50, a pair of boots for $5.62 4,
an umbrella for $2.12^, and a pair of gloves for $.87^ ;
wfeat was the cost of the whole? Ans. $13.125.
11. A grocer bought a barrel of sugar for $17.84, a box
of tea for $36.12, a cheese for $4, and a tub of butter for
$7.09; what was the cost of all ?
12. A merchant bought a quantity of goods for $458.25,
paid for duties $45; for freights $98.624, and for insur-
ance $16.40; how much was the whole cost?
Ans. $618.275.
13. Bought some sugar for $1.75, some tea for $.90,
some butter for $2.12^, some eggs for $.37|, and some spice
for $.25 ; what was the cost of the whole ? Ans. $5.40.
14. Paid for building a house $1045.75, for painting the
same $275.60, for furniture $648.87|, and for carpets
$105.10; what was the cost of the house and furnishing?
Ans. $2075.325.
15. A farmer receives 120 dollars 45 cents for wheat,
36 dollars 624 cents for corn, 14 dollars 9 cents for pota-
toes, and 63 dollars for oats ; how much does he receive foi
the whole]
16. A lady who went shopping, bought a dress for 7 dol-
lars 27 cents, trimmings for 874 cents, some tape for 6 cents,
some thread for 12^ cents, and some needles for 9 cents;
how much did she pay for all ? Ans. $8.42.
6
122 UNITEI ) STATES MONEY.
SUBTRACTION.
124. 1. From 246 dollars 82 cents 5 mills, take 175
dollars 27 cents.
OPERATION. ANALYSIS. Writing the less num-
$246.825 ber under the greater, dollars under
175.27 dollars, cents under cents, &c., we
subtract and point off in the result as
$71.555 Ans. j n subtraction of decimals. Hence
RULE. I. Write the subtrahend under the minuend,
dollars under dollars, cents under cents, &G.,
II. Subtract as in simple numbers, and place the point in
the remainder as in subtraction of decimals.
EXAMPLES FOB, PEACTICE.
(2) (3) (4) (5)
From $125.05 $327.105 $112.000 $43.375
Take 43.278 100.09 .875 2.06
Ans. $81.772 $227.015 $111.125 $41.315
6. From $3472.50 take $1042.125. Ans. $2430.375.
7. From $540 take $256.67. Ans. $283.33.
8. From $82.04 take $80.625. Ans. $1.415.
9. From 3 dollars 10 cents, take 75 cents.^4rcs.$2.35.
10. From 10 dollars, take 5 dollars 10 cts. Ans. $4.90.
11. From 100 dollars, take 50 dollars 50 cents.
12. From 1001 dollars 9 cents, take 300 dollars.
13. From 2 dollars, take 75 cents. Ans. $1.25.
14. From 96 cents, take 12J cents. Ans. $.835.
15. From 1 dollar take 25 cents. Ans. $.75.
16. From 50 cents take 37 cents 5 mills. Ans. $.125.
17. From 5 dollars, take 50 cents 8 mills. A ns. $4.492.
18. From 4 dollars, take J dollar 40 cents 5 mills.
19. Sold a horse for $200, which was $45.50 more than
he cost me; hcrw much did he cost me ? Ans. $154.50
SUBTRACTION. 1*28
20. A man bought a farm for $4640, and sold it for
$5027.50 ; how much did he gain ? Ans. $387.50.
2^. Borrowed $25 and returned $15.60 ; how much re-
mained unpaid ? Ans. 9.40.
22. A merchant having $10475, paid $2426 for a store,
and $5327.875 for goods; how much money had he left 1 ?
Ans. $2721.125.
23. Bought a sack of flour for $3.12^ ; how much
change must I receive for a 5 dollar bill ? Ans. $1.875.
24. Bought groceries to the amount of $1.875 ; how
much change must I receive for a 2 dollar bill ?
Ans. 12^ cents.
25. Paid; $3 7 5 for a pair of horses, and sold one of them
for $215.50}; how much did the other one cost me ?
Ans. $159.50.
26. I started on a journey with $50 and paid $10.62
railroad far$, $7.38 stage fare, $5.96 for board and lodging,
and $.75 fo porterage; how much money had I left V
Ans. $25.285.
27. A faflner sold some wool for $27.16, and a ton of hay
for $14.80. 'He received in payment a barrel of flour worth
$6.875, and Qie remainder in money ; how much money did
he receive? ' Ans. $35.085.
28. A woman sold a grocer some butter for $1.48, and
some eggs for $.94. She received a gallon of molasses worth
40 cents, a pound of tea worth 75 cents, and a pound of
starch worth 124 cents ; how much is still her due 1
Ans. $1.145.
29. A tailor bought a piece of broadcloth for $87.50,
and a piece of cassimere for $62.75. He sold both pieces
for $170.87^; how much did he gain on both 1 ?
Ans. $20.625.
124 UNITED STATES MONEY.
MULTIPLICATION.
125. 1. Multiply $26.145 by 34.
OPERATION.
ANALYSIS. We multiply as in sim-
ple numbers, always regarding the
104580 multiplier as an abstract number, and
78435 point off from the right hand of the
result, as in multiplication of decimals.
$888.930 Ans. Hence the following
RULE. Multiply as in simple numbers, and place the
point in the product as in multiplication of decimals.
EXAMPLES FOR PRACTICE. |
(2) (3) (4) j(5)
$327.48 $82.375 $160.09 $$7.875
15 46 { 87 123
6. What cost 8 cords of wood, at $3.50 ? fens. $28.
7. What cost 14 barrels of flejur, at $5.85 Sbarrel ?
8. What cost 25 bushels of cforn, at 75 cems a bushel ?
9. 4t $2.125 a yard, what wfll 18 yards otjsilk cost?
10. At $.8?5: apiece, what will be the cost o 9 turkeys?
11. A farmer sold 40 bushels .of potatoes & 37 cents a
bushel, and 2l barrels of apples at $2.25 ^barrel; how
much did he'r'eWive for both ? Ans. $62.25.
11. Bought 124 > *apres of land at $35.75 an acre, and
sold the whole for $6iOQp'; did I gain or lose, and how
much? Ans. $1567.
13. What will be the cost of 275 bushels of oats, at 42
cents a bushel ? Ans. $115.50.
14. A grocer bought 160 pounds of butter, at 14 cents
a pound, and paid 25 pounds of tea, worth 56 cents a pound,
and the remainder in cash; how much money did he pay?
DIVISION. 125
15. What will be the cost of 15 yards of broadcloth, at
$4.87 a yard] Ans. $73.125.
16. A grocer bought a tub of butter containing 84
pounds, at 12 i cents a pound, and sold the same at 15
cents a pound ; how much did he gain ? Ans. $2.10.
17. A farmer took 3 tons of hay to market, for which
he received $9.38 a ton. He bought 2 barrels of flour, at
$6.94 a barrel, and 12 pounds of tea, at $.625 a pound ;
how much money had he left ?
Ans. $6.76.
DIVISION.
126. 1. Divide $136 by 64.
64)$136.000($2.125 Ans.
128
~~80
64
ANALYSIS. We divide as in
~,QQ simple numbers, and as there is
i og a remainder after dividing the
dollars, we reduce the dividend
320 to mills, by annexing three ci-
320 phers, and continue the divis-
ion. Hence the following
KULE. Divide as in simple numbers, and place the point
in the quotient, as in division of decimals.
NOTE. 1. In business transactions it is never necessary to carry the division
further than to mills in the quotient.
EXAMPLES FOR PRACTICE.
(2) (3) (4) (5)
5)$43.50 10)$36.00 8)$371. 12)$169.50
$8.70 $3.60 $46.375 $14.125
126 UNITED STATES MONET.
6. Divide $13.75 by 11. Ans. $1.25.
7. Divide $162. by 36. Ans. $4.50.
8. Divide $246.30 by 15. Ans. $16.42.
9. Divide $1305. by 18. Ans. $72.50.
10. Divide $2.25 by 9. Ans. $.25.
11. Divide $658 by 280. Ans. $2.35.
12. Divide $195.75 by 29. Ans. $6.75.
13 Divide 1388 by 100. Ans. $13.88.
14. Divide $2675.75 by 278. Ans. $9.625.
15. Divide $68 by 32. Ans. $2.125.
16. Paid $168.48 for 144 bushels of wheat; what waa
the price per bushel ? Ans. $1.17.
17. Paid $2.80 for 35 pounds of sugar; what was the
price per pound ? Ans. $.08.
18. If 54 cords of wood cost $135, what is the price per
cord? Ans. $2.50.
19. Bought 125 bushels of oats for $62.50 ; what was
the cost per bushel ? Ans. $.50.
20. If 70 barrels of apples cost $175, how much will 1
barrel cost? Ans. $2.50.
21. If 100 acres of land cost $3156.50, how much will
be the cost of 1 acre] Ans. $31.565.
22. Paid $148.75 for 170 bushels of barley; how much
was the cost per bushel ? Ans. $.875.
23. If 13 pounds of tea cost $9.88, how much will 1
pound cost ?
24. Bought 2500 pounds of butter for $625 ; how much
was the cost per pound ? Ans. 25 cents.
25 Bought 2450 pounds of pork for $153.12^; how
much was the cost per pound 1 Ans. 6| cents.
26. Bought 4 barrels of sugar, each containing 200
pounds, for $72 ; what was the cost per pound ?
PROMISCUOUS EXAMPLES. 127
PROMISCUOUS EXAMPLES.
1. A merchant bought 14 boxes of tea for $560 ; but it
being damaged, he was obliged to sell it for $106.75 less
than he gave for- it ; how much did he receive a box ?
Ans. $32.375.
2. A farmer sold 120 bushels of wheat, at $1.12^ a
nushel, and received in payment 27 barrels of flour; what
did the flour cost him per barrel ?
3. If 35 yards of cloth cost $122.50, how much will 29
yards cost? Ans. $101.50.
4. If 4 tons of coal cost $35.50, how much will 12 tons
cost? Ans. $106.50.
5. If 29 pounds of sugar cost $3.625, how much will 15
pounds cosf? Ans. $1.875.
6. If 12 barrels of flour cost $108, how much will 18
barrels cost ? Ans. $162.
7. If 3 bushels of wheat cost $4.35, how much will 30
bushels cost ? Ans. $43.50.
8. A man bought a farm containing 125 acres, for
$2922.50 ; for how much must he sell it per acre to gain
$500 ? Ans. $27.38.
9. A farmer exchanged 50 bushels of corn worth 70
cents a bushel, for 28 bushels of wheat; how much was the
wheat worth a bushel. Ans. $1.25.
10. A person having $15000, bought 30 hales of cotton
each bale containing 940 pounds, at 10 cents a pound ; he
next paid $6680 for a house, and then bought 1000 barrels
of flour with what money he had left ; how much did the
flour cost him per barrel ? Ans. $5.50.
NOTE. For a full and complete development and application of Decimals and
United Statea money, the pupil is referred to the A.uthor's Progressive Practical
and Higher Arithmetic.
128 UNITED STATES MONEY.
BILLS.
127. A Bill, in business transactions, is a written state-
ment of articles bought or sold, together with the prices of
each, and the whole cost.
Find the cost of the several articles, and the amount or
footing of the following bills :
ao
CHICAGO, Sept. 20, 1861.
MR. J. C. SMITH,
JBo't. of SILAS JOHNSON,
36 pounds sugar at 8 cents a pound, $2.88
18 pounds coffee at 15 cents a pound, 2.70
24 pounds butter at 18 cents a pound, 4.32
10 dozen eggs at 12 cents a dozen, 1.25
4 gallons molasses at 44 cents a gallon, 1.76
Ans. $12.91.
(20
ROCHESTER, Jan. 25, 1862.
JOHN DABNEY, ESQ.,
Bo't. of BARDWELL & Co.,
14 pounds coffee sugar at 11 cents a pound, $1.54
6 pounds Y. H. tea at 62 cents a pound, 3.75
25 pounds No. 1 mackerel at 6 cents a pound, 1.50
5 bushels potatoes at 37 J cents a bushel, 1.875
3 gallons syrup at 80 cents a gallon, 2.40
7 dozen eggs at 16 cents a dozen, 1.12
Received Payment, Ans. $12.185
Bardwell & Co.,
per Adams
BILLS. 129
(3.)
MEMPHIS, Aug. 20, 1862
Mr. S. P. HAILE,
JBo't of PATTERSON & Co.,
20 chests Green Tea at $22.50
16 Black at 18.75
14 " Imperial at 32.87
15 sacks Java Coffee at 17.38
25 boxes Oranges at 4.62|
Received payment, $1586.575.
Patterson & Co.,
(40
OSWEGO, Sept. 4, 1861.
JAMES COROVAL & Co.,
Bd*t. of COLLINS & SON,
12 yards Broadcloth at $3.84
18 " Cassimere " 2.25
10 " Satinet " .87
42 " Flannel " .45
35 " Black Silk " 1.18
$155.53.
(5.)
BOSTON, April 10, 1862.
J. GK BENNET & SON,
Bc?t. of BUTLER, KINO & Co.,
14 Plows at $10.50
8 Harrows " 9.80
120 Shovels " .90
175 Hoes .62'
$442.775.
130 COMPOUND LUMBERS.
COMPOUND NUMBERS.
128. A Simple Number is either an abstract number,
or a concrete number of but one denomination. Thus, 48,
926; 48 dollars, 926 miles.
129. A Compound Number is a concrete number
whose value is expressed in two or more differfij^Renomi-
nations. Thus, 32 dollars 15 cents ; 15 days 4 hours 25
minutes.
130. A Scale is a series of numbers, descending or as
cending, used in operations upon numbers.
NOTE. In simple numbers and decimals the scale is uniformly 10; in compound
numbers the scales are varying.
CURRENCY.
I. UNITED STATES MONEY.
131. The currency of the United States is decimal cur-
rency, and is sometimes called Federal Money.
TABLE.
10 mills (m.) make 1 cent, ct
10 cents *' 1 dime, . . . . d.
10 dimes ** 1 dollar, $.
10 dollars " 1 eagle, B.
UNIT EQUIVALENTS.
ct. m.
d. 110
$ 110100
B 1 10 100 1000
1 10 100 1000 10000
SCALE uniformly 10.
COINS. The gold coins are the double eagle, eagle, halt
eagle, quarter eagle, three-dollar piece and dollar.
