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THE
TEACHERS' ASSISTANT;
OR,
A SYSTEM OF PRACTICAL ARITHMETIC:
WHEREIN
THE SEVERAL RULES OF THAT USEFUL SCIENCE ARE
ILLUSTRATED BY A VARIETY OF EXAMPLES
A LARGE PROPORTION OF WHICH ARE IN
FEDERAL MONEY.
THE WHOLE DESIGNED
TO ABRIDGE THE LABOUR OF TEACHERS, AND TO
FACILITATE THE INSTRUCTION OF YOUTH.
A NEW EDITION, WITH CORRECTIONS AND additions BY THE AUTHOR.
REVISED.
· COMPILED BY STEPHEN PIKE.
PHILADELPHIA:
PUBLISHED AND SOLD BY THOMAS DAVIS,
NO. 171 MARKET STREET.
1845

ENTERED according to Act of Congress, in the year 1838, by
M'CARTY & DAVIS,
in the Office of the Clerk of the District Court of the Eastern
District of Pennsylvania.
Signs.
# + 1 x +
::::
EXPLANATION OF CHARACTERS.
Significations.
equal; as 20s.-L. 1.
more; as 6+2=8.
less; as 8-2=6.
iato, with, or multiplied by; as, 6×2=12.
by (i. e. divided by) as, 6÷2=3; or, 2)6(3.
proportionality; as, 2 : 4 :: 6 : 12.
✔or Square Root; as, /64-8.
Cube Root; as, 3/64—4.
Fourth Root; as, 16-2, &c.
A Vinculum; denoting the several quantities
over which it is drawn, to be considered jointly
as a simple quantity

Mathe
PREFACE.
THE design of the following work is to furnish the seve-
ral rules of Arithmetic concisely expressed, together with
a variety of applicative examples, arranged in such order
that the learner may advance, by gradations, from what is
simple to what is more abstruse, and be unobstructed in
his progress by ignorance of particulars that he should
previously have known.
The compiler is aware that a number of works of a
similar nature is already in use, and that most of them are
possessed of considerable merit; yet he believes he has in
several respects improved upon them. Whether he has or
not, after making a few remarks, he will submit to the
judicious to determine.
Under each of the rules in the TEACHERS' ASSISTANT,
one or more wrought examples are given, which afford an
opportunity of explaining and illustrating them. Of the
examples for the application of the several rules, the
easiest occur first; such as are similar mostly succeed
each other; and all are delivered in as familiar terms as
could readily be employed. Federal money, as far as the
five primary rules are concerned, is treated of separately,
and agreeably to the manner in which it is used in trade,
mills being mostly rejected. Before entering upon Com-
pound Addition, a portion of Reduction is introduced,
which appears necessary, in order to explain that rule,
as well as Compound Multiplication and Compound Di-
vision.
Besides the foregoing particulars, a number more might
be adduced that are conceived to be worthy of attention;
such as the arrangement of the rules and examples in
Practice, Simple Interest, Tare and Tret, &c.; but these,
with the whole work, are referred to teachers and others
interested in the subject.
A 2
3

RECOMMENDATIONS.
Philadelphia, Sept. 23.
AGREEABLY to your request, I have examined Mr. PIKE's Treatise
of Arithmetic, and am much pleased with it. His mode of exemplify-
ing the rules is, I think, extremely well accommodated to the compre-
hension of juvenile pupils; while the general arrangement, extent, and
scientific execution of the work render it worthy of adoption in both
public and private seminaries.
Messrs. Johnson & Warner.
JAMES ABERCROMBIE, D.D.
Director of the Philadelphia Academy.
Philadelphia, Ninth mo. 26.
I HAVE examined the System of Arithmetic compiled by S. PIKE,
and am of opinion that it is well calculated for conveying to youth, in
a short time, a general knowledge of that science.
The commendable attention which the compiler has paid to a clear
elucidation of his subject, as well as his careful exclusion of any thing
which would unnecessarily perplex, entitles him to the thanks of those
who are engaged in the laborious task of imparting knowledge to youth.
BENJAMIN TUCKER.
Philadelphia, Sept. 16.
AFTER a careful inspection of Mr. S. PIKE's System of Arithmetic,
I give it a decided preference to every other I have yet seen, and shall
be glad of the publication of a work that, in my opinion, will deduct
from the labour of teaching, and conduce to the advantage of learners.
JOHNSON TAYLOR.
GENTLEMEN,
Philadelphia, Sept. 30.
I have no hesitation in declaring my belief, in concurrence with the
gentlemen who have already recommended S. PIKE's System of Arith-
metic, that its publication will conduce to the public and private utility
of the arithmetical student.
Messrs. Johnson & Warner.
Yours, &c.
SAMUEL B. WYLIE.
THE System of Arithmetic compiled by S. PIKE is, in my opinion,
a very judicious performance. The arrangement of the parts, the per-
spicuity of the rules, and the appropriate and familiar nature of the
examples, are peculiarly calculated to facilitate the progress of the
learner. I therefore give the work a decided preference to any other
on the subject, with which I am acquainted.
JOHN GUMMERE,
Principal and Teacher of Burlington Boarding-school.
4

CONTENTS
Numeration -
Simple Addition
Simple Subtraction
Simple Multiplication
Simple Division
Federal Money
Simple Reduction
Compound Addition
-
-
Compound Subtraction -
Compound Multiplication
Compound Division
Compound Reduction
The Single Rule of Three
The Double Rule of Three
PAGR
7
10
12
13
17
22
30
42
49
55
59
66
72
80
Practice
84
Tare and Tret
-
95
Simple Interest
99
Insurance, Commission, and Brokage,
109
Compound Interest
111
Discount
-
112
Equation
Barter
Loss and Gain
Fellowship
Exchange
Vulgar Fractions
Decimal Fractions
114
115
117
119
122
129
144
A 3
5
6
Involution
Square Root
Cube Root
CONTENTS.
PAGE
0
155
157
160
A general Rule for extracting the Roots of all
Powers
Alligation
Position
Arithmetical Progression
Geometrical Progression
162
163
168
171
•
174
Compound Interest by Decimals
Discount at Compound Interest
Annuities at Compound Interest
Annuities in Reversion
Perpetuities -
177
179
181
185
186
Permutation
187
Combination
188
Duodecimals
188
Promiscuous Questions -
193
!

ARITHMETIC.
ARITHMETIC is the art of computing by numbers. It
has five principal rules for its operations; viz. numera-
tion, addition, subtraction, multiplication, and division.
NUMERATION.
Numeration teaches to write or express numbers by
figures, and to read numbers thas written or expressed.
In treating of numbers, the following terms are em-
ployed: viz. unit, ten, hundred, thousand, and mil-
lion; as also billion, trillion, and some others.
the latter are seldom used.
A unit is a single one.
A ten is ten units.
A hundred is ten tens.
A thousand is ten hundreds.
A million is ten hundred thousands.
But
Note. As it takes ten hundred thousands to make a
million, when we express a number greater than a
thousand, and less than a million, we use tens of thou-
sands, or hundreds of thousands, or both, as the case
requires. Likewise, to express a number greater than
a million, we employ tens of millions, or hundreds
of millions, &c.
The following are the figures used in numeration,"
with their names above them.
One two three four five six
1
2
3
4
5 6
6
seven eight nine
7
8
9
Each of these figures represents the number which
its name denotes; but it is understood to be that num-
ber of units, or that number of tens, or that number of
hundreds, &c. according to its relative place which is
exemplified in the following tables.
B
7