The silver coins are the half and quarter dollar, dime and
half dime, and three-cent piece.
nickel coin is the cent.
MONEY AND CURRENCIES. 131
II. CANADA MONEY.
132. The currency of the Canadian provinces is deci-
mal, and the table and denominations are the same as those
of the United States money.
NOTE The decimal currency was adopted by the Canadian Parliament in 1868,
and the Act took effect in 1859. Previous to the latter year the money of Canada
was reckoned in pounds, shillings, and pence, the same as in England.
COINS. The new Canadian coins are silver and copper.
The silver coins are the shilling or 20-cent piece, the
dime, and half dime.
The copper coin is the cent.
NOTB. The 20-cent piece represents the value of the shilling of the old Cana-
da Currency.
III. ENGLISH MONEY.
133. English or Sterling money is the currency of
Great Britain.
TABLE.
4 farthings (far. or qr.) make 1 penny, , d.
12 pence " 1 shilling, 8.
20 shillings " 1 pound or sovereign.. . . or sov.
UNIT EQUIVALENTS.
d. far.
.. 1 = 4
, or SOT. 1 = 12 = 48
1 = 20 = 240 = 960
SCALE ascending, 4, 12, 20 ; descending, 20, 12, 4.
NOTH. Farthings are generally expressed as fractions of a penny ; thus, 1 far.,
sometimes called 1 quarter, (qr.) =}d.; 3 far.=%d.
Coixs. The gold coins are the sovereign (= 1) and the
half sovereign, (= 10s.)
The silver coins are the crown (= 5s.), the half crown,
(= 2s. 6d.), Ijie shilling, and the 6-penny piece.
The copper coin* are the penny, half-penny, and farthing.
132 COMPOUND NUMBERS.
WEIGHTS.
1 34. Weight is a measure of the quantity of matter a
body contains, determined according to some fixed standard.
I. TROY WEIGHT.
135. Troy Weight is used in weighing gold, silver,
and jewels; in philosophical experiments, &c.
TABLE.
24 grains (gr.) make 1 pennyweight, . . . pwt. or dwt.
20 pennyweights " 1 ounce, oz.
12 ounces " 1 pound, Ib.
UNIT EQUIVALENTS.
pwt. gr.
oz. 124
ib. 120480
1 12 240 5T60
SCALE ascending, 24, 20, 12 ; descending, 12, 20, 24.
II. AVOIRDUPOIS WEIGHT.
1 36. Avoirdupois Weight is used for all the ordinary
purposes of weighing.
TABLE.
16 drams (dr.) make 1 ounce, oz.
16 ounces " 1 pound, Ib.
100 Ib. " 1 hundred weight, .cwt
20 cwt., = 2000 Ibs., 1 ton, T.
UNIT EQUIVALENTS.
or,. dr.
1 - 16
cwt 1 16 256
T. 1 100 1600 25600
1 20 2000 32000 512000
SCALE ascending, 16, 16, 100, 20; descending, 20, 100, 10,
ia
WEIGHTS. 133
NOTE. The long or gross ton, hundred weight, and quarter were formerly in com-
mon use ; but they are now seldom used except in estimating English goods at the
U 8. custom-house, and in freighting and wholesaling coal from the Pennsylvania
mines.
LONG TON TABLE.
28 lb. make 1 quarter, qr.
4 qr. 112 lb. " 1 hundred weight, cwt.
20 cwt. 2240 lb. " 1 ton, T.
The following denominations are also in use:
56 pounds make 1 firkin of butter.
196 " " 1 barrel of flour.
200 " " 1 " " beef, pork, or fish.
280 " " 1 bushel, " salt at the N. Y. State salt works
32 " " 1 " * oats.
48 " 1 " " barley.
56 " " 1 " " corn or rye.
60 " " 1 " " wheat.
III. APOTHECARIES' WEIGHT.
137. Apothecaries' Weight is used by apothecaries
and physicians in compounding medicines ; but medicines
are bought and sold by avoirdupois weight.
TABLE.
20 grains [gr.] make 1 scruple sc. or 3
3 scruples " 1 dram, dr. or 3 .
8 drams " 1 ounce, oz. or .
12 ounces " 1 pound lb. or ft>
UNIT EQUIVALENTS.
sc. grr.
d. 120
oz. 1 3 60
n>. 1 8 24 480
1 12 96 288 5760
SCALE ascending, 20, 3, 8, 12; descending, 12,8,8,
20
134 COMPOUND NUMBERS.
138. COMPARATIVE TABLE OP WEIGHTS.
Troy. Avoirdupois. Apothecaries.
1 pound 5760 grains, = 7000 grains, 5760 grains,
1 ounce 480 437.5 " 480
175 pounds, 144 pounds, 175 pounds.
MEASURES OF EXTENSION.
139. Extension has three dimensions length, breadth,
and thickness.
A Line has only one dimension length.
A Surface or Area has two dimensions length and
breadth.
A Solid or Body has three dimensions length, breadth,
and thickness.
I. LONG MEASURE.
140. Long Measure, also called Lineal Measure, is
used in measuring lines or distances.
TABLE.
12 inches (in.) make 1 foot, ft
3 feet " lyard, yd.
5 \ yd., or 16 ft, " 1 rod, rd.
40 rods " 1 furlong, fur.
8 furlongs, or 320 rd., " 1 statute mile,. ml
UNIT EQUIVALENTS.
ft. in.
yd. 1 - 12
M. 1-3-36
tagm 1 5J 16 198
ml. 1 40 220 660" 7920
1 _. 8 320 1760 =- 5280 63360
SCALE ascending, 12 3, 5', 40, 8; descending, 8, 40, 5J, 8.
12.
MEASURES OF EXTENSION.
185
The following denominations are also in use :
j used by shoemakers in meas-
barleycorns make 1 inch, j uring he length of thefoot
( used in measuring the height
inches " 1 hand, < of horses directly over the
( fore feet.
" " Ispan.
" 1 sacred cubic.
9 " "
21.888 " "
3 feet "
6 .
1.15 statute miles"
1 pace.
1 fathom,
geographic
or
measurin S Depths
1 league.
) , , j of latitude on a meridian or
j L J \ of longitude on the equator.
8
60 '*
69.16 statute
360 degrees
NOTES. 1. For the purpose of measuring cloth and other goods sold by the yard,
the yard is divided into halves, quarters, fourths, eighths, and sixteenths. The old
table of cloth measure is practically obsolete.
** the circumference of the earth.
SURVEYORS' LONG MEASURE.
A Gunter's Chain, used by land surveyors, is 4
rods or 66 feet long, and consists of 100 links.
TABLE.
7.92 inches (in.) make 1 link, 1.
25 links " 1 rod, rd.
4 rods, or 66 feet, " 1 chain,... ch.
80 chains " 1 mile,.... mi.
UNIT EQUIVALENTS.
1. In.
rd. 1 7.92
ch. 1 25 198
!. 1 4 =- 100 792
1 _ 80 320 8000 63360
SCALE ascending, 7.92, 25, 4, 80 ; descending, 80, 4, 25, 7.92.
NOTK. The denomination, rods, is seldom used in chain measure, distances
bei ug taken In chains and links.
136
COMPOUND NUMBERS.
II. SQUARE MEASURE.
143. A Square is a figure having four equal sides, and
four equal angles or corners.
i yd. =3 ft. i square yard is a figure hav-
ing four sides of 1 yard or 3 feet
43 each, as shown in the diagram.
Its contents are 3X3=9 square
feet. Hence
Thus a square foot is 12 inches
i yd. =s ft. long and 12 inches wide, and the
contents are 12x12=144 square inches. A surface 20
feet long and 10 feet wide, is a rectangle, containing 20 X
10=200 square feet.
The contents or area of a square, or of any other figure
having a uniform length and a uniform breadth, is found,
by multiplying the length by the breadth.
144. Square Measure is used in computing areas or
surfaces ; as of land, boards, painting, plastering, paving,
&c.
TABLE.
144 square inches (sq. in.) make 1 square foot,
sq. ft
9 square feet " 1 square yard,
sq. yd.
30J square yards " 1 square rod,
sq. rd.
40 square rods " 1 rood,
R.
4 roods " 1 acre,
A.
640 acres " 1 square mile,
sq. mi,
UNIT EQUIVALENTS.
q. ft.
nq. in.
sq,yd. 1
144
B q. rd. 1 9
1276
R. 1 30} 272^
30204
A. 1 40 1210 10890
15681 CO
- mi 1 4 160 4840 435 fiO
G272640
1640 256dO 102400 3097GOO 27878400 4014480GOOO
MEASURES OF EXTENSION. 137
Artificers estimate their work as follows :
By the square foot : glazing and stone-cutting.
By the square yard : painting, plastering, paving, ceiling,
and paper-hanging.
By the square of 100 feet : flooring, partitioning, roofing,
slating, and tiling.
Brick-laying is estimated by the thousand bricks; also by
the square yard, and the square of 100 feet.
NOTES. 1. In estimating the painting of moldings, cornices, etc., the measuring
line is carried into all the moldings and cornices.
2. In estimating brick-laying by the square yard or the square of 100 feet, the
work is understood to be 1> bricks, or 12 inches, thick.
SURVEYORS' SQUARE MEASURE.
14L5. This measure is used by surveyors in computing
the area or contents of land.
TABLE.
625 square links (sq. 1.) make 1 pole, P.
16 poles " 1 square chain, ..sq. ch.
10 square chains ** 1 acre, A.
640 acres " 1 square mile, . . . sq. mi.
36 square miles (6 miles square) " 1 Township, Tp.
UNIT EQUIVALENTS.
P. sq. 1.
eq. ch. 1 625
A. 1 16 1000
Bq.mi. 1 . 10 160 10000
Tp. 1 640 6400 102500 64000000
1 36 23040 230400 3686400 2304000000
SCALE ascending, 625, 16, 10, 630, 36 ; descending, 36, 640,
10, 16, 625.
NOTKS. 1. A square mile of land is also called a section.
2. Canal and railroad engineers commonly use an engineers^ chain, which con-
ista of 100 links, each 1 foot long.
188
COMPOUND NUMBERS.
III. CUBIC MEASURE.
14LO. A Cube is a solid, or body, having six equal
square sides or faces.
If each side of a cube be 1 yard,
or 3 feet, 1 foot in thickness of
this cube will contain 3x3x1:=
9 cubic feet ; and the whole cube
will contain 3x3X3=27 cubic
[ , I, '|U^^ feet.
3 ft.i yd. A solid, or body, may have the
three dimensions all alike, or all different. A body 4 ft.
long, 3 ft. wide, and 2 ft. thick contains 4x3x2=24 cu-
bic or solid feet. Hence we see that
The cubic or solid contents of a ~body are found by multi-
plying the length, breadth, and thickness together.
147. Cubic Measure, also called Solid Measure, is
used in estimating the contents of solids, or bodies ; as
timber, wood, stone, &c.
1728
27
40
50
16
8
128
TABLE.
make 1 cubit foot, . . cu. ft.
u 1 cubic yard, cu. yd.
1 ton or load, ...T.
1 cord foot, . . . cd. ft.
1 cord of wood, . Cd.
( perch of )
1 -j stone or V Pch.
( masonry, )
SCALE ascending, 1728, 27, 40, 50, 16, 8, 128, 24f ; descend-
ing, 24f, 128, 8, 16, 50, 40, 27, 1728.
NOTES. 1. A cubic yard of earth is called a load.
2. Railroad and transportation companies estimate light freight by the space it
occupies in cubic feet, and heavy freight by weight.
cubic inches (cu. in.)
cubic feet
cubic feet of round timber, or )
" " hewn J
cubic feet
cord feet, or )
cubic feet f
cubic feet
MEASURES OF CAPACITY. 189
3. A pile of wood 8 feet long, 4 feet wide, and 4 feet high, contains 1 cord ; and
a cord foot is one foot in length of such a pile.
4 A perch of stone or of masonry is 16 > feet long, 1> feet wide, and 1 foot high.
MEASURES OF CAPACITY.
148. Capacity signifies extent of room or space.
All measures of capacity are cubic measures, solidity and
capacity being referred to different units, as will be seen by
comparing the tables.
Measures of capacity may be properly subdivided into
two classes, Measures of Liquids, and Measures of Dry
Substances.
I. LIQUID MEASURE.
149. Liquid Measure, also called Wine Measure, is
used in measuring liquids; as liquors, molasses, water, &c.
TABLE.
4 gills (gi.) make 1 pint, pt,
2 pints " 1 quart, qt.
4 quarts " 1 gallon, gal.
81i gallons " 1 barrel, bbL
2 barrels, or 63 gal. ' 1 hogshead, . . .hhd.
UNIT EQUIVALENTS.
Pt. gl.
qt 14
gaL 1 2 8
bbl 14882
hhd. 1 = 3H 126 252 1008
1 _ 2 =* 63 252 504 2G16
SCALE ascending, 4, 2, 4, 31$, 2; descending, 2, 81$, 4, 2, 4*
140 COMPOUND NUMBERS.
15O. The following denominations are also in use:
86 gallons make! barrel of beer.
54 " or H barrels " 1 hogshead "
42 " " 1 tierce.
2 hogsheads, or 120 gallons, <{ 1 pipe or butt
2 pipes, or 4 hogsheads, *' 1 tun.
NOTES. 1. The denominations, barrel and hogshead, are used in estimating tne
capacity of cisterns, reservoirs, vats, &c.
2. The tierce, hogshead, pipe, butt and tun, are the names of casks, and do not
express any axed or definite measures. They are usually gauged, and have their
capacities in gallons marked on them.
3. Ale or beer measure, formerly used hi measuring beer, ale and milk, is almost
entirely discarded.
II. DRY MEASURE.
151. Dry Measure is used in measuring articles not
liquid ; as grain, fruit, salt, roots, ashes, &c.
TABLE.
2 pints (pt.) make 1 quart, . . . .^" qt.
8 quarts " 1 peck, pk.
4 pecks " 1 bushel,. bu. or bush.
UNIT EQUIVALENTS.
qt. pt.
pk. 1 - 2
bu . 1 _ 8 - 16
14 82 64
SCALE ascending, 2, 8, 4; descending, 4, 8, 2.
NOTES. 1. In England, 8 bu. of 70 Ibs each are called a quarter, used in measuring
grain. The weight of the English quarter is ^ of a lon ton.
2. The wine and dry measures of the same denomination are of different capacl
ties. The exact and the relative size of each may be readily Been by the following
TIME. 141
COMPARATIVE TABLE OF MEASURES OF CAPACITY.
Cu.in.in Cu. in. in Cu.in.in Cu.in.in
one gallon, one quart. one pint. one gill.
Wine measure, 231 57f 28 7/ 2
Drymeasnre ; (*pk.,)268J 67| 33| 8|
3. The ber gallon of 282 inches is retained in nse only by custom. A bushel
commonly estimated at 2150.4 cubic inches.
MEASURE OF TIME.
153. Time is the measure of duration.
TABLE.
60 seconds (sec.) make 1 minute, ^ .min.
60 minutes " 1 hour, ..h.
24 hours " 1 day, ...... da.
7 days ' 1 week, wk.
365 days " 1 common year, ..... .yr.
866 days " 1 leap year, yr.
12 calender months '* 1 year, .yr.
100 years " 1 century, -^0.
CJN'IT EQUIVALENTS.
min. sec.
h. 1 - 60
dn. 1 60 8600
wk . 1 24 1440 88400
1 7 168 10080 604800
yr. mo. f 365 8760 525600 81536000
112 ( 366 8784 527040 31622400
SCALE ascending, 60, 60, 24, 7; descending, 7, 24, 60,60.
H2
COMPOUND NUMBERS.
The calendar year is divided as follows :
No . of month . Season .
Names of months
1
2
Winter,
j Jamiary,
( February,
8
4
5
Spring,
( March,
1 April,
(May,
6
7
8
Summer,
(June,
] July,
( August,
9
10
11
Autumn,
( September,
< October,
( November,
12
Winter,
December,
Abbreviations. No. of days.
Jan.
Feb.
Mar.
Apr.
Jun.
Aug.
Sept.
Oct.
Nov.
Dec,
31
28 or 2*
81
30
31
30
81
81
30
31
80
81
865or3i/6
NOTES. 1. The exact length of a solar year is 365 da. 5 h. 48 min. 46 sec. ; but
fbr convenience it is reckoned 11 min. 14 sec. more than this, or 365 da. 6 h.
265> da. This % day, in four years makes one day, which, every fourth, bissex-
tile, or leap year, is added to the shortest month, giving it 29 days. The leap year*
are exactly divisible by 4, as 1856, 1860, 1864.
The number of days in each calendar month may be easily remembered by
committing the following lines :
" Thirty days hath September,
April, June, and November ;
All the rest have thirty -one,
Save February, which alone
Hath twenty-eight ; and one day more
We add to it one year in four."
2. In most business transactions 30 days are called 1 month.
3. The centuries are numbered from the commencement of the Christian era ,
the months from th commencement of the year ; the days from the commence-
ment of the month, and the hours from the commencement of the day, (12 o'clock,
midnight.) Thus, May 23d, I860, 9 o'clock A. M., is the 9th hour of the 23d day
of the 5th month of the 60th year of the 19th century.
CIRCULAR MEASURE. 143
CIRCULAR MEASURE.
155. Circular Measure, or Circular Motion, is used
principally in surveying, navigation, astronomy, and geogra-
phy, for reckoning latitude and longitude, determining loca-
tions of places and vessels, and computing difference of
time.
Each circle, great or small, is divisible into the same
number of equal parts, as quarters, called quadrants,
twelfths, called signs, 360ths, called degrees, &c. Conse-
quently the parts of unequal circles, although having the
same names, are of unequal lengths.
TABLE.
60 seconds (") make 1 minute,. . . . '.
60 minutes u 1 degree, . . . . .
80 degrees " 1 sign, S.
12 signs, or 360 " 1 circle, 0.
UNIT EQUIVALENTS.
t n
1 60
a 1 60 3600
. 1 30 1800 108000
1 12 860 21600 1296000
SCALE ascending, 60, 60, 30, 12 ; descending, 12, 30, 60, 60.
NOTES. 1. Minutes, of the earth's circumference are called geographic or nauti-
cal miles.
2. The denomination, signs, is confined exclusively to Astronomy.
8. A degree has no fixed linear extent. When applied to any circle, it is alwayi
j-g--g- part of the circumference. But, strictly speaking, it is not any part of a
circle.
4. 90* make a quadrant or right-angto.
5 60" make a soxtaat or of a circl*.
144: COMPOUND NUMBERS.
MISCELLANEOUS TABLES.
156. COUNTING.
12 units or things make 1 dozen.
12 dozen " 1 gross.
12 gross " 1 great gross.
20 units t4 1 score.
157. PAPER.
24 sheets ..... malse .... 1 quire.
20 quires 1 ream.
2 reams 1 bundle.
5 bundles " 1 bale.
BOOKS.
The terms folio, quarto, octavo, duodecimo, &c., indicate
the number of leaves into which a sheet of paper is folded.
A sheet folded in 2 leaves is called a folio.
A sheet folded in 4 leaves " a quarto, or 4to.
A sheet folded in 8 leaves ** an octavo, or 8vo.
A sheet folded in 12 leaves " a 12mo.
A sheet folded in 16 leaves " a 16mo.
A sheet folded in 18 leaves ,. M an 18mo.
A sheet folded in 24 leaves " a 24mo.
A sheet folded in 32 leaves * a 32mo.
COPYING.
75 words make 1 folio or sheet of common law.
90 " " 1 " " " " chancery.
1 60. An Aliquot Part of a number is such a part as
will exactly divide that number; thus, 3, 5, 7 are aliquot
parts of 15.
NOTE. An aJ*<ptat part may bo a whole or mixed number, while A factor must b
whole number.
ALIQUOT PARTS.
145
ALIQUOT PARTS OF ONE DOLLAR.
161.
50 cents = J of 1 dollar.
33i cents = $ of 1 dollar.
25 cents = i of 1 dollar.
20 cents = 1 of 1 dollar.
16 cents = i of 1 dollar.
12* cents = of 1 dollar.
10 cents =03 of 1 dollar.
8i cents = T ' 5 of 1 dollar.
6i cents = T ^ of 1 dollar.
5 cents =' of 1 dollar.
1 63. PARTS OF 81 IN NEW YORK CURRENCY.
4 shillings =
2 shillings 8d. =
2 shillings =
1 shil. 4 pence =
1 shilling =
6 pence =
1 63. PARTS OF $1 IN NEW ENGLAND CURRENCY.
3 shillings
2 shillings
1 shillings 6d.
164.
10 hund. Ibs. =
1 shilling
9 pence
6 pence
J ton.
5 hund. Ibs. = J ton.
4 hund. Ibs. = i ton.
ALIQUOT PARTS OF A TON.
2 hund. 2 qrs.
2 hund. Ibs.
1 hund. Ibs.
= Si
= ton.
1 63. ALIQUOT PARTS OF A POUND AVOIRDUPOIS.
8 ounces = \ pound. I 2 ounces |- pound.
4 ounces = { pound. 1 ounce = T 1 g pound.
166.
ALIQUOT PARTS OF TIME.
Parts of 1 year.
Parts of 1 month.
6
months
2 year.
15
days
= month.
4
months
i year.
10
days
= J month.
3
months
i year.
6
days
= i month.
2
months
=
i year.
5
days
= i month.
li
mouths
I year.
3
days
=r y 1 ^ month.
1J
months
^^
J year.
2
days
zn y'-j month.
1
month
= ^2 year.
1
day
= -^Q month
146 COMPOUND NUMBERS.
REDUCTION.
1 67. Reduction is the process of changing a number
from one denomination to another without altering its value.
168. Reduction Descending is changing a number of
one denomination to another denomination of less unit value,
and is performed by multiplication ; thus : $1 = 10 dimes
= 100 cents = 1000 mills; 1 yard = 3 feet 36 inches.
1. Reduce 6 gal. 2 qt. 1 pt. to pints.
OPERATION. ANALYSIS. Since in 1 gal.
6 gal. 2 qt. 1 pt. there are 4 qt. in 6 gal. there
4 4 qt.X6=-24 qt and the 2
qt in the given number, ad-..
26 <!* ded, makes 26 qt in 6 gal. 2 ;
qt Since in 1 qt. there are '
Ans. 53 pt 2 ^ in 26 & there 2 Pt X
26=52 pt and the 1 pt in
the given number added, make 53 pints in the given com-
pound number. As either factor may be used as a multipli-
cand, ( 61 )i we may consider the numbers in the descend-
ing scale as multipliers. Hence the following
RULE. I. Multiply the highest denomination of the giver*
compound number ~by that number of the scale which will
reduce it to the next lower denomination, and add tj the
product the given number, if any , of that lower denomination .
II. Proceed in the same manner with the result obtained
in each lower denomination, until the reduction is brought to
the denomination required.
EXAMPLES FOR PRACTICE.
2. In 8 Ib. 10 oz. how many ounces ? Ans. 138 oz.
3. In 12 6s. 9d. how many pence 1 Ans. 2961d.
' , In 4 yd. 1 ft. 10 in. how many inches 1
T n 3 mi. 5 fur. 26 rd. how many rods 1
REDUCTION. 147
6. In 18s. 8d. 3 far. bow many farthings 1
Ans. 899 far.
7. Reduce 3 Ib. 9 oz. 12 pwt. to pennyweights.
8. In hhd. 15 gal. 2 qt. how many pints ?
9. Reduce 4 da. 5 hr. to minutes. Ans. 6060 min.
10. Reduce 10 bu. 1 pk. 6 qt. to pints. Ans. 1308 pt.
11. Reduce 14 A. 3 R. 20 sq. rd. to square rods.
12. Reduce 4 cd. 3 cd. ft. 9 cu. ft. to cubic inches.
13. Reduce 4 yr. 7 mo. to hours. Ans. 39600 hr.
14. Change 2 T. 11 cwt. to pounds. Ans. 5100 Ib.
15. Change U Ib. 9 oz. 10 pwt. to grains.
16. Change 5 lb.-6 43 23 10 gr. to grains.
17. Change 3 mi. 6 fur. to feet. Ans. 19800 ft.
18. In 40 chains how many links 1 Ans. 4000 1.
X19. In 28 sq. rd. 12 sq. yd. 4 sq. ft. how many square
inches'? . Ans. 1113840 sq. in.
20. In 16 A. 4 sq. ch. 8 P. 80 sq. 1. how many square
links? Ans. 1645080 sq. 1.
21. In 12 tons of round timber how many cubic inches 7
22. In 8 bbl. 26 gal. how many pints ? Ans. 2224 pt.
23. Reduce 4 pipes to quarts. Ans. 2016 qt.
\24. Reduce 23 bu. 3 pk. to pints. Ans. 1520 pt.
. Reduce 8 S. 18 40' to minutes. Ans. 15520'.
26. Reduce 15 to seconds. Ans. 54000".
27. Reduce 2 months to minutes. Ans. 86400 min.
28. Change 2 reams 10 quires to sheets.
29. In 40 score how many single things'? Ans. 800.
30. In 14 great gross how many dozens ?
In 30 20' 24" how many seconds ?
32. In the 8 Autumn months how many hours 1
33., In the three Summer months how many minutes ?
34. In 75 cords how many cubic feet ?
148 COMPOUND NUMBERS.
169. Reduction Ascending is changing a number of
one denomination to another of greater unit lalue, and is
performed by Division; thus, 1000 mills =100 cents =$1.
1. Reduce 53 pints to gallons.
OPERATION. ANALYSIS. Dividing the given
2)53 number of pints by 2, because -
there are ^ as many quarts as
4)26 qt.+l pt. pintS) we O k tain 26 qt. plus a re-
~"7 i _i_o f mainder^of 1 pt. We next divide
26 qt, .by 4, because there are |
Ans. 6 gal. 2 qt. 1 pt as many allon as quarts, and wo
obtain 6 gal. and a remainder of 2 qt. Th^ lastjquotient, with
the several remainders annexed, forms the answer.
2. Reduce 4902 inches to rods.
OPERATION. ANALYSIS We divide suc-
12)4902 cessively by the numbers in
the ascending scasTe in the
16|)408 ft.-j-6 in. sam e manner as in the pre-
ceeding example. But in di-
~ viding the 408 ft. by 16^,
' _ we first reduce 408 ft. to
24 rd.-f- 2 7y 4 =12 ft. halves by multiplying by 2,
and we have 816 halves ; and
Am. 24 rd. 12 ft. 6 in. rcducing 16 , to m ^ we
have 33 hakes. Then dividing 816 by 33 we obtain 24 rd.
plus a remainder of 24 halves to 12 ft. which, with the proceed-
ing remainder annexed to the last quotient, gives the answer.
RULE. I. Divide the given number Ly that number of
Hie scale which wilt reduce it to the next higher denomina-.
lion.
II. Divide the quotient by the next higher number in
scale j and so proceed to //><; highest denomination required*
The last qi/of?i')if } with the several remainders annexed in a
reversed order, will be the ansiccr.
REDUCTION. 149
*
EXAMPLES FOR PRACTICE.
3. How many pounds in 3460 ounces ?
Ans. 216 Ib. 4 oz.
4. How many shillings in 556 farthings ?
Ans. 11s. 7d.
5. "How many yards in 1242 inches 1
6. How many gallons in 2347 pints ?
7.. Reduce 23547 troy grains to pounds.
Ans. 4 Ib. 1 oz. 1 pwt. 3
8. Reduce 1597 quarts to bushels.
Ans. 49 bu. 3 pk. 5 qt.
9r Reduce 107520 oz. avoirdupois to pounds.
10. In 28635 sec. how many hours ?
Ans. 7 hr. 57 min. 15 sec.
11. In 10000"/ow many degrees ? &
'Ans.' 2 46' 40".
ftl2. In 11521 gr. apothecaries weight how many pounds ?
Ans. 21b 1 gr-
13. In 3561829 seconds how many weeks?
14. Reduce 67893 cu. ft. to cords.
v 15. In 1491 pounds how many hundred weight?
5 16. In 12244 pints how many hogsheads ?
17. In 25600 sq. rd. how many acres? Ans. 160 A.
18. How many miles in 51200 rd. ? Ans. 160 mi.
19. How many barrels in 6048 gills? Ans. 6 bbl.
20. In 316800 inches how many miles ? Ans. 5 mi.
21. In 1728 how many gross ? Ans. 12 gross.
22. In 4060 how many score ? Ans. 203 score.
A23. Reduce 1435 feet to fathoms.