NUMERATION.
TABLE FIRST.
TABLE SECOND.
Hundred million
Ten million
Million
Hundred thousand
Ten thousand
Thousand
Hundred
Ten
Unit
Hundreds of millions
Tens of millions
Millions
Hundreds of thousands
Tens of thousands
Thousands
Hundreds
Tens
to Units
#1
1 1 1, 1 1 1,
1 1
2 2 2,2 2 2 2 2
These tables show that in using figures to express
numbers, they are placed in a horizontal row-the first
figure at the right hand representing one or more units,
the next tens, the next hundreds, &c. Thus a 1 is one
unit, or one ten, or one hundred, &c. according to the
place in which it stands; and in like manner, a 2 is two
units, or two tens, or two hundreds, &c. The same
rule determines the value of each of the other figures.
In reading numbers, the units and tens are taken to-
gether. 1 ten and 1 unit are read eleven; 1 ten and 2
units, twelve; 1 ten and 3 units, thirteen, &c.: 2 tens
and one unit are read twenty-one; 3 tens and 1 unit,
thirty-one, &c. Thus the number expressed by the
row of figures in table first is read—one hundred and
eleven millions, one hundred and eleven thousands,
one hundred and eleven. That expressed by the
figures in table second is read-two hundred and twen
ty-two millions, two hundred and twenty-two thou
sands, two hundred and twenty-two.
The succeeding tables will further illustrate the subject
TABLE THIRD.
Tens
Units
Hundreds
Thousands
Tens of thousands
1
1 2
1 2 3
1,2 3 4
1 2,3 4 5
1 2 3,4 5 6
2 3 4,5 6 7
1 2,3 4 5,6 7 8
1 2 3,4 5 6,7 8 9
Hundreds of thousands
Millions
Tens of millions
Hundreds of millions

1
1
A
1
One
Twelve
One hundred and twenty-three
1 thousand 234
12 thousand 345
123 thousand 456
1 million 234 thousands 567
12 millions 345 thousands 678
123 millions 456 thousands 789
1

NUMERATION.
9
In writing numbers which have no units, or no tens,
or no hundreds, &c. the order observed in the foregoing
tables must be maintained by filling the vacant places
with a character called a nought or cypher, (0) which,
of itself, represents no number. See
TABLE FOURTH.
Tens
Thousands
Hundreds
Units
10
Ten
Tens of thousands
Hundreds of thousands
Millions
Tens of millions
Hundreds of millions
100
1, 0 0 0
1 0, 0 0 0
1 0 0,0 0 0
0 0 0
1, 0 0 0,
1 0, 0 0 0, 0 0 0
1 0 0, 0 0 0, 0 0 0
2 0 0,00 0, 0 0 2
3 0 0,00 3, 0 3 0
4 0 4, 0.4 0, 4 0 0
5 5 0, 5 0 0, 0 0 0
-
One hundred
1 thousand
10 thousand
100 thousand
1 million
10 millions
- 100 millions
200 millions and 2
- 300 millions 3 thou. and 30
404 millions 40 thou. 4 hun.
550 millions 500 thousand
EXAMPLES.
-
Read the following numbers, or write them in words.
Note.-Making a point or dot after every third
figure, counting from the units place, greatly facilitates
the reading of large numbers.
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 30,
31, 32, 40, 43, 44, 50, 55, 56, 60, 67, 68, 70, 71, 79,
80, 82, 88, 94, 92, 100, 101, 111, 112, H3, 114,
120, 128, 130, 132, 200, 203, 210, 300, 320, 332,
400, 500, 600, 700, 800 900, 1000, 2001, 3010, 4020,
5200, 10250, 23450, 356789, 6789402, 76450791,
20156789, 1304136784.
Write the following numbers in figures.
Ten. Twelve. Fifteen. Seventeen. Twenty-six. Thirty-
nine. Fifty-two. Seventy-four. Eighty-one. Ninety-
six. One hundred and fifteen. Two hundred. Three
hundred and twenty. Nine hundred and nine. One
thousand two hundred. Seven thousand seven hun-
B 2

10
SIMPLE ADDITION,
dred and thirty. One hundred and forty thousand.
Seven hundred thousand five hundred and sixty-three.
Seventeen millions. Eighty-four millions two thou-
sand and forty-nine. Two hundred millions and fifteen.
SIMPLE ADDITION.
Addition teaches to collect several numbers into one.
The number formed by adding several numbers is
called the amount or sum of those numbers.
RULE.
Place the numbers one under another, with units un-
der units, tens under tens, &c. then, beginning with the
units, add up all the columns successively, and under
each column set down its amount. But if either of the
amounts (except the last) be more than 9, set down its
right hand figure only, and add the number expressed
by its left hand figure or figures into the next column.
The whole amount of the last column must be set down.
PROOF.
Perform the addition downwards.
EXAMPLES.
41 33
4 8 3 2
4 2 1 1
S523
5130
4320
3022
9 7 4 3
4 0 6 0 1 0
8321
7244
2 0 7 2 40
Amount 1 9 6 8 7 3 0 3 4 2
6 2 2 7 0 0
1
2
1 2
24
41
80
1 2 3
3
3
1 3
3 6
6 0
90
322
2
4
20
4 1
70
7 0
2 3 1
1
5
30
90
5 0
6 0
202
9 64
804
670
70 1
20
375
670
9 50
300
37
4 1 2
942
100
204
43 1
566
820
200
702
560
7 1 9
170
320
500
290

SIMPLE ADDITION.
11
1 2 3 4
3210
1369
3220
4 0 1 2
1 4 0 0 4
4500
372 1
5079
9020
4 5 3 2
5011
2904
5800
2046
3043
1 5 6 0 4
3200 9
56005
1 3 20 8
11
7300
3211
2 5 3 20
2 5 3 2 2 1
1 3 1 2 04
1 0 5 1 3 2
7300
2 4 200
3 0 9 2 20
1408000
8060
1 9 0 0 0
1 6 0 4 0
1 2 5 1 0 0
676000
2010 0 0 0
2 1 3 0 0 0 0 0
1 0 0 0 0 0 0 0
Add the following numbers, viz. 14, 18, 99, 45, 28,
27, 19, 38, 16, 39, 48, 29, 260, 148.
Add, six hundred and forty, seventy-nine, eighty,
one hundred, two hundred and ten, four hundred and
fifty.
Add, nineteen thousands, fifty thousands, one million.
one hundred and one, one hundred and twenty-five.
APPLICATION.
1. If John give Charles twenty nuts, and James give
him fifty-six, and Joseph give him ninety-five, how
many will he have?
Answer 171.
2. A person went to collect money, and received of
one man ninety dollars; of another, one hundred and
forty dollars; of another, one hundred and one dollars;
and of another, twenty-nine dollars. How much did
he collect in all?
Ans. 360 dollars.
3. Deposited in bank, fifty dollars in gold; three hun-
dred dollars in silver, and five thousand dollars in notes.
What is the whole amount deposited? Ans. 5350 dols.
4. The distance from Philadelphia to Bristol is 20
miles; from Bristol to Trenton, 10 miles; from Trenton
to Princeton, 12 miles; from Princeton to Brunswick,
18 miles; from Brunswick to New York, 30 miles. How
many miles from Philadelphia to New York? Ans. 90.
5. A merchant bought of one person 50 barrels of
flour for 300 dollars; of another person, 75 barrels for
525 dollars; and of another person, 125 barrels for 1000

12
SIMPLE SUBTRACTION.
dollars. How many barrels did he buy, and how much
did he pay for the whole?
Ans. 250 barrels, and paid 1825 dollars.
SIMPLE SUBTRACTION.
By Subtraction we ascertain how much greater one
number is than another: or what remains when a less
number is taken from a greater.
RULE.
Place the less number under the greater, with units
under units, tens under tens, &c. Then, beginning at
the units place, take each lower figure from the one
above it, and set down what remains. But if either of
the lower figures be greater than the upper one, con-
ceive 10 to be added to the upper,* then take the lower
from it, and set down the remainder. When 10 is thus
added to the upper figure, there must be 1 added to the
next lower figure.
PROOF.
Add the remainder to the less number, and their
amount will be equal to the greater.
EXAMPLES.
From
Take
2 5 6 8
1 3 2 6
4 4 2 1 5 4 1 0 3 0 0 7 6
3 8 79
2000806
Remainder 1 2 4 2
5 4 2
5 390 29 270
4 3. 9 5
2 1
8 7
152
1 7
4 9
141
453
362
241
23
764 2
6 7 2 9
2 0 4 3
1 7 0 3
6 5 0 4
304
1 8 6 0
950
3 2 0 6
107
3 2 0 1 6
12045
98700
2 5 290
5 0 0 6 1 27 6 5 0 4 0 0 3
49952 1
7 0 0 0 3 0 2
Some prefer taking the lower figure from 10, adding the remainder
to the upper, and setting down their amount.