24. Reduce 10000 sheets of paper to reams.
Ans. 20 reams 16 quires 16 sheets.
25. Reduce 27878400 sq. ft. to square miles.
150 COMPOUND NUMBERS.
au
PROMISCUOUS EXAMPLES IN REDUCTION.
1. Reduce 4 dollars 67 cents to cents. Ans. 467 cents.
2. Reduce 3724 mills to dollars. Ans. $3.724.
3. Reduce 9690 cents to dollars. Ans. $96.90.
4. Reduce 8 dollars to mills. Ans. 8000 mills.
5. In 91751 farthings how'iSkft^painKfe^ **
Ans. 95 Us. 5d^3 far.
>6. In 3 Ib. 4 oz. 7 pwt. how many grains 1
^7. In 3 tons of cheese how many pounds ?
8. How much will 4 cheese cost, each weighing 36
pounds, at 9 cents a pound 1 Ans. $12.96.
9. How much would 2 Ib. 8 oz. 12 pwt. of gold dust be
worth, at 72 cents a pwt. ? Ans. $409.44.
10. Bought 1 T. 15 cwt.,36 Ib. o sugar at 7 cents a
pound; howNjtoh di^jtcost? >' . ,N^.4ws. $247.52.
11. Paid $25,$rf<Sr otnr ho|^acf of molasses, and sold
it all at 50 cents a gallon ; how much was the whole gain ^
VL2. How many pounds in t> barrels of flour ?
13. How many bushels of oats in a load weighing 1280
pounds] Ans. 40 bu.
14. How many bushels of wheat in a load weighing 2175
pounds ? Ans. 36 bu. 15 Ib.
15. A grocer bought 3 barrels of flour at $6 a barrel,
and sold it out at 4 cents a pound how much did he gain
on the whole ? Ans. $5.52.
\16. In a board 12 feet long and 2 feet wide, how many
square feet? Ans. 24 sq. ft.
" 17. In a block of marble 6 feet long and 3 feet square,
how my cubic feet? ^ Ans. 54 cu. fect.-
18. In a pile of woocT^ feet long 6 feet high and 3 feet
wide, how many cubic feet ? how many cords ?
'\ C\ l1 \ Ans. 468 cu. ft. ; or 3 Cd. 84 cu. ft.
c
REDUCTION. 151
19. In 259200 cubic inches of hewn timber how many
tons ? Ans. 3 T.
^^0. How many square rods in a field 90 rods long and 75
rods wide ? How many acres ? Ans. 42 A. 30 sq. rd.
21. A pond ot water measures 3 fathoms 2 feet 9 inches
in depth ; how many inches deep is it ? Ans. 249 in.
22. What will 3 miles of telegraph cable cost at 12 cents
afoot? Ans. $1900.80.
23. What is the age of a man 3 score and 5 years old 1
Ans. 65 years.
24. How much will I receive for a load of wheat weigh-
ing 2760 pounds at $1.50 per bushel 1 Ans. $69.
25. How many cubic feet in a stick of timber 32 feet
long 2 feet wide and 1 foot thick ? Ans. 64 cu. ft.
\26i> How many square feet in one acre ?
^ 1^7. In 176 yards how many rods ? Ans. 32 rd.
28. A pile of wood is^lCjJ^jJgjjg, 8 feet high, and 8
feet' w"k!ef now much is it worth at $3.50 a cord ?
Ans. $28. '
29. What would be the value of a city lot 40 feet wide
and 120 feet long, at 2 cents a square foot ? Ans. $96.
80. A grocer bought 4 barrels of cider, at $2 a barrel, and
after converting it into vinegar, he retailed it at 15 cents a
gallon ; how much was his whole gain. Ans. $10.90.
31. At 6 cents a pint how much molasses can be bought
'for $4.26? Ans. 8 gal. 3 qt. 1 pt.
32. An* innkeeper bought a load of 40 bushels of oats,
at 36 cents a bushel, and- retailed them at 25 cents a peck j,
how much did he make on the load? Ans. $25.60.
23. What will be the cost of a hogshead of wine at 8
cents a gill? Ans. $161.28.
34. In 120 gross how many score ? Ans. 864 score.
152 COMPOUND NUMBERS.
85. If a man walk 4 miles an hour, and 10 hours a day,
how many miles can he walk in 24 days? Ans. 960 mi.
26. What will be the cost of 2 bu. 1 pk. G qt. of tiinj
thy seed, at 10 cents a quart? Ans. $7.80.
87. What would be the value of a silver goblet, weigh- -^
ing 8 oz. 14 pwt., at $.15 a pwt. ? Ans. $26.10.
88. What .will 16 reams of paper cost at 20 cents a quire! ^
Ans. $64.
39. If 1 bushel of wheat make 45 pounds of flour, how ^
many pounds will 500 bushels make ? How many barrels ]
Ans. 114 bbl. 156 pounds.
40. Bought a gold chain, weighing 2 oz. 18 pwt. at $.90
a pwt.; how much did it cost? Ans. $52.20.
41. How many minutes more.iare there in the Summer
than in the Autumn months ? Ans. 1440 min.X
t^ j
\^ 42. How much will it cost to dig a -cellar 24 ft; long ;
18 ft. wide and 6 feeUkeD^ULcent a cufoc Foot ?
^^^^^^^^^^^^^^"*^**
"- 43. How many boxes, each containing 12 pounds, can bt
filled from a hogshead of sugar containing 9 cwt.?
Ans. 75 boxes.
*^ 44. What will be the cost of 5 bales of cloth, each bak
containing 15 pieces, and each piece measuring 26 yards,
at $1.75 a yard?
s^.45. If a cannon ball goes at the rate of 10 miles a min-j
ute, how many miles would it go, at the same rate, in 2
hours? Ans. 12QO miles.
*^ 46. At 11 cents a pound what will be the cokt of 3 cwt.
"2 qr. 21 Ib. of coffee? ^te.\$40.81.
47. If a man earn $30 a month, how much will he earn
in 5 years? ^^Ans. $1800.
r^
W
ADDITICtff. ' 153 -
ADDITION. -
1 7O. Compound numbers are added, subtracted, multi-
plied, and divided by the same general methods as are em-
ployed in simple numbers. The only modification of the
operations and rules is that required for borrowing, carry-
ing, and reducing by a vaiying, instead of a uniform scale.
1. What is the sum of 36 bu. 2 pk. 6 qt. 1 pt., 25 bu.
1 pk. 4 qt., 18 bu. 3 pk. 7 qt. 1 pt., 9 bu. Opk.^^t. 1'pU
OPERATION. ^ AyjA'siM AiTHnging
* ?* qt. pt. the rtumbere in columns,
or; -i 4 A plaei.ig units of the same
o 7 i draTO under each
021 ,"we first iukl the^
units in the right hand
Ans. 90 0. 4 1 v,.mn, otiowest denom-
ination, and find the
amount to be 3 pints, which is equal to 1 qt 1 pt. We write
the 1 pt. under the column of pints, and add the i qt.to the col-
umn of quarts. We find the amount of the second column to
be 20 qt. which is equal to 2 pk. 4 qt Writing the 4 qt under
the column of quarts, we add the 2 pk. to the column of pecks.
Adding the column of pecks in the same manner, we find the
amount to be 8 pk. equal to 2 bu. Writing pk. under the col-
umn of pecks, we add the 2 bu. to the column of bushels. Add-
ing the last column, we fird the amount to be 90 bu. which we
write under the left hand denomination, as in simple numbers.
Hence the following
RULE 1. Write the numbers so tliat tlwse of the same
unit value will stand in tJie same column.
II. Beginning at tJie riylit hand, add each denomination
as in simple numbers, carrying to each succeeding denomi-
nation one for as many units as it takes of the denomination
added j to make one of the next higher denomination.
154
COMPOUND NUMBERS.
EXAMPLES FOR PRACTICE.
(2.)
(3.)
.
S.
d.
far.
ft)
z
3 .
'f) er
47
10
9
1
10
10
4
1 12
25
6
4
3
9
5
2 10
36
18
2
14
4
16
12
00
10
6
P-r
7
1 00
8
7
3
1
6
3
2 15
Ans.lSQ
3
3
3
32
7
5
2 13
(4) (5)
hhd.
gal.
qt.
pt.
T. cwt. Ib.
oz.
dr.
24
21
3
1
3 12
15
10
11
102
42
2
16
20
7
9
38
9
1
5 9
6
12
42
50
1
18
17
14
00
207
60
3
10 15
59
1
00
(6)
(7)
da.
h.
min.
sec.
Ib. oz.
pwt.
gr.
27
14
40
36
16
11
18
.21
106
% 20
14
25
26
9
15
10
16
12
50
45
11
10
00
8
52
16
39
18
4
6
12
00
(8)
(9)
mi.
fur.
rd.
yd. ft.
in.
P.
sq.yd. sq.ft.
2
5
25
4 1
10
12
20
5
1
3
30
1 2
7
9
15
6
4
16
5
4
15
10
7
10
6
8
2 2
11
20
26
3
ADDITION. 155
10. What is the sum of 2S. 12, 40', 25"; 5S. 9, 27',
88"; 16 10' 50"; IS, 16?
11. What is the sum of 44A. 2E. 24P., 10A. OE. 20P.,
25A. IE. 6^. 36P.? Ans. 86A. IE.
12. What is the sum of 25 rd. 12 ft. 5 in., 28 rd. 9 ft
10 in., 18 rd. 10 ft., 12 rd. 14 ft. 9 in.?
Ans. 2 fur. 5 rd. 14 ft.
13. What is the sum of 5 Cd. 6 cd. ft. 9 cu. ft., 4 Cd. 3
cd ft. 12 cu. ft., 10 Cd. 14 cu. ft., 2 Cd. 7 cd. ft.?
Ans. 23 Cd. 2 cd. ft. 3 cu. ft.
^14. What is the sum of 40 yd. 2 ft. 10 in., 37 yd. 1 ft. 9
in., 28 yd. 11 in., 10 yd. 2 ft., 15 yd. ?
Ans. 132 yd. 1 ft. 6 in.
^15. What is the sum of 13 Cd. 60 cu. ft. 164 cu. in., 25
Cd.75 cu. ft., 18 Cd^25 cu. ft. 540 cu. in., 8 Cd. 1030 cu.
in.? Ans. 65 Cd. 33 cu. ft. 6 cu. in.
\^16. A grocer bought 4 hhd. of sugar ; the first weighed
11 cwt. 2 qr. 21 lb.; the second 10 cwt. 1 qr. 16 lb.; the
third 10 cwt. 22 lb.; and the fourth 9 cwt. 3 qr. How
much did the whole weigh ? Ans. 2T. 2 cwt. 9 lb.
^17. A man has a farm divided into three fields; the first
-^contains 26 A. 2 E. 30 P. ; the second, 48 A. 27 P. ; an
the third, 35 A. 2 E. How many acres in the farm ?
Ans. 110 A. 1 E. 17 P.
18. If a printer one day use 2 bundles 1 ream 10 quires
of paper, the next day 3 bundles 1 ream 12 quires, 20
sheets, and the next, 4 bundles 9 quires, how much does
he use in the three days ?
Ans. 10 bundles 1 ream 11 quires 20 sheets.
19. A tailor used, in one year, 3 gross 6 doz. 10 buttons,
another year, 2 gross 9 doz. 9 buttons, and another year,
4 gross 7 doz. ; how many did he use in the three years'?
156 COMPOUND NUMBERS.
SUBTRACTION.
171. From 24 Ib. 6 oz. 5 pwt. 12 gr. take 14 Ib. 9 oz.
10 pwt. 7 gr.
OPERATION. ANALYSIS. Writing the
oz. pwt. gr. subtrahend under the
6 5 12 . , , . .. c
Q -. p. - minuend, placing units or
the same denomination
Ans. 98 15 5 under each other, we sub-
tract 7 gr. from 12 gr.
and write the remainder, 5 gr., underneath. Since we cannot
subtract 10 pwt. from 2 pwt, we add 1 oz. or 20 pwt. to the^.
5 pwt. and subtract 10 pwt from the sum, 25 pwt, and write
the remainder, 15 pwt, underneath. Having added 20 pwt or
1 oz. to the minuend, we now add 1 oz. to the 9 oz. in the sub-^
trahend, making 10 oz ; but as we cannot take 10 oz. from 6 oz.
we add 1 Ib, or 12 oz. to the 6 oz. making 18 oz. and subtract-
ing 10 oz. from 18 oz. we write the remainder, 8 oz. under the
denomination r* ounces. Having added 1 Ib. to the minuend,
we now add 1 ib. to the 14 Ib. in the subtrahend, and subtract-
ing 15 Ib. from 24 Ib. as in simple numbers, we write the re-
mainder, 9 Ib. under the denomination of pounds. Hence
RULE. I. Write the subtrahend under the minuend,***
so that units of the same denomination shall stand under*
each other.
II. Beginning at the right hand, subtract each denomi-
nation separately, as in simple numbers.
III. If the number of any denomination in the subtra-
hend exceed that of the same denomination in the minuend,
add to the number in the minuend as many units as make
one of the next higher denomination, and then subtract j in
this case add 1 to the next higher denomination of the sub-
trahend before subtracting. Proceed in the same manner
mth oac'h denomination.
SUBTRACTION. 157
EXAMPLES FOR PRACTICE.
(2) (3)
cwt. qr. Ib. oz. dr. lihd. gal. qt. pt.
From 18 1 14 9 8 7 28 2 1
Take 5 2 20 6 10 3 42 3
12
2
19
2
14
3 48
3
1
(4)
(5
)
fh
K
3 .
g
T
bu. pk.
qt.
pt.
12
7
3
1
11
104 2
T.
6
Jr**
8
5
4
2
15
56 3
4
1
(6)
(7)
mi.
fur.
rd.
yd.
ft. in.
A.
R.
p.
40
5
30
3
2 10
400
2
25
14
6
15
4
1 01
325
1
30
(8)
(9
)
wk.
da.
hr. i
nin.
sec.
S.
r
n
10
4
16
40
22
6 25
45
38
4
5
12
45
50
4 28
40
50
(10)
(11
)
T. c
!Wt.
qr.
Ib.
oz.
Cd. cd.ft.<
JU.ft.
cu.i
Q.
14
5
2
18
9
120 4
6
520
10
14
3
12
14
94 T
12 1
500
2)
(13)
(1J
yd.
ft.
in.
Cd. cu.ft
eq.yd.
sq.fl
i sq.
in
74
2
6
325 80
27 '
6
91
3
9
2
9
128 112
14
8
12(
)
158 COMPOUND NUMBERS.
15. From 125 mi. 6 fur. take 90 mi. 4 fur. 25 rd.
Ans. 35 mi. 1 fur. 15 rd.
16. A man bought 1 hhd. of molasses, and sold 42 gal.
3 qt. 1 pt. ; how much remained 1 Ans. 20 gal. 1 pt.
17. A person bought 9 T. 14 cwt. 3 qr, of coal, and
having burned 4 T. 15 cwt. sold the remainder ; how much
did he sell ? Ans. 4 T. 19 cwt. 3 qr.
18. If from a tub of butter containing 1 cwt. 21 Ib
there be sold 24 Ib. 8 oz. how much remains ?
Ans. 96 Ib. 8 oz.
19. From a pile of wood containing 42 Cd. 5 cd ft. there
was sold 16 Cd. 6 cd. ft. 12 cu. ft. ; how much remained ?
Ans. 25 Cd. 6 cd. ft. 4 cu. ft.
20. If from a field containing 37 A. 3 R. 26 P. there be 4
taken 14 A. 2 R. 30 P., how much will there be left ?
21. A farmer having raised 50 bu. 2 pk. of wheat, kept
for his own use 25 bu. 3 pk.; how much did he sell ?
Ans. 24 bu. 3 pk.
22. The distance from New York to Albany is 150 miles;
when a man has traveled 84 mi. 6 fur. 30 rd. of the dis-
tance, how much farther has he to travel ?
Ans. 65 mi. 1 fur. 10 rd.
23. What is the difference in the longitude of two places
one 71 20' 26", and the other 44 35' 58" West?
Ans. 26 44' 28".
24. If from a 'hogshead of molasses 10 gal. 2 qt. be
drawn atone time, (T gal. 3 qt. at another, and 14 gal. at
another, how much will remain ? Ans. 28 gal. 3 qt.
85. From a section of land containing 640 acres, there
was sold at one time 140 A. 2. R. 36 P., at another time
200 A. 1 R., and at another time 75 A. 28 P. . how much
remained ? Ans. 223 A. 3. R. 16 P.
MULTIPLICATION. 159
MULTIPLICATION.
172. 1. A farmer lias 8 fields, each containing 4 A. 2
R. 27 P.; how much land in all ?
OPERATION. ANALYSIS. In 8 fields are 8 times as
A- R. P. much land as in 1 field. We write the
' multiplier under the lowest denomination
of the multiplicand, and proceed thus ; 8
gy -^ jg times 27 P. are 216 P., equal to 5 R 16
P.; and we write the 16 P, under the
number multiplied. Then, 8 times 2 R. are 16 R., and 5 R ad-
ded make 21 R., equal to 5 A. 1 R ; and we write the 1 R un-
der the number multiplied. Again, 8 times 4 A. are 32 A. and
5 A. added make 37 A., which we write under the same de-
nomination in the multiplicand, and the work is done. Hence
RULE. I. Write the multiplier under the lowest denom-
ination of the multiplicand.
II. Multiply as in simple numbers, and carry as in ad-
dition of compound numbers.
EXAMPLES FOR PRACTICE.
(20 (3.)
hhd. gal. qt. pt. bu. pk. qt. pt.
6 20 21 9261
3 4
Ans. 18 61 3 1 38 3 2
(4.) (5.)
lb. oz. pwt. gr. T. cwt. Ib. oz.
12 8 14 16 10 15 20 8
5 6
63 7 13 8 64 14 23
160 COMPOUND NUMBERS.
6. Multiply 14 A. 2 R. 26 P. by 8.
AM. 117 A. 1 R. 8 P.
7. Multiply 6 yd. 2 ft. 9 in. by 12. Ans. 83 yd.
8. Multiply 7ft)- 8 . 5 3 . 13- 10 gr. by 7.
Ans. 54ft>. |. 6 3. 1 3. 10 gr.
9. Multiply 24 bu. 1 pk. 6 qt. by 10.
10. Multiply 9 cu. yd. 15 cu. ft. 520 cu. in. by 5.
Am. 47 cu. yd. 22 cu. ft. 872 cu. in.
11. Multiply 84 12s. 6d. 2 far. by 9.
12. If a pipe discharge 4 hhd. 20 gal. 3 qt. of water in
1 hour, how much will it discharge in 5 hours, at the same
rate ? Am. 21 hhd. 40 gal. 3 qt.
13. If a load of coal by the long ton weigh 1 T. 4 cwt.
2 qr. 20 Ib. what will be the weight of 6 loads ?
Ans. 7 T. 8 cwt. 8 Ib.
14. If 1 acre of land produce 26 bu. 3 pk. 4 qt. of
wheat, how much will 11 acres produce ?
15. If a man travel 30 mi. 4 fur. 20 rd. in 1 day, how
far will he travel in 9 days, at the same rate 1
,16. What is the weight of 3 dozen silver spoons, each
dozen weighing 2 Ib. 10 oz. 12 pwt. 14 gr. ?
Ans. 8 Ib. 7 oz. 17 pwt. 18 gr.
17. If a wood chopper can cut 2 cd. 6 cd. ft. 8 cu. ft. oi
wood in a day, how many cords can he cut in 10 days ?
18. In 20 barrels of potatoes, each containing 2 bu. 8
pk. 6 qt., how many bushels ? Ans. 58 bu. 3 pk.
19. A grocer bought 14 barrels of sugar, each weighing
5 cwt. 1 qr. 15 Ib.; how much did the whole weigh?
20. If the sun, on an average, change his longitude 59'
9" each day, how much will be the change in 25 days?
21. If 1 qt. 1 pt. 3 gi. of wine fill 1 bottle, how much
will be required to fill 3 dozen bottles of the same capacity ?
MULTIPLICATION. 161
22. If a yard of cloth cost 2 10s. 9d. how much will
18 yards cost ? Ans. 45 13s. 6d.
23. If a person average 8 hr. 20 min. 40 sec. of sleep
daily, how much will he sleep in 30 days ?
Ans. 10 da. 10 hr. 20 min.
24. How many cords of wood in 8 piles, each containin
40 cd. ft. 104 cu. ft. 432 cu. in. 1
Ans. 46 Cd. 4 cd. ft. 2 cu. ft.
25. If each silver table-spoon weigh 1 oz. 12 pwt. 16 gr.,
what is the weight of 1 dozen spoons ?
26. If the moon's average daily motion is 33 10' 35",
how much of her orbit does she traverse- in 21 days ?
27. How much land in 12 lots, each containing 2 A. 120
P.? Ans. 33 A.
28. How many bushels of wheat jn 48 sacks, each con-
taining 165 pounds ? Ans. 132 bu.
29. If a locomotive move 4 fur. 36 rd. in one minute,
how far will it move in one hour 1 Ans. 36 mi. 6 fur.
30. If a family consume 2 gal. 1 qt. 1 pt. of molasses in
1 week, how much will they consume in 1 year ?
Ans. 1 hhd. 60 gal. 2 qt.
31. If it take a man 5 hr. 42 min. 50 sec. to saw
cord of wood, how long will it take him to saw 16 cords ?
Ans. 91 hr. 25 min. 20 sec.
32. How many bushels of apples can be put into 75 bar-
rels, each barrel containing 3. bu. 1 pk. ?
Ans. 243 bu. 3 pk.
33. If a man can build 3 rd. 12 ft. 10 in. of wall in 1
day, how much can he build in 10 days ?
I Ans. 37 rd. 12^. 4 in.
34. If a man can mow 2 A. 96 P. of grass in a day, how
much can 27 men mow, at the same rate?
- D 17;
-. <- . -
162 COMPOUND NUMBERS.
DIVISION.
173. If 4 acres of land produce 102 bu. 2 pk. 2 qt. of
wheat, how much will 1 acre produce ?
OPERATION. ANALYSIS. One acre will pro-
bn. pk. qt. pta. (i uc e J as much as 4 acres. Wri-
4)102 3 2 ting the divigor on the left o{ tbe
oc 2 ft 1 dividend, we divide 102 bu. by 4,
and we obtain a quotient of 25 bu.,
and a remainder of 2 bu. We write the 25 bu. under tbe de-
nomination of bushels, and reduce the 2 bu. 'o pecks, making 8
pk., and the 3 pk. of the dividend added makes 11 pk. Divi-
ding 11 pk. by 4, we obtain a quotient of 2 pk. and a remain-
der of 3 pk. ; writing the 2 pk. under the order of pecks, we
next reduce 3 pk. to quarts, adding the 2 qt of the dividend,
making 26 qt, which divided by 4 gives a quotient of 6 qt. and
a remainder of 2 qt Writing the 6 qt. under the order of
quarts, and reducing the remainder, 2 qt, to pints, we have 4
pt, which divided by 4 gives a quotient of 1 pt, which w
write under the order of pints, and the work is done.
2. A* farmer put 182 bu. 1 OPERATION.
pk. of apples into 46 barrels 5 46) j^ P |v 2 ^
how many bu. did he put in- ' 92
40
4
When the divisor is largo lfii(3
we divide by long division, as -j^gg
shown in the operation. From _
these examples we derive the 23
following 8
184(4 qt.
184
- Any. 2 bu. 3 pk. 4 qt.
DIVISION. 163
RULE. I. Divide the highest denomination as in simple
numbers, and each succeeding denomination in the same
manner, if there be no remainder.
II. If there be a remainder after dividing any denomina-
tion, reduce it to the next lower denomination, adding in the
given number of that denomination, if any, and divide as
before.
III. The several partial quotients will be the quotient
required.
EXAMPLES FOR PRACTICE.
( 3 ) W
A. R. P. Ib. oz. pwt. gr.
2)95 2 30 3)52 4 16 18
47 3 15 17 5 12 6
(5)
. wk. da. h. min. sec. bu.
7)33 5 23 45 10 6)88
4 5 20 32 10 14 3 2
(7) (8)
ft). 3- & gr. gal. qt. pt.
5)28 9 4 f 2 5 9)376 3 1
5 9 2 17 41 3 1
(9) (10)
hhd. gal. qt. pt. A. R. P.
12)9 28 2 9)129 2 25
49 2 1 14 1 25
(11) (12)
mi. fur. rd. ft. in. Ib. oz. pwt. gr
7)217 5 19 12 6 11)185 1 19 13
31 81 6 6 - 16 9 19 23
164 COMPOUND NUMBERS.
13. Divide 185. 17s. 6d. by 8.
Ans. 23. 4s. 8d. 1 far.
14. Divide 16 ft,. 13 oz. 10 dr. by 6.
Ans. 2 Ib. 12 oz. 15 dr.
15. Divide 358 A. 1 R. 17 P. 6 sq. yd. 2 sq. ft. by 7.
Ans. 51 A. 31 P. 8 sq. ft.
16. Divide 192 bu. 3 pk. 1 qt. 1 pt. by 9.
Ans. 21 bu. 1 pk. 5 qt. 1 pt.
17. Divide 9 hhd. 28 gal. 2 qt. by 12.
. .. Ans. 49 gal. 2 qt, 1 pt.
18. Divide 328 yd. 1 ft. 3 in. by 21.
Ans. 15yd. 1 ft. 11 in.
19. Divide 36S>. 11 f. 4 3: 23. 7 gr. by 11.
Ans. 3R. 4 f . 23. 13, 17 gr.
20. Divide 16 cwt. 3 qr. 18 Ib., long ton weight, bj 32.
21. If a steamboat run 174 mi. 26 rd. in 14 hours, how
far does she run in 1 hour ?
22. A farm containing 322 A. 2 R. 10 P. is to by divi-
ded equally among 13 persons ; how much will each .have ?
Ans. 24 A. 3 R., 10 P.
23. A cartman drew 38 cd. 5 cd. ft. 6 cu. ft. o'f wood, at
80 loads ; how much did he average per load ?
Ans. 1 cd. 2 cd. ft. 5 cu<*ft.
24. If 24 barrels of flour cost 98. 16s., how much will
1 barrel cost 1 Ans, 4. 2s. 4d.
25. If a vessel sail 163 16' 12" in 27 days, how far
does she sail on an average per day ?
Ans. 5 40' 36".
26. If 3 dozen spoons weigh 9 Ib. 8 oz. 12 gr., how much
does each spoon weigh ? Ans. 3 oz. 4 pwt. 11 gr.
PKOMISCUOUS EXAMPLES. 165
PROMISCUOUS EXAMPLES.
1. A farmer raised 200 bu. 2 pk. of barley, 175 bu. 3 pk.
of corn, 320 bu. 1 pk. of oats, and 225 bu. 2 pk. of rye;
what was the whole quantity of grain raised 7
2. A person having bought 325 A. 2 R. of land, sold
150 A. 1 R. 25 P. of it; how much had he remaining?
3. What is the whole weight of 72 hogsheads of sugar,
each weighing 12 cwt. 3 qr. 1 Ans. 45 T. 18 cwt.
4. If a railroad car run 148 miles 4 fur. in 8 hours,
what is the average rate of speed per hour 1
5. A grocer having purchased 98 cwt. 2 qr. of sugar,
sold 10 cwt. 1 qr. 20 Ib. to one man, and 18 cwt. 16 Ib. to
another; how much remained unsold-?
6. Bought 12 tea-spoons, each weighing 16 pwt. 20 gr.,
an'd 6 table-spoons, each weighing 1 oz. 12 pwt. ; what was
their total weight ? Ans. 1 Ib. 7 oz. 14 pwt.
7. A farmer raised 24 T. 17 cwt. of hay; he sold 5 loads,
each weighing 1 T. 8 cwt. 21 Ib. ; how much has he re-
maining ? Ans. 17 T. 15 cwt. 95 Ib.
8. A jeweler having 36 Ib. 10 oz. 14 pwt. of silver, uses
21 Ib. 6 oz. of it, and then manufactures the remainder into
8 tea-pots ; what is the weight of each ?
Ans. 1 Ib. 11 oz. 1 pwt. 18 gr.
9. A man purchasing 2 A. 140 sq. rd. of land, reserves
| an acre for his own use, and divides the remainder in 4
equal lots ; how much does each lot contain ?