MULTIPLICATION.
13
Take one hundred and fifty-six from three hundred
and twenty-five.
Subtract fifteen thousands five hundred and nine from
twenty thousands six hundred and fifty-four.
Subtract twenty-five from ten thousands.
APPLICATION.
1. Charles has thirty-two marbles, and John has
twenty-five: how many has Charles more than John?
Ans. 7.
2. William is seventeen years old, and James is nine:
how much older is William than James? Ans. 8 years.
3. Charles had twenty-five apples, but gave his brother
twelve of them: how many had he left? Ans. 13.
4. A person had in bank 9000 pounds, but drew out
1112 pounds: how much money had he remaining in
bank?
Ans. 7888 pounds.
5. My friend owed me one hundred and fifty dollars,
but has paid me ninety dollars: how much does he still
Ans. 60 dollars.
owe me?
ADDITION AND SUBTRACTION.
1. If I add 500, 627, and 1000, and subtract from their
amount 900, what number will remain ? Ans. 1227.
2. A person borrowed of me, at one time, 62 dollars;
at another time, 150 dollars; and at another time, 200
dollars. He has now paid me 300 dollars. How much
does he still owe me?
Ans. 112 dollars.
3. Subtract 267 from 345, and add 150 to the re-
mainder.
Facit 228.
4. A person had in his desk 1000 dollars. He took
out 120 dollars to pay a debt. He afterwards put in 75
How much was there then in the desk?
Ans. 955 dollars
dollars.
SIMPLE MULTIPLICATION.
Multiplication teaches to find what a number amounts
to when repeated a given number of times.
The number to be multiplied is called the multipli-
cand.
The number to multiply by is called the multiplier.

14
MULTIPLICATION.
The number produced by multiplying is called the
product.
The multiplier and multiplicand are also called fac-
tors.
The scholar should commit the following Table
to memory before he proceeds further.
MULTIPLICATION TABLE.
Twice 13 times 4 times 5 times 6 times 17 times
1make2 1makc3 1make4 1make5 1make6 1make7
∞ 20
4 2
62
82 10 2 12 2
14
3
63
93
12 3
15 3
18 3
21
4
8
4
12 4
16 4
20 4
24 4
28
5
10 5
15 5
20 5
25 5
30 5
35
6
12 6
18 6
24 6
30 6
366
42
7
14 7
21 7
28 7
35 7
42 7
49
8
16 8
24 8
328
40
8
48 8
56
9
18 9
27 9
36 9
45
9
54 9
63
10
2010
3010
40 10
5010
60 10
70
11
22 11
3311
4411
5511
66 11
77
12
2412
3612
48 12
6012
72/12
84
8 times
9 times
10 times 11 times
12 times
1 make 8 1 make 9
1make 10 1 make 11
1 make 12
2
16 2
18
2
20 2
22 2
24
3
24 3
27 3
30 3
33 3
36
32 4
36 4
40 4
44 4
48
5
40 5
45 5
50 5
55 5
60
6
4S 6
54 6
60 6
66 6
72
7
56 7
63 7
70
7
77 7
84
64 8
72 8
80 8
88 8
96
72 9
81 9
90 9
99 9
108
10
80 10
90 10
100 10
110 10
120
11
8811
99 11
11011
121 11
132
12
96 12
108 12
120 12
132 12
144
When the multiplier does not exceed 12, work by
RULE I.
Set the multiplier under the right hand figure or
figures of the multiplicand. Then, beginning with the
units, multiply all the figures of the multiplicand, in
succession, and set down the several products. But if

MULTIPLICATION.
15
Si
either of the products (except the last) be more than 9,
set down its right hand figure only, and add its left hand
figure or figures to the next product. -The whole of
the last product must be set down.
PROOF.
Multiply by double the multiplier, and the product
will be double the former product.
EXAMPLES.
Multiplicand 2 4 3 2 7 4 2 0 0 5 2 4 0 0 9 2
Multiplier
2
3
4
1 2
Product 4 8 6 8 2 2 8 0 2 0 8 4 8 1 1 0 4
1 2
2
38
3
6 0
4
1 12
245
209
3
6
2
2 1 1 0 6 6 9 0 9 0 0 8 1 9 9 2 1 3 G 1
7
co
4 8 7 3 2 97990
3
1 2
9
10
6
8 4 6 0 5 3 4 0 0 99
10
1 2
6 7 0 7 0 5 3 4 6 2 1 4 4 0 9 5 6 0 4 2 0 0 0 9
2
1 1
9
When the multiplier exceeds 12, work by
RULE II.
Multiply by each figure of the multiplier separately,
first by the one at the right hand then by the next, and
so on, placing their respective products one under ano-
ther, with the right hand figure of each product directly
under that figure of the multiplier by which it is pro-
duced. And these products together; and their amount
will be the product required.
Note.-When cyphers occur at the right hand of
either or both of the factors, omit them in the opera-
tion, and annex them to the product.

16
MULTIPLICATION.
EXAMPLES.
Multiplicand 26043
Multiplier
432
5208 6
78 129
1 0 4 1 7 2
2043 20
5 07 20 0
40 8 6 4
143024
1021 60
1 0 3 6 3 1 1 0 4 000
Product 1 1 2 5 0 5 7 6
3. Multiply
25 by
13 Facit
325
4.
125 by
36
4500
5.
5231 by
145
758495
6.
129186 by
98
12660228
7.
23430 by
230
5388900
8.
756 by
2000
1512000
9.
5400420 by 23000
124209660000
10.
2104
1418516800
11.
1622460400
674200 by
5401 by 300400
Note. When the multiplier is the exact product of
any two factors in the multiplication table, the opera-
tion may be performed thus: multiply by one of the
factors, and then multiply the number produced by the
other factor.
EXAMPLES.
Multiply 3412 by 21.
3 4 1 2
3
3412
7
1 0 2 3 6
23884
77
3
Product 7 1 6 5 2
Product 7 1 6 5 2
3. Multiply 43102 by 66 Facit
2844732
4.
5.
12071 by 99
871075 by 42
1195029
36585150
75812112
526473 by 144
APPLICATION.
1. Richard has 125 nuts, and George has 6 times that
number. How many has George?
Ans. 750.
9. There are 20.boxes of raisins with 14 pounds in

SIMPLE DIVISION.
17
?
each box. How many pounds are there in all? Ans. 280.
3. The price of one orange is 9 cents: how many
cents will five oranges come to, at the same price?
Ans. 45.
4. There are 12 pence in one shilling. How many
pence are there in 40 shillings?
Ans. 480.
ADDITION AND MULTIPLICATION.
1. Multiply 25 by 10, and 36 by 14, and 124 by 45.
Add the several products, and tell their amount.
Ans. 0334.
2. There are ten bags of coffee weighing each 120
pounds; and 12 bags weighing each 135 pounds. What
is the weight of the whole? Ans. 2820 pounds.
3. A merchant bought five pieces of linen containing
25 yards each, and 2 pieces containing 24 yards each,
and 1 piece containing 26 yards. How many yards
were there in the whole?
Ans. 199.
SUBTRACTION AND MULTIPLICATION.
1. Multiply 342 by 22, and from the product sub-
Facit 7124.
tract 400.
2. There are 15 bags of coffee, each of which weighs
112 pounds. The bags which contain the coffee weigh
22 pounds. How much would the coffee weigh without
the bags ?
Ans. 1658 pounds.
3. There are 12 chests of tea, each of which weighs
96 pounds. The chests which contain the tea weigh
each 20 pounds. What would the tea weigh without
the chests?
Ans. 912 pounds.
DIVISION.
By division we ascertain how often one number is
contained in another.
The number to be divided is called the dividend.
The number to divide by is called the divisor.
The number of times the dividend contains the divi-
sor is called the quotient.
If, on dividing a number, there be any overplus, it is
called the remainder.