.4ns. 95 sq. rd.
10. How many pounds of sugar in 28 barrels, each con-
taining 3 cwt. 1 qr. 17 Ib. ? Ans. 9576 pounds.
11. If from a piece of land containing 5 A. 3 R., 2 A.
72 P. be taken, how many square rods will remain 1
166 COMPOUND NUMBERS.
12 Divide a tract of land containing 1299500 square
rods into 25 farms of equal area ; how many acres will
there be in each ?
Ans. 324 A. 3 R. 20 P.
13. A merchant buys 3 hogsheads of molasses at 30 cents
a gallon, and sells it at 45 cents ; how much does he gain
on the whole ?
14. What is the cost of 3 chests of tea, each weighing
2 cwt. 2 qr. 18 lb., at $.84 a pound ? Ans. 225.12.
15. How many steps of 30 inches each must a person
take in walking 12 miles?
16. If a man buy 10 bushels of chestnuts, at $3 a bushel,
and sell them at 10 cents a pint, how much is his whole
gain? Ans. $34.
17. How many times will a wheel 13 ft. 4 inches in cir-
cumference turn round in going 12 miles?
Ans. 4752.
18. If 8 horses eat 12 bu. 3 pk. of oats in 3 days, how
many bushels will 20 horses eat in the same time ?
Ans. 31 bu. 3 pk. 4 qt.
19. How much sugar at 9 cents a pound must be given
for 2 cwt. 43 lb. of pork at 6 cents a pound ?
Ans. 162 pounds.
20. How many cubic feet in a room 18 feet long, 16 feet
wide, and 10 feet high ?
21. A person wishes to ship 720 bushels of potatoes in
barrels, which shall hold 3 bu. 3 pk. each, how many bar-
rels must he use ? Ans. 192.
22. How many rods of fence will inclose a farm a mile
square ? Ans. 1280 rods.
23. If granite weigh 175 pounds a cubic foot, what ia
the weight of a cubic yard ? Ans. 2 T. 7 cwt. 25 lb.
CANCELLATION. 167
CANCELLATION.
174:. Cancellation is the process of rejecting equal
factors from numbers sustaining to each other the relation
of dividend and divisor.
It has been shown ( 70 ) that the dividend is equal to
the product of the divisor multiplied by the quotient.
Hence, if the dividend can be resolved into two factors,
one of which is the divisor, the other factor will be the
quotient.
1. Divide 72 by 9.
OPERATION. ANALYSIS. We see in this
Divisor. 0)0 X$ Dividend. example, that 72 is composed
- of the factors 9 and 8, and
8 Quotient. that the factor 9, is equal to
the divisor. Therefore we reject the factor 9, and the remain-
ing factor, 8, is the quotient.
174. Whenever the dividend and divisor are each
composite numbers, the factors common to both may first
be rejected without altering the final result.
2. What is the quotient of 48 divided by 24 ?
OPERATION. ANALYSTS. We first indi-
48 $X$X2 wte the operation to be per-
-^-:=~2 Ant. " IbVmed; by wfrtiifg the dividend
above a line, and the divisor
below it. We resolve 48, into the factors 3, 8 and 2, and 24 in-
to the factors 3, and 8. We next cancel the factors 3, and 8,
which are common to the dividend and divisor, and we have
left the factor 2, in the dividend, which is the quotient.
NOTE. When all the factors or numbers in the dividend are cancelled, 1 should
bo retained.
163 CANCELLATION.
If any two numbers, one in the dividend and one
in the divisor, contain a common factor, we may reject
that factor. ^
3. In 15 times 63, how many t'imes 457
OPERATION. ANALYSIS. In this example we see
that 5 will divide 15 and 45 ; so we
Ans re J ec * 5 as a factor of 15, and retain
the factor 3, and also as a factor of 45,
and retain the factor 9. Again 9 will
divide 9 in the divisor, and 63 in the
dividend. Dividing both numbers by 9, 1 will be retained in
the divisor, and 7 in the dividend. Finally the product of 3 X
7 = 21, the quotient.
4. What is the quotient of 25x18X^X4, divided by
15X4X9X3?
OPERATION.
2 ANALYSIS. In
O this, as in the
3 3 preceedingez-
s ample, we re-
ject all the factors that are common to* both dividend and
divisor, and we have remaining the facjtoi^S^a.^^Hiiyjs^. and
the factors 5, 2, and 2 in the dividend. Completing the work,
we have 2 3=6|, Av$. ^
From the precee^jfog^examples and illustrations we de-
rive the following :
BuLTT-T-^Brff. ^^^^^^f^tay^-mw^r
above a horizontal line, and the numbers composing the di-
visor below it.
II. Cancel all the factors common to both dividend and
divisor.
III. Divide the product of the remaining factors of the
dividend by the product of the remaining factors of the di-
visor , and the result will be the quotient.
CANCELLATION.
169
Nones. Ir Rejecting a factor from any number is dividing the numbr by that
factor.
2- When a factor is cancelled, the unit, 1, is supposed to take its place.
3. One factor in the dividend will cancel only one equal factor hi the diviaor.
4. If all the factors or numbers of the divisor are cancelled, the product of th
remaining factors of the dividend will be the quotient.
5. By many it is thought more convenient to write the factors of the dividend on
the right of a vertical line, and the factors of the divisor on the left.
EXAMPLES FOR PRACTICE.
1. Divide the product of 12x8x6 by 8x4X3.
FIRST OPERATION.
3X2
-=6 Ans.
SECOND OPERATION.
If
6 Ans.
2. Divide the product of 25x18x4x3,
FIRST OPERATION.
&X20X4XJ 5X3X4 60
04
\ -
84 Ans.
3. Divide the uroduct of 36x10X7 by 14x5x9.
. 4.
4. What is the quotient of 21X8X40X3 divided by
12X7X20? Ans. 12.
5. What is the quotient of 64x18x9 divided by 30 X
27X4? Ans. 3f
6. Divide the product of 120x44x6 by 60x11X8.
Ans. 6
8
170 CANCELLATION.
7. Multiply 200 by 60, and divide the product by 50
multiplied by 48. Ans. 5.
8. Multiply 8 times 32 by 6 times 27, and divide the
product by 9 times 96. Ans. 48.
9. What is the quotient of 21x8x60x8x6 divided by
7X12X3X8X3? Ans. 80.
10. What is the quotient of 18x6x4x42 divided by
4X9X3X7X6? Ans. 4.
11. If 18X5X^X66 be divided by 40x22x6, what is
the quotient? Ans. 10^.
12. The product of thj numbers 26, 11, and 21, is to be
divided by the product of the numbers 14 and 13 ; what is
the quotient ? Ans. 33.
13. The product of the numbers 48, 72, 28 and 5, is to be
divided by the product of the numbers 84, 15, 7 and 6;
what is the quotient ? Ans. 9^.
14. How many tons of hay at $9 a ton, must be given for
27 cords of wood, at $4 a cord ? Ans. 12 tons.
15. How many bushels of corn, worth 60 cents a bushel,
must be given for 25 bushels of rye, worth 90 cents a
bushel? Ans. 37^ bushele.
16. How many peaches worth 2 cents eaoh must be given
for 48 oranges, at 3 cents* each ? Ans. 72
17. How many days work, at 75 cents a day, will pay for
30 pounds of coffee, at 15 cents a pound ? Ans. 6 days.
18. How many, suits of clothes, at $18 a suit, can be made
from 5 pieces of cloth, each piece containing 24 yards, at
$3 a yard ? Ans. 20 suits.
19. How many tubs of butter, each containing 48 pounds,
at 14 cents a pound, must be given for 3 boxes of tea, each
containing 42 pounds, worth 60 cents a pound ?
Ant. ll
CANCELLATION. 171
20. How many days work, at 84 cents a day, will pay
for 36 bushels of corn worth 56 cents a bushel 1 ?
Ans. 24.
21. A farmer exchanged 45 bushels of potatoes worth 30
cents a bushel, for 15 pounds of tea; what was the tea
worth a pound? Ans. 90 cents.
22. A grocer bought 120 pounds of cheese, at 9 cents a
pound, and paid in molasses, at 45 cents a gallon ; how
many gallons of molasses paid for the cheese 1
Ans. 24 gallons.
23. Gave 12 barrels of flour, at $7 a barrel, for hay
worth 818 a ton ; how many tons of hay was the flour
worth? Ans. 4 tons.
24. Sold 8 firkins of butter, each weighing 56 pounds,
at 15 cents a pound, and received in payment 3 boxes of
tea, each containing 40 pounds; how much was the tea
worth a pound ? Ans. 56 cents.
25. A man took 6 loads of apples to market, each load
containing 14 barrels, and each barrel 3 bushels. He sold
them at 50 cents a bushel, and received in payment 9 bar-
rels of sugar, eac^. weighing 210 pounds ; how much was
the sugar worth a pound 1 An-s. 6| cents.
26. A grocer sold 12 boxes of soap, each containing 51
pounds, at 10 cents a pound ; he received in payment a
certain number of barrels of potatoes, each containing 3
bushels, at 30 cents a bushel ; how many barrels did he
receive ? Ans. 68 barrels.
27. A man sold 4 loads of barley, each load containing
60 bushels, at 70 cents a bushel, and received in payment
2 pieces of cloth, each piece containing 35 yards, how much
was the cloth worth a yard ? -4ns. $2.40.
172 ANALYSIS.
ANALYSIS.
176. Analysis, in arithmetic, is the process ot solving
problems independently of set rules, by tracing the relations
of the given numbers and the reasons of the separate steps
of 'the operation according to the special conditions of
each question.
177. In solving questions by analysis, we generally rea-
son from the given number to unity, or 1 ; and then from
unity, or 1, to the required number.
178. United States money is reckoned in dollars, dimes,
cents, and mills, one dollar being uniformly valued in all
the States at 100 cents ; but in most of the States money is
sometimes still reckoned in pounds, shillings and pence.
NOTB. At the time of the adoption of our decimal currency by Congress, in
1786, the colonial currency, or bills of credit, issued by the colonies, had depreciated
in value and this depreciation, being unequal in the different colonies, gave rise to
the different values of the State currencies ; and this variation continues wherever
the denomination of shillings and pence are in use,
Georgia Currency.
Georgia ; South Carolina, $l=4s. 8d. 56d.
Canada Currency.
Canada, Nova S,cotia, $l=5s.=60d.
New England Currency.
NewEnglanl Sta es, Indiana, Illinois, J
Missouri, Virginia, Kentucky, Tennes-> $1 6s. 72d.
see, Mississippi, Texas, )
Pennsylvania Currency.
New Jersey, Pennsylvania, Delaware, ) ^ ^ $l_7s 6d. 90d.
New York Currency.
New York, Ohio, Michigan, ) *, fta or ,
North Carolina, f * ]
In many of the States it is customary to give the retail price
of articles in shillings and pence, and the cost of the whole in
dollars and cents.
ANALYSIS. 173
The following will be found an easy, shoit, and practical
method of reducing currencies to dollars and cents.
EXAMPLES FOR PRACTICE.
1. What will be the cost of 36 bushels of apples, at 3
shillings a bushel, New England Currency ?
OPERATION. ANALYSIS. Since 1
6 bushel costs 3 shillings,
36X3 = 108s. I $0 3G bushels will cost 36
108^-6 = $18 Or I 3 times3s.,or36x3~l08s.;
~~ and as 6s. make 1 doUar,
18, Ans. New Engknd cuvrencV|
there are as many dollars in 108s. as 6 is contained times in 108,
or 108-r-6=i
2. What will 112 bushels of barley cost, at 5s. 6d. per
bushel, New York currency ?
OPERATION.
7
Or
XX ft
* ANALYSIS. We mul-
U tiply the number of
bushels by the price,
$77 and divide the result by
$77 Ans. the value of 1 dollar as
in the first example, reducing both the price and 1 dollar to pence,
and we obtain $77. Or, when the price is an aliquot part of a
shilling, the price may be reduced to an improper fraction for a
multiplier, thus; 5s. 6d 5 As. 3-8., the multiplier. The value
of a dollar being 8s., we divide by 8 as in the operation. Hence
To find the cost of articles in dollars and cents, when the price
is in shillings and pence,
Multiply the commodity by the price, and divide the pro-
duct by the value of one dollar in the required currency,
reduced to the same denominational unit as the price.
174 ANALYSIS.
3. What will 180 cords of wood cost at 8s. 4d. per cord,
Pennsylvania currency?
OFEKATION. ANALYSIS. Multiply
2
Or,
100
$200
4 the quantity by the
price in pence, and* di-
vide the product by the
value of 1 dollar in
$200, Ans. , .,
pence ; or, reduce the
shillings and pence, both of the price and of the dollar, to the
fraction of a shilling before multiplying and dividing, thus ;
8s. 4d.=8^s. "= 2 3 5s> ^ ne mu ltip ner ' The value of the dollar
being 7s. 6d. =7s. ^s. we divide by ^ as in the operation.
4. What will be the cost of 7 J yards of cloth, at 6s. 8d.
New York currency ?
OPERATION.
, - or AA ANALYSIS. We reduce the quan-
' tity and the price to improper frao
$6.25 Ans. tions > before multiplying.
NOTE. When there is a remainder in the dividend, it may be reduced to cents
and mills by annexing two or three ciphers and continuing the division.
5. What will 7 hhd. of molasses cost at Is. 3d. per quart,
Georgia currency 1 ?
OPERATION. ANALYSIS. In this example we
% first reduce 7 hhd. to quarts, by mul-
63 tiplying by 63, and 4, and then mul-
^ tiply by the price, either reduced to
pence or to an improper fraction, and
2 I Q45 00 divide by the value of 1 dollar re
' duced to the same denomination aa
$472.50 Ans. the price.
ANALYSIS. 175
6. Sold 8 firkins of butter, each containing 56 pounds at
Is. 3d. per pound, and received in payment tea at 6s. 8d.
per pound; how many pounds of tea would pay for the butter?
OPERATION. ANALYSIS. The operation in
28 this is similar to the preceding
3 examples, except that we divide
9 the cost of the butter by the price
of a unit of the article received in
Ans. 84 pounds. , , , .,
payment, reduced to the same de-
nominational unit as the price of a unit of the article sold. The
result will be the same in whatever currency.
7. What will be the cost of a load of oats containing 64
bushels at 2s. 6d. a bushel, New York currency ?
Ans. $20.
8. At 9d. a pound, what will be the cost of 120 pounds
of sugar, New England currency? Ans. 15.
9. What will be the value of a load of potatoes, meas-
uring 35 bushels, at 2s. 3d. a bushel, Penn. currency?
Ans. $10.50.
10. What will be the cost of 240 bushels of wheat, at
9s. 4d. a bushel, Michigan currency 1 Ans. &2SO.
11. In New Jersey currency ? Ans. $298.66|.
12. In IHinois currency? Ans. $373.33^.
13. In South Carolina currency ? Ans. $480.
14. In Virginia currency ? Ans.
15. In Ohio currency ? Ans.
16. In Canada currency ? Ans. $448.
17. How many days work at 7s. 6d. a day, must be given
for 5 bushels of wheat at 10s. a bushel ? Ans. 6f days.
18. What will be the cost of 5 casks of rice, each weigh-
ing 168 pounds, at 3d. per pound, South Carolina currency 1
Ans. $45.
176 ANALYSIS.
19. How many pounds of sugar^at 9d. per pound, must
be given for 18 bushels of apples, at 2s. 7d. per bushel ?
Ans. 62 pounds.
20. Bought 3 casks of catawba wine, each cask contain-
ing 64 gallons, at 7s. 9d. per quart, Ohio currency ; what
was the cost of the whole 1 Ans. $744.
21. What will it cost to build 150 rods of wall, at 3s. 8d.
per rod, Canada currency ? Ans. $110.
22. How many pounds of butter, at 18d. a pound, must
be given for 12 pounds of tea, at 5s. 4d. a pound 1
Ans. 42| pounds.
23. What will be the cost of 4 hogsheads of molasses, at
Is. 2d. per quart, Mississippi currency 1 Ans. $196.
24. A farmer exchanged 28 bushels of barley / worth 5s.
8d. a bushel, with his neighbor, for corn worth 7s. a bushel;
how many bushels of corn was the barley worth ?
Ans. 22| bushels.
25. What will a load of wheat, measuring 45 bushels, be
worth at lls. a bushel, Kentucky currency ?
Ans. $82.50.
26. What will 12 yards of Irish linen cost, at 4s. 9d. a
yard, Pennsylvania currency? J.TI&. $7.60.
27. Bought the following bill of goodg of f radewell &
Co. ; how much did the whole amount to, New York cur-
rency ?
4 yards of cloth at 5s. 6d. per yard,
9 " calico, - - " - Is. 4d.
10 " ribbon, - 2s. 3d.
6 gallons molasses, - " 4s. 8d. per gallon, "
3 J pounds of tea, - " 6s. per pound.
Ans. $13.1875.
PERCENTAGE. 177
PERCENTAGE.
179. Per cent is a term derived from the Latin words
per centum, and signifies by the hundred, or hundredths, that
is, a certain number of parts of each one hundred parts, of
whatever denomination. Thus, by 5 per cent, is meant 5
cents of every 100 cents, $5 of every $100, 5 bushels of
every 100 bushels, &c. Therefore, 5 per cent, equals 5
hundredths^.OS^jf^T^. 8 per cent, equals 8 hun-
dredths = .08= T $ = 2 2 ~.
1 8O. Percentage is such a part of a number as is in-
dicated by the per cent.
181. The Base of percentage is the number on which
the percentage is computed.
1 82. Since per cent, is any number of hundredths, it
is usually expressed in the form of a decimal. or a common
fraction, as in the following
* TABLE.
Decimals. Common Fractions. Lowest Term*
1 per cent .01 T ^ T ^
2 per cent " .02 " T 3s " A
4 per cent " .04 " ^ ^
5 per cent " .05 " T ^ J^
6 per cent. ,.06 " T fo * ^
7 per cent " 07 " T ^ " jfa
8 per cent. " .08 " T fo " &
10 per cent. .10 " J^ ft
16 percent " .16 ^ " 2 \
20 per cent ' .20 " ^ J
25 per cent " .25 " ^ a ^
50 per cent. ' .50 " T & " %
100 per cent. " 1.00 " |jj ' " I
178 PERCENTAGE.
. To find the jfig^centage of any number.
1. A man having $12*0, paid out 5 per gent, of it for
groceries ; how much did he pay out 1
OPERATION.
$120
.05
_ ANALYSIS. Since 5 per cent, is T 5 o """
$6.00 .05, he paid out .05 of $120, or $120X05
=$6. Hence the
RULE. Multiply the given number or quantity by the
rate per cent, expressed decimally, and point off as in dec-
imals.
EXAMPLES FOR PRACTICE.
2. What is 4 per cent, of $300 ? Ans. $12.
3. What is 3 per cent, of $175? Ans. $5.25.
4. What is 5 per cent, of 450 pounds ?
5. What is 6 per cent, of 65 gallons ? Ans. 3.9 gal.
6. What is 9 per cent, of 200 sheep ? Ans. 18 sheep.
7. What is 7 per cent, of $97? Ans. $6.79.
8. What is 10 per cent, of $12.50 ? Ans. $1.25.
9. What is 40 per cent, of 840 men? Ans. 336 men.
10. What is 25 per cent, of 740 miles ?
11. A man having $4000, invests 25 per cent, of it in
land; what sum does he invest? Ans. $1000.
12. A man bought 1500 barrels of apples, and found on
opening them that 12 per cent, of them were spoiled ; how
many barrels did he lose ? Ans. 180 barrels.
13. A farmer having 180 sheep, sold 45 per cent, of them
and kept the remainder; how many did he sell and how
many did he keep 1 Ans. He kept 99.
14. Having deposited $1275 in bank, I draw out 8 per
cent, of it; how much remains? Ans. $1173.
COMMISSION. 179
COMMISSION.
1 84. An Agent, Factor, or Broker, is a person who
transacts business for another.
1 5. A Commission Merchant is an agent who buya
and sells goods for another.
186. Commission is the fee or compensation of an
agent, factor, or commission merchant.
187. To find the commission or brokerage on any
sum of money.
1. A commission merchant sells butter and cheese to the
amount of $1540 ; what is his commission at 5 per cent. ?
OPERATION. * ANALYSIS. Since the com-
$1540X-05=$77, Ans. mission on $1 is 5 cents or
.05 of a dollar, on $1540 it is $1540X.05=$77. Hence the
RULE. Multiply the given sum by the rate per cent,
expressed decimally ; the result will be the commission or
brokerage.
EXAMPLES FOR PRACTICE.
2. What commission must be paid for collecting $3840,
at 3 per cent. ? Ans. $115.20.
3. A commission merchant sells goods to the amount of
$5487.50; what is his commission, at 2 per cent. ?
Ans. $109.75.
4. An agent buys 5460 bushels of wheat at $1.50 a
bushel ; how much is his commission for buying, at 4 per
cent.? Am. $327.60.
5. A commission merchant sells 400 barrels of potatoes
at $2.25 a barrel, and 345 barrels of apples at $3.20 a bar
rel ; how much is his commission for selling, at 5 per cent. ?
6. An age/nt sold my house and lot for $6525 ; what wasi
his commission at 2 per cent. ]
180 PERCENTAGE.
"Y* PROFIT AND LOSS.
18 8. "Profit and Loss are commercial terms, used to
express the gain or loss in business transactions, which is
usually reckoned at a certain per cent, on the prime or first
cost of articles.
1 89. To find the amount of profit or loss, when
the cost and the gain or loss per cent, are given.
1. A man bought a horse for $135, and afterward sold
him for 20 per cent, more than he gave ; how much did he
gain?
OPERATION.
ANAWESIS. Since $1 gains 20
$135 X- 20 ^ 2 ?, ^ ns - cents, or 20 per cent., $135 will
gain $135X-20=$27. Hence the
RULE. Multiply the cost by the rate per cent, expressed
decimally-.
EXAMPLES FOR PRACTICE.
2. Bought a horse for $150, and sold him at 15 per
cent, profit; how much was my gain? Ans. $22.50.
3. Bought 25 cords of wood at $3.50 a cord, and sold
it so as to gain 33 per cent. ; how much did I make ?
Ans. $28.87*.
4. Paid 7 centra pound for 2480 pounds of pork, and
afterward lost 10 per cent, on the cost, in selling it ; how
much was my whole loss ? Ans. $17.36.
5. Bought 1000 bushels of wheat at $1.25 a bushel,
and sold the flour at 18 per cent, advance on the cost of the
wheat ; how much was my whole gain ? Ans. $225.
6. A grocer bought 6 barrels of sugar, each containing
220 pounds, at 7J cents a pound, and sold it at 20 per cent
profit; how much was the whole gain ? Ans. $19.80.
SIMPLE INTEREST.
181
SIMPLE INTEREST.
190. Interest is a sum paid for the use of money.
191. Principal is the sum for the use of which in-
terest is paid.
1 92. Rate per cent, per annnm is the sum per cent,
paid for the use of $100 annually.
NOTE. The rate per cent, is commonly expressed decimally, as hnndredtha.
(182.)
193. Amount is the sum of the principal and in-
terest.
194. Simple Interest is the sum paid for the use
of the principal only, during the whole time of the loan
or credit.
195. Legal Interest is the rate per cent, establish-
ed by law. It varies in different States, as follows :
Minnesota, 'jS)p er cent.
Mississippi, (3.
Missouri, .6
New Hampshire,. .6
New Jersey, 6
New York, ft/
North Carolina, ... 6
Ohio, 6
Pennsylvania, .... 6
Rhode Island .... 6
South Carolina,...; 7
Alabama, ..... .'. . .8 per cent.
Arkansas, ........ 6
California,. ...... 10
Connecticut, ...... 6
'Delaware, ........ 6
Dist of Columbia,. 6
Florida, .......... 8
...... 7
...... .6
Indiana, ......... 6
Iowa, ............ 7
Kentucky, ....... 6
Louisiana ........ 5
Maine. . ; ..... . ...6
Maryland, ........ 6
Massachusetts, .... 6
Michigan, ........ 7
Tennessee, ....... 6
Texas, .......... .8 ',
U.S. (debts),....^. "
Vermont, ....... ..6/ "
Virginia, ....... jfrf
\F "
Wisconsin,
NOTES. 1. The legal rate in Canada, Nova Scotia, and Ireland is 6 per cent., and
In England and France 5 per cent.
2. When the rate per cent, is not specified in accounts, notes, mortgages, con-
tracts, &c., the legal rate is always understood.
182 PERCENTAGE.
CASE I.
196. To find the interest on any sum, at any rate
per cent., for years and months.
1. What is the interest on $140 for 3 years 3 months,
at 7 per cent. ?
OPERATION.
$140
.07 ANALYSIS. The interest on
$140, for 1 yr., at 7 per cent,
$9.80 int. for 1 year. [ a .07 of the principal, O r $9.-
_ _^ 80, and*?h% interest for 3 yr.
245 3 mo. is 3 T %3| times the
2940 interest for one yr., or^$9.80
X 3 J, which is $31.
Ans. $31.85 Int. for 3 yr. 3 mo. Hence, the following
RULE. I. Multiply the principal fylthe rate per cent. t
and the product will be the interest for 1 year.
II. Multiply this product fy the time^in years and frac-
tions of a year, and the result will be the required interest.
EXAMPLES FOR PRACTICE.
2. What is the interest on $48.50 for 2 years 6 months,
at 6 per cent. ? Ans. $7x275.
3. What is the interest on $325.41 for 3 years.!
months, at 5 per cent. 1 Ans. $54.235.
4. What is the interest on $279.60 for 1 year 9 months,
at 7 per cent. 7 - Ans. $34.251.
5. What is the amount of $26.84 for 2 yr. 6 mo., at 5
per cent. 1 Ans. $30.195.
6. What is the amount of $200 for 1 yr. 9 mo., at 7
percent? Ans. $224.50.
7. What is the interest on $750 for 1 year^3 months,
at 5 per cent. 1 Ans. $46.875.
SIMPLE INTEREST. 183
CASE II.
197. To find the interest on any sum, for any
time, at any rate per ccjnt.
Obvious Relations between Time aud Interest.
I. The interest on any sum for 1 year, at 1 per cent., is
.01 of that sum, and is equal to the principal with the sep-
eratrix removed two places to the left.
II. A month being -^ of a year, J 2 of the interest on
any sum for 1 year is the interest for 1 month.
III. The interest on any sum for 3 days is ^=-^=.1
of the interest for 1 month, and any number of days may
readily be reduced to tenths of a month by dividing by 3.
IV. The interest on any sum for 1 month, multiplied
by any given time expressed in months and tenths of a
month, will produce the required interest.
^\. What is the interest on $306 for 1 yr. 6 mo. 12 da.,
at 7 per cent ?
OPERATION. ANALYSIS. Removing the
] p*. 6 mo. 12 da. = 18.4 mo. seperatrix in the given princi-
12)$3.060 P al two places to the left,
we have $3. 06, the interest on
$.255 the given sum for 1 year at 1
per cent. (I). Dividing this
by 12, we have $.255, the inter-
2040 GSt f r * montll > at ! P er cent *
(!!) Multiplying this quo-
tient by 18.4, the time ex-
pressed in months and deci-
mals of a month. (HI), we
$32.8440 Ans. have $4.692, the interest on the
given sum for the given time, at 1 per cent. (IV). And multi-
plying this product by 7, the rate per cent, we have $32.844, the
required interest. Hence,
VUl
1
184 PERCENTAGE.
RULE. I. Remove the separatrix in the given principal
two places to the left ; the result will be the interest f or \
year at 1 per cent.
II. Divide this interest by 12 f the result will be the in-
terest for 1 month, at 1 per cent.
III. Multiply this interest by the given time expressed in
months and tenths of a month ; the result will be the interest
for the given time, at 1 per cent.
IV. Multiply this interest by the given rate ; the product
wiU be the interest required.
EXAMPLES FOR PRACTICE.
2. What is the interest on $34.25 for 3 yr. 8 mo. 15 da.,
5 per cent. ? Ans. $6.35.
3. What is the interest on $260 for 9 mo. <f da., at 6 per
cent.? Ans. $11.826,
4. What is the interest on $450, at 6 per cent, for 10
mo. 18 days? Ans. $23.85.
5. What is the interest on $372 for 1 yr. 10 mo. 15 days,
at 7 per cent. ? Ans. $48.825.