18
SIMPLE DIVISION.
The dividual is a partial dividend, or so many of the
dividend figures as are taken to be divided at one time,
and which produce one quotient figure.
When the divisor does not exceed 12, work by
RULE I.
See how often the divisor is contained in the first left
hand figure or figures of the dividend.* If it be con-
tained an exact number of times, set down that number;
and then see how often it is contained in the next
figure. But if it be contained any number of times
with a remainder, set down the number of times, and
conceive the remainder to be prefixed to the next figure;
then see how often the divisor is contained in these,
and proceed as before till the whole is divided.
PROOF.
Multiply the quotient by the divisor, and to their
product add the remainder, (if any,) and the result will
be equal to the dividend.
EXAMPLES.
Dividend
Divisor 3)963 5)2960
12)112813 12)970811280
}
Quotient 321
592
9401+1
80900940
2)864
4)1416
5)56160
6)12180
12)115218
8)7284016
3)9635410
12)850811550
9)24600134
11)405320004
11. Divide
46323 by
9 Facit
5147
12.
13.
1430400 by 7
6730214 by 10
204342 Rem. 6.
673021
4.
*The multiplication table shows how often any number, not exceeding
12, is contained in any other number not exceeding 144; as that 4 is
contained in 12 three times, because 3 times 4 are 12; 10 is contained in
115 eleven times with 5 over; because 11 times 10 are 110, which, with
5, make 115.

SIMPLE DIVISION.
19
When the divisor exceeds 12, work y
RULE II. or
LONG DIVISION.
Take for the first dividual as few of the left hand
figures of the dividend as will contain the divisor, try
how often they will contain it, and set the number of
times on the right of the dividend-multiply the divi-
sor by this number-subtract its product from the di-
vidual, and to the remainder affix the next figure of the
dividend, to form a second dividual: or if this be not
sufficient, set a cypher on the right of the dividend,
and affix the next figure, and so on, till a sufficient
number of figures are affixed-try how often the divi-
sor is contained in this second dividual, and proceed as
before. Continue this process till all the dividend
figures are employed as above directed; or till the num-
ber they form, when affixed to a remainder, is not large
enough to contain the divisor.
When the work is done, the figures on the right of
the dividend form the quotient.
PROOF. As under Rule I.
EXAMPLES.
Quotient
Divisor Dividend
320)12864016081(40200050
42)9870(235
S4
147
126
210
210
1280
640
640
1608
1600
$
Remainder 81
1. Divide
4633 by 41 Facit
113
2.
3.
4.
5.
6.
11111
2303 by 49
47
465 by
27
17 Rem. 6
40231 by
75
536
31
253622 by 422
601
13699840 by 342
40058
1
7.
4586841 by 3467
1323
S.
46447786 by 1234
37640
26

20
SIMPLE DIVISION.
Note 1. Cyphers on the right of the divisor may be
omitted in the operation, observing to separate as many
figures from the right of the dividend, which annex to
the remainder.
EXAMPLES.
1. Divide 146340 by 5400. Facit 27, remainder 540.
54|00)1463|40(27
108
383
378
540
2. Divide
76173 by
320 Facit 238 Rem. 13
3.
867894 by
300
2892
294
4.
15463420 by
1600
9664
1020
5.
3689
4765
6.
11214
99607765 by 27000
1345680000 by 120000
Note 2. When the divisor is the exact product of
any two factors in the multiplication table, the division
may be performed thus:-Divide first by one of the
factors agreeably to rule 1; then divide the quotient by
the other factor in the same manner.
When a remainder occurs in the first operation and
none in the last, it is the true one: but a remainder in
the last operation must be multiplied by the first divi-
sor, and its product added to the first remainder (if any)
for the true remainder.
EXAMPLES.
1. Divide 46508974 by 96. Facit 484468. Rem. 46.
8)46508974
12)5813621-6 first remainder.
484468-5 last remainder.
8
40
6
46 true remainder.

SUBTRACTION AND DIVISION.
21
2. Divide
3.
34320 by 99 Facit 346 Rem. 66
20208 by 48
421
4.
52818
48
5704392 by 108
APPLICATION.
1. As division is a short method of discovering how
often one number is contained in another, how often is
3 contained in 3699?
Ans. 1233 times.
2. How many times is 25 contained in 132 ?
Ans. 5 times with 7 over.
Ans. 40.
How
3. There are 12 pence in one shilling. How many
shillings are there in 480 pence ?
4. The price of a pair of shoes is 2 dollars.
many pair may be had for 56 dollars?
Ans. 28.
5. Fifty-four apples are to be divided, equally, be-
tween two boys. How many must each boy have?
Ans. 27.
6. Suppose a man travel 40 miles a day: how many
days will he be in travelling 240 miles?
ADDITION AND DIVISION.
Ans. 6.
1. If I add 167, 394, and 447; and divide their amount
by 12: what number will result?
Ans. 84.
2. A person has in money, 5000 dollars; in bank-
stock, 3500 dollars; and in merchandise, 12500 dollars.
He intends to divide this property, equally, among his
3 sons. What will be the share of each son?
Ans. 7000 dollars.
3. Suppose a farmer, who has a plantation of 520
acres, buys an adjoining one of 375 acres, and divides
the whole into five equal portions: how many acres
will there be in each portion?
Ans. 179.
SUBTRACTION AND DIVISION.
1. Subtract 2468 from 5796, and divide the remain-
der by 26.
Result 128.
2. William bought 12 pears: he kept 6 of them, and
divided the rest between his two sisters.
did each sister receive?
How many
Ans. 3.
3. A man, at his decease, left property, amounting
to 12426 pounds. He directed in his will that 1000
pounds should be given to his niece; and that the

22
FEDERAL MONEY.
remainder of the property should be divided, equally,
between his two nephews. What is the share of each
nephew?
Av. . 5713 pounds.
MULTIPLICATION AND DIVISION.
1. Multiply 145 by 12, and divide the product by 6.
Result 290.
2. To find how many dollars are contained in any
number of pounds, we multiply the pounds by 8, and
divide their product by 3. How many dollars are
there in 456 pounds?
Ans. 1216.
3. To find how many pounds are contained in any
number of dollars, we multiply the dollars by 3, and
divide their product by 8. How many pounds are
there in 8576 dollars?
FEDERAL MONEY,
Ans. 3216.
OR MONEY OF THE UNITED STATES.
The denominations of Federal Money are; Eagle,
Dollar, Dime, Cent, and Mill.
10 mills (m.) make
10 cents
1 cent, cts.
1 dime.
10 dimes (or 100 cts.) 1 dollar, D. or $
1 eagle.
10 dollars
These denominations have precisely the same rela-
tive values as those of unit, ten, hundred, &c. and are
also similarly ranged in Numeration. Federal money
is therefore added, subtracted, multiplied, and divided
by the same rules that are given for Simple Addition,
Subtraction, Multiplication, and Division.
It must be remarked, however, that in writing sums
of Federal Money, parts of a cent are generally used
instead of mills; and that, in reading those sums, nei-
ther the eagles nor dimes are mentioned: the former
being considered as tens of dollars; and the latter as
tens of cents.
The parts or fractions of a cent, used instead of mills,
are expressed by two numbers, placed one above the
other, with a line drawn between them. The under
number denotes the part; and the upper one informs
how many of that part are designed to be expressed: as,

NUMERATION OF FEDERAL MONEY.
23
SL
one fourth; & three fourths: one third; two thirds;
a half.
NUMERATION OF FEDERAL MONEY.
In writing sums of Federal Money, the cents are
placed on the right of the dollars, and are separated from
them by a point. If there are not more than 9 cents in
the sum, a cypher is put in the tens' place; and if there
are no cents, two cyphers are used.
If the point which separates the dollars from the cents
be removed or supposed to be removed, the sum may
be considered as cents only: and when the sum is cents
only, if two figures be separated from the right, all on
the left of these will be dollars. See the following
TABLE.
Thousands of dollars
Hundreds of dollars
Eagles or tens of dollars
Dollars
Dimes or tens of cents
Cents
1 2
23
5 6 0
642
6004
3
"
3 4
-
O
1 dol. 34 cents, or 134 cents
10. 12 dols. 10 cents, or 1210 cents
20 23 dols. 20 cents, or 2320 cents
07. 560 dols. 7 cents, or 56007 cents
0 64. 642 dols. 6 cts. and 4, or 642064 c.
1 2 6004 dols. 12 cts. and 2, or 6004121 c.
8 9126 dols. 18 cts. and , or 912618 c.
9 1 2 6, 1
EXAMPLES.
To be read by the learner as dollars and cents, and
also as cents only.
1.49 3.26 4.75 9.18 17.90 21.09 14.02
125.00 426.00 900.00 340.06 3911.10 4006.18
76420.01 19560.00 11904.10 4896.733 400.004
4500.06.
c 2
244
ADDITION OF FEDERAL MONEY.
The following to be written in figures.
Seventeen dollars and fifty-two cents. Forty-nine
dollars and seventeen cents. Eighty-four dollars and
ten cents. Sixty dollars and twelve and a half cents.
Two hundred and fourteen dollars and six cents and a
half. Three hundred dollars. One thousand dollars.
Seven thousand dollars and four cents.
ADDITION OF FEDERAL MONEY.
RULE.
Place the sums one under another, with dollars under
dollars and cents under cents; then, if there are no
fractions, proceed in the same manner as in Simple Addi-
tion, observing to separate the cents of the amount from
the dollars thereof, by placing a point between them.
When fractions occur, find their amount in fourths ;*
consider how many cents these fourths will make; add
them with the cents in the right hand column, and pro-
ceed as before directed.
Proof; as in Simple Addition.
Note.-To find how many cents there are in any
number of fourths of a cent, divide them by 4, and the
quotient will be cents.
EXAMPLES.
D. cts. ทาง.
D. cts.
D.
cts.
40 15 5
42 06
182
,
,
46 12
"
21 10 0
156
20
4
12/2
9
و
340 59
89
064
9
2
20
09
250 25
2140
00
?
50,
17
200 00
4000
50
2
$ 177, 64 9
989
10
6233
87
و
و
و
* In Addition, Subtraction, and Division of Federal Money, all frac-
tions less than a fourth are omitted, and every fraction greater than a
fourth is reckoned a half, three fourths, or a whole cent, according to
its value: so that in these three operations, no fractions are used except-
ing fourths a half being counted two fourths. But in Multiplication it
is often material that no fraction be omitted, and that all fractions should.
be estimated at their real value.

SUBTRACTION OF FEDERAL MONEY.
25
D. cts.m.
D. cts. D.
cts.
D.
cts.
5, 40
2
21 14
140
064
1
183
2
,
4
10 0
56
10
350 19
3
12 5
95
75
200 00
"
و
و
و
12 56年
​45
12/
5 2
04
56 15
350 08
95, 1
88
و
و
15
520 12
و
145 12/1/20
3500, 183
"
も
​8. Add the following sums; viz. 45 dollars; 156 dol-
lars; 1000 dollars, and 750 dollars.
9. Add 48 dollars 20 cents; 14 dollars 58 cents: 100
dollars, and 500 dollars.
10. Add 4 cents; 10 cents; 55 cents; 15 cents, and
11 cents.
11. Add 12 cents; 183 cents; 564 cents; 20 cents;
95 cents, and 42 cents.
APPLICATION.
1. Bought a hat for 4 dollars; a pair of shoes for 2
dollars 25 cents; a pair of stockings for 1 dollar 50
cents; and a pair of gloves for 75 cents. What is the
cost of the whole?
Ans. S dollars 50 cents.
2. Bought a Bible for 1 dollar; an English Reader
for 75 cents; an Introduction for 50 cents; a slate for
314 cents; a slate pencil for 1 cent; and a copy book
for 12 cents. How much do they all amount to?
Ans. 2 dollars 694 cents.
3. Suppose I buy a barrel of sugar for 30 dollars 87½
cents; a bag of coffee for 22 dollars 183 cents; and a
bushel of salt for 1 dollar 12 cents: what sum must I
pay for the whole?
Ans. 54 dollars 184 cents.
SUBTRACTION OF FEDERAL MONEY.
RULE.
Place the less sum under the greater, with dollars
ruder dollars, cents under cents, &c.: then if there are
o fractions, proceed as in Simple Subtraction; observ-
ing to separate the dollars from the cents, in the remain-
der. If there is a fraction in the upper sum, and none
in the lower, set it down as part of the remainder, and
proceed as before directed. If there is a fraction in

26
SUBTRACTION OF FEDERAL MONEY.
:
each of the sums, and the lower less than the upper,
subtract the lower from the upper, and set down the
difference. If there is a fraction in the lower sum, and
none in the upper, subtract it from 4, and set down the
difference in this case there must be 1 added to the
right hand figure of the cents, in the lower sum, before
it is subtracted from the one above it. If there is a
fraction in each of the sums, and the lower greater than
the upper, subtract the lower from 4, add the difference
to the upper, and set down the amount.
In this case,
as in the last, there must be one added to the right hand
figure of the lower cents.
Proof; as in Simple Subtraction.
EXAMPLES.
D. cts. m.
D.
cts.
D. cts.
D. cts
54, 67, 5
5.6
75
35
183
25
182
>
"
40 01 2
41
25
21
10
14
22/
,
"
,
,
14 66, 3
15
50
,
14, 082
10, 964
D. cts.
D.
cts.
D. cts. m.
D. cts.
,
65 49
55, 144
520 31
2
210
121/
2
14,
9, 10, 8
07 6
35 20
,
14, 12
$ 10, 34%
310,
18
,
22
D. cts.
35 12
00
D.
cts.
D. cts.
D.
cts.
49 184
50
00
456
45
20, 064
20 12/
451
202
13. Subtract 456 dollars from 1000 dollars.
14. Subtract 45 cents from 64 cents.
15. Subtract 375 dollars 18 cents from 400 dollars.
APPLICATION.
1. Bought goods to the amount of 545 dollars 95
cents, and paid at the time of purchase, 350 dollars.
How much remains to be paid? Ans. 195 dols. 95 cts.
2. A merchant bought a quantity of coffee, for which
he paid 560 dollars. He afterwards sold it for 610 dol-

MULTIPLICATION OF FEDERAL MONEY.
27
lars 873 cents. How much did he gain by the trans-
action?
Ans. 50 dollars 873 cents.
3. If a storekeeper sells goods for 102 dollars, which
cost 125 dollars 75 cents: how much will he lose by the
sale?
Ans. 23 dollars 75 cents.
MULTIPLICATION OF FEDERAL MONEY.
:
RULE.
Set the multiplier under the sum to be multiplied:
then, if there is no fraction, proceed as in Simple Mul-
tiplication observing to separate the cents from the
dollars in the product. If there is a fraction in the sum,
multiply it, and find how many cents are contained in
its product: then multiply the cents of the sum, and
add to their product the cents contained in the product
of the fraction, and proceed as before directed.* Or, if
the multiplier exceed 12, multiply the sum, omitting
the fraction: then multiply the fraction, and add the
number of cents contained in its product to the product
of the rest of the sum.
Proof: as in Simple Multiplication.
Note. To multiply a fraction of a cent, and find
how many cents are contained in its product-multiply
the upper number of the fraction, and divide its pro
duct by the under one, and the result will be the num-
ber of cents.
EXAMPLES.
D. cts.
D. cts.
12
50
10, 56
>
4
2
D. cts.
140, 182/
10
D. cts.
10, 87/1/1
125
50,00
21, 121
1401, 863
5435
2174
D. cts. m.
D.
>
9 10, 3
12,
cts.
75
D. cts.
1087
2
145, 184
7
62/
1359,37