6. What is the interest on $221.75 for 3 yr. 7 mo. 6 da.,
at 7 per cent. ? . Ans. $55.88.
8. What is the interest on $267.27 for 6 mo. 24 days, at
6 per cent. ? Ans. $9.086.
9. What is the interest on $365 for 2 mo. 3 days, at 6
per cent. ? Ans .$3.83.
10. What is the interest on $785.10, for 1 yr. 6 months
18 days, at 5 per cent. ? Ans. $60.845.
11. On $450 for 3 yr. 7 months, at 8 per cent.?
2. What is the interest on $600 for 2 yr. 8 mo., at 7
r cent. ? * Ans. $112.
13. What is the amount of $1000 for 9 mo. 15 days, at
7 per cent.? Ans. $1055.414.
INTEREST. 186
14. What is the interest on $860 for 6 mo. 6 days,, at 6
per cent. ? * i -x l| 4 A .^ras. $26.66.
15. What istthe interest on $137.45 for 8 mo. 27 days,
at 6 per cent. ? I
^^J.6. Find the amount of $8 Jo for 1 yr. 6 mo. at 3 per
"'cent.'? Ans. $914.375.
v!7. Find the amdfint of $350 v for 9 mo., at 4 per cent 1
\ % X Ans. $360.497.
18. Find'the amount o|%8.50 for 1 yr. 9 mo. 12 da., at
6 per cent. ? Ans. $9.409.
19. Find the amount of $457 for 1 yr. 4 mo. 24 da., at
6 per cent. ? . Ans. $495.388.
20. Find the amount of $650 for 3 yr. 10 mo. 21 days at
7 per cent. ? ^ Ans. $827.009.
21. What is the interest on $79 for 15 mo., at 7 pei
cent. ? Ans. $6.912 x .
22. Find the amount of $.86 for 5 mo., 7 per cent.
V Ans. $.885.
23. What is the interest on $78.75 for 1 yr. 9 mo., at 4
per cent. ? Ans. $5.5125.
24. What is the interest on $1750 for 30 days, at 9 per
cent. ? Ans. $13.125.
25. What is theterest on $3654.25 for 33 days, at 10
per cent. ? Ans. $33.497.
26. Find the amount of $269.50 for 120 days, at 7 pei
cent.? Ans. $275.788.
7. Find the amount of $1625 for 1 yr. 6 mo., at 8 per
t.1 Ans. $1820.
NOTE. For a full treatise of Percentage in all its applications to the businesi
transactions of life, and also for the developement and application of thoae sub-
jects ordinarily treated by arithmetic, the pupil is referred to the Author's Pro*
fressive Practical, and Progressive Higher Arithmetics.
186 PROMISCUOUS. EXAMPLES.
PROMISCUOW EXAMPLES.
1. Multiply the difference between 876042 and 8342GO
by 176. Ans. 7353632.
2. To 47320 add three* times the ^difference between
46270 and 31032. Ans. 93034.
3. From 212462+432046, take/rf? 7240 230124.
4. Divide the sum of 4802+560 10 +20342 by 4 times
the difference between 1200 and 1082.
Ans. 171ft*.
5. What is the difference 1 between 1824624+15624
and 896042 12342? Ans. 956548.
6. What is the difference between 3426 x 284 and
2001041 Ans 772880.
7. What is the difference between 3931476-^-556 and
14x875? Ans. 5179.
8. How many timesan 36 be subtracted from 1 1772 ?
Ans. 327.
9. How many times can 8 x 27 be taken from 1554768 ?
10. Divide 420x216 by 43756 42851.
Ans. lOOyW
11. Multiply 3 times the sum of^624+1036 by
times the difference of 375296. Ans. 2682840.
12. What is the difference between 5 times 2.5, and
5x2.5? Ans. 11.25.
13. Multiply 4.05 + .025+1.8 by 21.875.
14. Divide 5 by .8 x .025. . Ans. 250.
15. How many times can 1.05 be taken from 4.725 ?
Ans. 4.5 times.
16. To .02 times 32.5 add 5.7 times 16.0412.0026.
Ans. 23166318.
PROMISCUOUS EXAMPLES. 187
17. What is the difference between .675 .15 and .23
X.009? Ana. 4.49793.
18. A farmer sold a horse for $140, a cow for $25, and
28 sheep at $2,50 a head ; how much more did he receive
for the horse than for the cow and sheep ? Ans. $45.
19. A young lady having $75, went out shopping, and
bought 14 yards of silk for a dress, at $1,50 a yard, a
shawl for $15,75, a bonnet for $8, a pair of gloves for
$1.125, and a pair of shoes for $1,75; how much money
had she remaining 1 Ans. $27.37|.
20. A grocer bought 12 firkins of butter, each contain-
ing 56 pounds, at 14 cents a pound ; he afterward sold 5
firkins, at 16 cents, and 7 firkins, at 18 cents a pound ;
how much was his whole gain 1 Ans. 21.28.
21. A miller sold 256 barrels of flour, at $6.80 a barrel,
which was $475.60 more than the wheat from which it
was made, cost him ; what was the cost of the wheat ?
Ans. $1265.20.
22. An estate worth $25640, has demands against it to
* the amount of $9376 ; after these claims are paid, the
remainder is to be divided equally among 5 individuals ;
how much will each receive ? Ans. $3252.80
23. If 15 tons of hay cost $311.70, how much will 1
ton cost? Ans. $20.78.
24. Paid $1.24 for 15.5 pounds of beef; how much was
the price per pound 1 Ans. $.08.
25. A farmer exchanged 21 bushels of wheat, at $2 a
bushel, for cloth worth $3 a yard ; how many yards did
he receive 1 Ans. 14 yards.
26. A man having labored for a farmer 1 year, at $15
a month expended the year's wages for cows, at $18 each;
how many cows did he buy ? Ans. 10
188 PROMISCUOUS EXAMPLES.
27. What will be the cost of 3 hogsheads of sugar, each
weighing 15 cwt.,.at 8 cents a pound ? Ans. $360.
28. How many bushels of wheat, at $1.12 a bushel, can
be bought for $81.76 ? Ans. 73.
29. If 140 barrels of apples cost $329, how much is the
cost per barrel ? Ans. $2.35.
30. At $.825 per bushel, how many bushels of corn
can be bought for $264? Ans. 320.
31. If 25 yards of cloth can be bought for $125.25, how
many yards can be bought for $751.50 1 Ans. 150.
32. If 150 bushels of wheat cost $435, how much will
311 bushels cost? Ans. $901.90.
33. If 250 pounds of tea cost $135, what is the price
per pound 1 Ans. $.54.
34. If 13 spoons be made from 2 Ib. 10 oz. 9 pwt. of
silver, what will be the weight of each ?
Ans. 2 oz. 13 pwt.
35. If a man travels 20 mi. 3 fur. 36 rd. in a day, how"
far will he .travel in 61 days at the same rate ?
Ans. 1249 mi. 5 fur. 36 rd.
36. If I put 376 gal. 3 qt. 1 pt. of cider into 9 equal
casks, how much do I put into each cask 1
37. If a family use 1} pounds of tea in 1 month, how
much would they use in 1 year ? Ans. 13i pounds.
38. What would be the cost of 565 pounds of butter
at 12i cents a pound? Ans. $70.625.
39. At $4.25 per bushel how much clover-seed can be
bought for $11.6875 1 Ans. 2.75 bushels.
40. At -Jg- of a dollar a pound, what will be the cost
of 12 pounds of sugar ? Ans. $.75.
41. At | of a dollar a yard, what will be the cost of
40| yards of cloth? Ans. $15.30.
PROMISCUOUS EXAMPLES. 189
42. How many cubic yards of earth must be thrown
from a cellar 40 ft. long, 30 ft. wide, 6 ft. deep; and what
will be the cost of the excavation, at 12^ cents a cubic
yard ? Ans. 2662- cubic yards ; $33.33i.
43. If 6 pounds of cheese cost $, how much will 10
pounds cost? Ans. $li.
44. HDW much wheat at SI. 25 a bushel, must be given
for 50 bushels of corn at $.70 a bushel 1
45. At 10 cents a pint, how much will 189 gallons of
molasses cost? Ans. $151.20.
46. At 15 cents a pound, how much will -fa of a pound
of coffee cost ? Ans. 2f cents.
47. If 3 gallons of molasses cost $ , how many gal-
lons can be bought for $4 1 Ans. 14f .
48. At $7j a firkin, how many firkins of butter can be
bought for $33 ? Ans. 4f .
49. If of a yard of cloth cost $4, what will one yard
cost "? Ans. $24.
50. At $3 a barrel, how many barrels of cider can be
bought for $8 ? Ans. 2}| barrels.
51. What part of 100 pounds is 16 pounds?
Ans. -^g.
52. How much wood in a load 10 ft. long, 3| ft. wide*
and 4 ft. high ? Ans. 1 Cd. 12 cu. ft.
53. How many tons of coal may be bought for $346.125
at $9.75 per ton ? . Ans. 35.5 tons.
54. What is the interest on $136.80 for 1 yr. 11 mo.,
at 7 per cent.? Ans. $18.354.
55. What will be the cost of .6 of a gallon of wine, at
$.65 a gallon 1 Ans. $.39.
56. A owns 4 of a flouring mill, and sells, f of his shard
to B ; what part of the whole has he left ?
190 PROMISCUOUS EXAMPLES.
57. If 2 yards of cloth cost $6f , how much will 9 yards
cost? . Ans. $30.
58. What will | of | of a barrel of flour cost at $7^
per barrel 1 ? Ans. $2|.
59. If 1 acre of land yield 1 T. 9 cwt. 1 qr. 22 Ib. oi
hay, how much will 18 acres yield ?
GO. A speculator bought 1575 barrels of potatoes, and
upon opening them, he found 15 per cent.of them spoiled;
how many barrels did he lose ? Ans. 236.25.
61. How many steps of 30 inches each, must a person
take in walking 10 miles? Ans. 21120.
62. A man bought 12 bushels of chestnuts, at $4.50 a
bushel, and sold them at 12 cents a pint ; how much was
his whole gain? Ans. $88.16.
63. What is the interest on $300, for 10 mo. 21 days,
at 6 per cent. ? Ans. $16.05.
64. An agent in Chicago, purchased 5450 bushels of
wheat, at $.82 a bushel ; what was his commission at 2
per cent, on the purchase money 1 Ans. $89.38.
65. A vessel loaded with 4500 bushels of corn, was
overtaken by a storm at sea, and it was found necessary
to throw overboard 25 per cent, of her cargo ; what was
tfie whole loss, at 60 cents a bushel ? Ans. $675.
66. A grocer bought 2 hogsheads of molasses, at 37^
cents a gallon, and sold it at 20 per cent, advance on the
cost ; how much was his whole gain? Ans. $9.45.
67. If f of acres of land is worth $60, what is the
value of 1 acre? Ans. $84.
68. If ly bushels of wheat sow an acre of land, how
many acres will 12 bushels sow? Ans. 9 acres.
69. If a farm is worth $3840, how much is f of it
worth ? Ans. $2400.
PROMISCUOUS EXAMPLES. 19]
70. If 17 kegs of nails weigh 27 cwt, 3 qrs. 23 Ibs.
3oz., long ton weight, how much will 1 keg weigh 1 ?
71. If a bushel of apples cost of a dollar, how many
may be bought for f of a dollar ?
72. Divide of f by j of | ? Ans. J.
73. What is the amount of $620 for 4 yr. 3 mo.,
t 6 per cent. ? Ans. $778.10.
74. What is the brokerage on $5462, at 4 per cent.,
75. How many pounds of butter at 13| cents a pound,
must be given for 1230 pounds of sugar at 8 cents a
pound? Ans. 728f pounds.
76. Divide 168 bu. 1 pk. 6 qt. of corn equally among
35 persons. Ans. 4 bu. 3 pk. 2 qt.
77. What will be the cost of lathing and plastering
overhead, a room 36 feet long and 27 feet wide, at 28
cents a square yard 1 Ans. $30.24.
78. How much land at $2.50 an acre, must be given in
exchange for 360 acres, at $3.75 an acre ?
79. What is the amount of $564.58, for 3 yr. 5 mo.
12 da., at 6 per cent ? Ans. $681.448.
80. How much sugar at 9 cents a pound, should be
given for 6^ cwt. of tobacco, at 14 cents a pound 1
81. How many times may a jug which holds J of a
gallon, be filled from a cask containing 128 gallons'?
82. A man having $25000, invested 30 per cent, of it in
bonds and mortgages, 45 per cent, of it in bank stocks,
and the remainder in railroad stock; how much did he
invest in railroad stock ?
Ans. $6250.
83. How many times can a box holding 4 bu. 3 pk
2 qt, be filled from 336 bu. 3 pk. 4 qt. ?
Ans. 70.
192 PKOMISCUOUS EXAMPLES.
84. How many cords of wood in 17 piles, each 11 feet
long, 4 feet wide, and 6 feet high 1
85. If the price of 1 acre of land is $32f , what is the
value of | of an acre ? Ans. $28f |-.
86. What number of times will a wheel 14 ft. 10 in.
in circumference, turn round in traveling 11 mi. 6 fur
15rd. 12ft. 6 in.? Ans. 4200.
87. A man bought a farm of 136 acres, at $94 an acre ;
he paid $475 for fencing and the improvements, and
then sold it for 14 per cent, advance on the whole cost ;
how much was his whole gain ?
Ans. $1856.26.
88. If 36.48 yards of cloth cost $54.72, how much will
14.25 yards cost ? Ans. $21.375.
80. If $13.342 will pay for 17.5 bushels of barley,
how many bushels can be bought for $76.24 1
Ans. 100 bushels.
90. A lady having $40.50, spent 40 per cent, of it for
dry goods; how much had she left?
Ans. $24.3b.
91. A gentleman bought a house and lot for $6425 in
the course of five years it increased in value 110 per cent,
how much was the property then worth 1
Ans. 813492.50.
92. How much will a broker charge to chahge $560
uncurrent money for currehTmt)lfey;"^^3^)cr^eet. 1
Ans. $16.80.
93. If 4 hogsheads of wine cost $181.44, what will be
the cost of 1 pint ? Ans. 9 cents.
94. What will 5 casks of rise cost, each weighing 165
pounds, at 3d per pound, Georgia currency ?
Ans. $44.198-
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