28
de
DIVISION OF FEDERAL MONEY.
8. Multiply $500
D. cts.
by
4 Product 2000,00
9.
$42 564 cts. by
3
127,684
10.
25 cts. by
3
75
11.
37 cts. by
5
1,87
12.
$4 184 cts. by
12
50,25
13.
$10 333 cts. by
10
103,333
14.
$5 663 cts. by
20
113,33
15.
$29 38 cts. by 96
2820,48
16.
17.
18.
$102 19 cts. by 120
12262,80
$31 17 cts. by 208
6484,40
$25 184 cts. by 25
629,68
APPLICATION.
1. How much will 11 oranges come to, at 12 cents
each?
Ans. 1 dol. 37 cts.
2. What will 10 loaves of bread come to, at 64 cents
a loaf?
Ans. 62 cts.
3. What will 8 cords of wood amount to, at 4 dollars
50 cents a cord?
Ans. 36 dollars.
4. Sold 213 barrels of flour, for 6 dollars 25 cents per
barrel. What is the amount? Ans. 1331 dols. 25 cts.
5. Bought 308 pounds of coffee at 21 cents a pound.
What is the amount?
Ans. 64 dols. 68 cts.
6. How much will 132 pieces of linen come to, at 17
dollars 37 cents each?
Ans. 2293 dols. 50 cts.
DIVISION OF FEDERAL MONEY.
RULE.
Divide as in Simple Division. When a remainder
occurs, multiply it by 4, and add the number of fourths
that are in the fraction of the sum (if any) to its pro-
duct divide this product by the divisor, and its quo-
tient will be fourths; which annex to the quotient of
the sum.
Proof: as in Simple Division.

DIVISION OF FEDERAL MONEY.
29
D. cts.
£)45, 22
D. cts.
3)63, 18
EXAMPLES.
D. cts. D. cts.
25)629, 68(25, 18
50
22, 61
21, 064
129
125
D. cts.
8)85, 00
D. cts.
m.
00
12)740, 41, 2
46
25
10, 624
'
61, 70, 1
218
200
D. cts.
4)25, 24
D. cts. m.
11)56, 50, 7
18
4
25)75(3 fourths
75
D. cts
8. Divide 56 dols. 15
cts. by
10 Quotient 5,61%
9.
96 dols.
by 5
19,20
10.
156 dols.
by
4
39,00
11.
346 dols.
by
43,25
12.
1465 dols. 92
cts. by
2
'732,964
13.
14.
500 dols. 734
58 dols. 14
cts. by
9
55,632
cts. by 38
1,53
15.
417 dols. 96
cts. by 129
3,24
16.
7550 dols.
by 125
60,40
17.
18.
4640 dols. 184
cts. by 15
309,341
8
28 dols. 80 cts. by 360
APPLICATION.
1. To divide 52 dollars 68 cents, equally, among 6
persons, what sum must be given to each ?
Ans. 8 dols. 78 cents.
2. If 8 pounds of coffee cost 2 dollars 4 cents, what
is the price of 1 pound?
Ans. 25 cts.
3. Bought 29 yards of fine linen for 65 dollars 25
cents. What was the price per yard? Ans. 2 Juls. 25 cts.
4. Paid 58 dollars 75 cents for 235 yards of muslin.
What was it per yard?
Ans. 25 cts.

30
SIMPLE REDUCTION.
5. If 103 bushels of wheat cost 225 dollars 57 cents;
how much is it a bushel?
Ans. 2 dols. 19 cts.
6. Soid 144 yards of fine linen for 90 dollars. How
much is that per yard?
Ans. 62 cts.
PROMISCUOUS EXAMPLES.
1. If I add the following sums, viz. 556 dollars 183
cents; 825 dollars 12 cents; and 1000 dollars; and
subtract from their amount 125 dollars: what sum will
result?
Ans. 2256 dols. 314 cts.
2. If I subtract 125 dollars 183 cents from 456 dol-
lars 75 cents, and multiply the remainder by 4, what
sum will result?
Ans. 1326 dols. 25 cts.
3. A person has 200 dollars. He owes his tailor 65
dollars 872 cents; his shoemaker, 25 dollars 75 cents;
and his hatter, 18 dollars. What sum will he have re-
maining, after paying these debts? Ans. 90 dols. 37½ cts.
4. Purchased 10 bushels of potatoes at 564 cents per
bushel; 2 bushels of corn at 872 cents per bushel; and
2 barrels of flour at 8 dollars per barrel. What is the
amount of the whole?
Ans. 23 dols. 37 cts.
5. Calculate the amount of articles in the following
bill:
J. JONES,
Bought of S. Smith,
D. cts.
19 yards of lace, at 2, 37 per yard
-
14 yards of ribbon, at
24 ditto ditto at
8 pair of gloves, ` at
184
25
13 fans,
- at
27 per pair
13 each
2 pair of knots, - at
25
per pair
Amount $ 58, 161 cts.
SIMPLE REDUCTION.
Reduction is the changing of a sum or quantity, from
one denomination to another, without increasing or
lessening its value.
Simple reduction is the reducing of sums or quanti-
ties which have but a single denomination.

SIMPLE REDUCTION.
31
When a sum or quantity is to be changed to a lower
denomination than its own, work by
ULE
-Iultiply the sum or quantity by that number of the
lower denomination which makes one of its own.t
/See notes 1, 2, and 3.)
If there are one or more denominations between the
denomination of the given sum or quantity, and that
to which it is to be changed: first change it to the next
lower than its own, and then to the next lower, and so
See notes 4 and 5.
on.
ENGLISH MONEY.
The denominations of English Money are pound,
shilling, penny, and farthing.
·
4 farthings (qr.) make 1 penny
12 pence
20 shillings
-
d.
- 1 shilling S.
- 1 pound
F
Farthings are written as fractions, thus:
one farthing.
two farthings, or a halfpenny.
three farthings.
EXAMPLES.
Note 1.-To reduce pounds to shillings, multiply
them by 20, because every pound makes 20 shillings.
Reduce 15 pounds to shillings. Facit 300 shillings.
15
20
300 shillings.
2. Reduce 256 pounds to shillings.
Facit 5120 s.
*The reason of this rule is plain: for if it take twenty shillings to
make one pound, it must take 5 times 20 shillings to make five pounds;
and to find how many 5 times twenty are, we may either multiply 20 by
5, or 5 by twenty. Likewise, if it take 4 pecks to make one bushel, it
must take 6 times 4 pecks to make 6 bushels, &c.
+ One denomination is said to be lower than another, when it is of less
value; and higher, when it is of greater value: thus, a shilling is a lower
denomination than a pound; and a higher denomination than a penny.
D

32
SIMPLE REDUCTION.
Note 2.-To reduce shillings to pence, multiply
them by 12, because every shilling makes 12 pence.
3. Reduce 60 shillings to pence.
Facit 720 d.
4. Bring 120 shillings to pence.
Facit 1440 d.
Note 3. To reduce pence to farthings, multiply
them by 4, because every penny makes 4 farthings.
5. Reduce 350 pence to farthings. Facit 1400 qrs.
6. Change 4560 pence to farthings. Facit 18240 qrs.
Note 4.To reduce shillings to farthings, change
them first to pence, and then change those pence to
farthings.
7. Reduce 10 shillings to farthings.
Facit 480 qrs.
8. Bring 115 shillings to farthings. Facit 5520 qrs.
Note 5.-To reduce pounds to farthings, reduce
them first to shillings; then change those shillings
to pence; and then change those pence to farthings.
9. Reduce 5 pounds to farthings. Facit 4800 qrs.
10. Bring 76 pounds to farthings. Facit 72960 qrs.
FEDERAL MONEY.
The denominations of Federal Money have
already been given.
EXAMPLES.
Note 1.-To reduce dollars to cents, multiply them
by 100; or, which is the same thing, annex two cyphers
to their number.
1. Reduce fifty dollars to cents.
2. Reduce 125 dollars to cents.
3. Bring 5560 dollars to cents.
Facit 5000 cts.
Facit 12500 cts.
Facit 556000 cts.
Note 2.— To reduce cents to fourths or quarters of
a cent, multiply them by 4;-to halves, multiply
them by 2;—to thirds, multiply them by 3, &c.
4. Reduce 25 cents to fourths or quarters of a cent.
Facit 100 fourths.
5. Reduce 256 cents to fourths of a cent.
Facit 1024 fourths.
6. Reduce 45 cents to half cents. Facit 90 halves.
7. Bring 145 cents to thirds of a cent. Facit 435 thirds.
Note 3. To reduce dollars to fourths, halves, or
thirds of a cent, &c.—first bring them to cents; and
then bring those cents to fourths, or halves, &c.

SIMPLE REDUCTION.
33
8. Reduce 12 dollars to fourths of a cent.
Facit 4800 fourths.
9 Bring 122 dollars to halves of a cent.
Facit 24400 halves.
10. Bring 54 dollars to thirds of a cent.
Facit 16200 thirds.
Note 4.-To reduce dollars to mills, multiply them
by 1000; or, which is the same thing, annex three
cyphers to their number.
11. Reduce 26 dollars to mills.
12. Bring 150 dollars to mills.
Facit 26000 m.
Facit 150000 m.
AVOIRDUPOIS WEIGHT.
By this weight are weighed things of a coarse, drossy
nature, that are bought and sold by weight, and all
metals but silver and gold.
The denominations of Avoirdupois Weight are ton,
hundred weight, quarter, pound, ounce, and dram.
16 drams (dr.) make 1 ounce
16 ounces
02.
1 pound
lb.
1 quarter of a cwt.
gr.
C.wt.
T.
4 quarters, or 112 lb. 1 hundred weight
28 pounds -
20 hundred weight
- 1 ton
EXAMPLES.
1. Reduce 27 tons to hundred weights. Facit 540 C.wt.
2. Bring 45 hundred weight to quarters.
3. Bring 250 quarters to pounds.
4. Reduce 76 pounds to ounces.
5. Bring 40 ounces to drams.
6. Reduce 8 tons to pounds.
7. Bring 2 tons to drams.
TROY WEIGHT.
Facit 180 qr.
Facit 7000 78.
Facit 1216 oz.
Facit 640 dr.
Facit 17920 lb.
Facit 1146880 dr.
By this weight jewels, gold, silver, and liquors are
weighed.
The denominations of Troy Weight are pound,
ounce, pennyweight, and grain.
24 grains (gr.) make 1 pennyweight dwl.
20 pennyweights
12 ounces
Oz.
1b.
1 ounce
1 pound
EXAMPLES.
Facit 6912 oz.
1. Reduce 576 pounds to ounces.

SIMPLE REDUCTION.
Change 1740 ounces to pennyweights.
Facit 34800 dwts.
3. Bring 145 pennyweights to grains. Facit 3480 gr.
4. Reduce 15 pounds to pennyweights. Facit 3600 dwts.
5. Bring 75 pounds to grains.
Facit 432000 gr.
APOTHECARIES WEIGHT.
By this weight apothecaries mix their medicines, but
buy and sell by avoirdupois weight.
The denominations of Apothecaries Weight are pound,
ounce, dram, scruple, and grain.
20 grains (gr.) make 1 scruple 9
3 scruples
8 drams
12 ounces
1 dram
1 ounce
DIONOL
1 pound t
EXAMPLES.
1. Reduce 56 pounds to ounces.
2. Reduce 142 ounces to drams.
3. Bring 84 drams to scruples.
4. Bring 16 scruples to grains.
5. Reduce 8 ounces to scruples.
6. Bring 14 pounds to grains.
LONG MEASURE.
Facit 672
Facit 1136
Facit 252
Facit 320 gr.
Facit 1929
Facit 80640 gr.
Long Measure is used for lengths and distances.
The denominations of Long Measure are degree,
league, mile, furlong, pole, yard, foot, and inch.
12 inches (in.) make
3 feet
5 yards
40 poles (or 220 yds.)
8 furlongs (or 1760 yds.)
3 miles
60 geographic, or miles -
or}
69 statute
1 foot
ft.
1 yard
yd.
1 rod, pole, or perch P.
1 furlong
-fur.
1
mile
M.
1 league
L.
1 degree
- deg.
Note.--A hand is a measure of 4 inches, and used in
measuring the height of horses.
A fathom is 6 feet, and used chiefly in measuring ne
depth of water.
EXAMPLES.
1. Reduce 20 leagues to niles.
Facit 60 m.

SIMPLE REDUCTION.
35
2. Reduce 75 miles to furlongs.
3. Bring 42 furlongs to poles.
4. Bring 50 poles to yards.
5. Bring 16 yards to feet.
6. Bring 49 feet to inches.
7. Bring 10 yards to inches.
8. Reduce 3 leagues to poles.
CLOTH MEASURE,
Facit 600 fur.
Facit 1680 P.
Facit 275 yd.
Facit 48 ft.
Facit 588 in.
Facit 360 in.
Facit 2880 P.
By this measure cloth, tapes, &c. are measured.
The denominations of Cloth Measure are English ell
Flemish ell, yard, quarter of a yard, and nail.
4 nails (na.) make 1 quarter of a yard
4 quarters
3 quarters
5 quarters
1 yard -
1 ell Flemish
1 ell English
EXAMPLES.
1. Reduce 46 English ells to quarters.
2. Bring 5 Flemish elis to quarters.
3. Bring 22 yards to quarters.
4. Bring 40 qrs. to nails.
5. Bring 51 English ells to nails.
qr.
yd.
E. Fl.
E. E.
Facit 230 qr.
Facit 15 gr.
Facit SS qr.
Facit 160 na.
Facit 1020 na.
LAND MEASURE, OR SQUARE MEASURE.
This measure shows the quantity of lands.
The denominations of Land Measure are acre, rood,
square perch, square yard, and square foot.
144 square inches make 1
make 1 square foot
9 square feet
304 square yards -
40 square perches
4 roods
ft.
1 square yard
yd.
1
square perch
1 rood
P.
R.
-
1 acre
A.
EXAMPLES.
Facit 160 .
1. Reduce 40 acres to roods.
2. Reduce 15 roods to square perches. Facit 600 P.
3. Bring 28 square perches to square yards.
Facit 847 sq. yd.
4. Bring 42 square yards to square feet.
Facit 378 sq. ft
5. Bring 6 square feet to square inches.
Facit 864 sq. in.
D

36
SIMPLE REDUCTION.
6. Bring 12 acres to square perches.
Facit 1920 sq. P.
LIQUID MEASURE.
This measure is used for beer, cider, wine, &c.
The denominations of Liquid Measure are tun, pipe
or butt, hogshead, gallon, quart, and pint.
2 pints (pt.)
make
4 quarts
63 gallons
2 hogsheads
1 quart
1 gallon
1 hogshead
26.
gal.
hd.
1 pipe or butt pi. o bt.
2 pipes (or 4 hogsheads) 1 tun
T
Note. By a law of Pennsylvania, 16 gallons make
one half barrel; 31 gallons one barrel; 64 gallons one
double barrel; 84 gallons one puncheon; 42 gallons one
ticrce.
EXAMPLES.
1. Reduce 45 tuns to pipes.
2. Reduce 25 pipes to hogsheads
3. Bring 9 hogsheads to gallons.
4. Bring 40 gallons to quarts.
5. Bring 21 quarts to pints.
6. Bring 35 gallons to pints.
7. Bring 3 tuns to gallons.
DRY MEASURE.
Facit 90 pi.
Facit 50 hhd.
Facit 567 gal.
Facit 160 qt.
Facit 42 pt.
Facit 280 pt.
Facit 756 gal.
This measure is used for grain, fruit, salt, &c.
The denominations of Dry Measure are bushel, peck,
quart, and pint.
2 pints (pt.) make
1 quart
qt.
S quarts
4 pecks -
1 peck
1 bushel
pe.
Вге
EXAMPLES.
Facit 68 pe.
Facit 320 qi.
Farit 50 pt.
Facit 96 pt.
Facit 768 pt.
1. Reduce 17 bushels to pecks.
2. Reduce 40 pecks to quarts.
3. Bring 25 quarts to pints.
4 Bring 6 pecks to pints.
5. Bring 12 bushels to pints.
TIME.
The denominations of Time are year, month, week,
day, hour, minute, and second.

SIMPLE REDUCTION.
37
60 seconds (sec.)
60 minutes
24 hours
7 days
make
1 minute min.
1 hour
H.
1 day
D.
1 week
W.
Y
52 weeks, 1 day, and 6 hours, or
or} 1 year
365 days, and 6 hours
12 months (mo.)
1 year
Note.-The six hours in each year are not reckoned
till they amount to one day: hence, a common year
consists of 365 days, and every fourth year, called leap
year, of 366 days.
The following is a statement of the number of days
in each of the twelve months, as they stand in the
calendar or almanac :
The fourth, eleventh, ninth, and sixth,
Have thirty days to each affix'd:
And every other thirty-one,
Except the second month alone,
Which has but twenty-eight in fine,
Till leap year gives it twenty-nine.
EXAMPLES.
1. Reduce 8 years to months.
Facit 96 mo.
2. Bring 6 years to weeks, (supposing 52 weeks to
rake a year.)
3. Bring 3 years to days, (supposing
make a year.)
4. Reduce 25 weeks to days.
5. Reduce 12 days to hours.
6. Bring 14 hours to minutes.
7. Bring 9 minutes to seconds.
8. Bring 4 weeks to minutes.
When a sum or quantity is to be
denomination than its own, work by
RULE 2.*
Facit 312 W.
365 days to
Facit 1095 D.
Facit 175 D.
Facit 288 H.
Facit 840 min.
Facit 540 sec.
Facit 40320 min.
changed to a higher
Divide the given sum or quantity by that number of
its own denomination which makes one of the denomi-
* The reason of this rule may be seen by considering that as it takes
twenty shillings to make one pound, there must be just as many pounds
in any number of shillings as there are twenties in that number; and that
to find how many twenties there are in any number we divide it by 20.

38
SIMPLE REDUCTION.
nation to which it is to be changed. (See notes 1, 2,
and 3.)
When there are one or more denominations between
the denomination of the given sum or quantity and that
to which it is to be changed, first change it to the one
next higher than its own, and then to the next higher,
and so on. (See notes 4 and 5.)
Remainders are of the same denomination as
the sum or quantity divided. (See examples 2, 6, 8,
10, and 12, in English Money.)
EXAMPLES
ENGLISH MONEY.
Note 1.-To change shillings to pounds, divide
them by 20, because 20 shillings make 1 pound.
1. Bring 60 shillings to pounds.
20)60(3£
Facit 3£
60
2. Bring 135 shillings to pounds.
20)135(6£ 15s.
120
15s.
or
2|06|0
3 £
Facit 6 15s.
or
2013 5
6£ 15s.
Facit 6£
Facit 22£ 6s.
3. Bring 120 shillings to pounds.
4. Bring 446 shillings to pounds.
Note 2.—To bring pence to shillings, divide them
by 12, because 12 pence make 1 shilling.
5. Bring 72 pence to shillings.
6. Bring 195 pence to shillings.
Facit 6.s.
Facit 16s. 3d.
Note 3.-To bring farthings to pence, divide them
by 4, because 4 farthings make 1 penny.
7. Bring 36 farthings to pence.
8. Bring 763 farthings to pence.
Facit 9d.
Facit 190d. 3 qrs.
Note 4. To bring pence to pounds, bring them first
to shillings, and then bring those shillings to pounds.
9. Bring 480 pence to pounds.
Facit 2£
10. Bring 9655 pence to pounds. Facit 40£ 4s. 7d.
Note 5.-To bring farthings to pounds, bring
them first to pence, then bring those pence to shil-
lings, and then bring those shillings to pounds.

SIMPLE REDUCTION.
39
11. Bring 3840 farthings to pounds.
12. Bring 6529 farthings to pounds.
Facit 4£
Facit 6 16 s. 0 d. 4
FEDERAL MONEY.
Note 6. To reduce cents to dollars, divide them
by 100: or, which is the same thing, separate two
figures from the right of their number: and all on
the left of these will be dollars.
Facit 6 dollars.
Facit 12 dols. 50 cts.
Facit 45 dols. 75 cts.
1. Bring 600 cents to dollars.
2. Bring 1250 cents to dollars.
3. Bring 4575 cents to dollars.
Note 7.-To change fourths of a cent to cents, di-
vide them by 4-To change halves of a cent to cents,
divide them by 2-To change thirds of a cent to
cents, divide them by 3, &c.
4. Bring 20 fourths of a cent to cents.
5. Bring 125 fourths of a cent to cents.
6. Bring 75 half cents to cents.
cts.
Facit 5 cents.
Facit 31
Facit 37 cts.
Facit 144 cts.
7. Bring 432 thirds of a cent to cents.
Note 8. To change mills to dollars, divide them
by 1000: or, which is the same thing, separate three
figures from the right of their number, and all on
the left of these will be dollars.
S. Bring 4000 mills to dollars.
Facit 4 dols.
9. Bring 25750 mills to dollars. Facit 25 dols. 75 cts.
10. Bring 96532 mills to dollars.
Facit 96 dols. 53 cts. 2 m.
AVOIRDUPOIS WEIGHT.
1. Bring 75 cwt. to tons.
2. Bring 56 qrs. to cwt.
3. Bring 840lb. to quarters.
4. Bring 86 oz. to lb.
5. Bring 176 drams to oz.
6. Bring 958 qr. to tons.
7. Bring 9856 lb. to cwt.
5
Facit 3 T. 15 cwt.
Facit 14 cwt.
Facit 30 qrs.
Facit 5 lb. 6 oz.
Facit 11 oz.
Facit 11 T. 19 cwt. 2 qr.
TROY WEIGHT.
1. Bring 672 oz. to pounds.
2. Bring 145 dwt. to ounces.
3. Bring 560 gr. to dwt.
4. Bring 960 dwt. to lb.
5. Bring 9624 gr. to lb.
Facit SS cwt.
Facit 56 lb.
Facit 7 oz. 5 dwt.
Facit 23 dwt. 8 gr.
Facit 4 lb.
Facit 1 lb. 8 oz. 1 dwt.

40
SIMPLE REDUCTION.
APOTHECARIES WEIGHT.
1. Bring 672 ounces to pounds.
2. Bring 336 drams to ounces.
3. Bring 91 scruples to drams.
4. Bring 89 grains to .
5. Bring 192 to Z.
6. Bring 12660 gr. to .
Facit 56 b
Facit 42
Facit 303 1
Facit 499 gr
Facit 8
Facit 2 H 23 3
LONG MEASURE.
1. Bring 60 miles to leagues.
2. Bring 567 furlongs to miles.
3. Bring 640 poles to furlongs.
4. Bring 286 yards to poles.
5. Bring 52 feet io yards.
6. Bring 588 inches to feet.
7. Bring 2880 poles to leagues.
S. Bring 75 inches to yards.
Facit 20 L.
Facit 70 M. 7 fur.
Facit 16 fur.
Facit 52 P.
Facit 17 yds. 1 ft.
Facit 49 ft.
Facit 3 L.
Facit 2 yds. 0 ft. 3 in.
CLOTH MEASURE.
1. Bring 60 qr. to French ells.
2. Bring 464 qrs. to English ells.
3. Bring 750 qrs. to Flemish ells.
4. Bring 46 qrs. to yards.
5. Bring 480 nails to quarters.
Facit 10 E. Fr.
Facit 92 E. E. 4 gr.
Facit 250 E. FI.
Facit 11 yds. 2 qr.
Facit 120 qr.
6. Bring 95 nails to English ells. Facit 4 E. E. 3 qr. 3 na.
LAND MEASURE.
1. Bring 286 roods to acres.
Facit 71 A. 2 R.
2. Bring 360 square perches to roods. Facit 9 R.
3. Bring 4719 square yards to square perches.
Facit 156 P.
4. Bring 756 square feet to square yards. Facit 84 yds.
5. Bring 1728 square inches to square feet. Facit 12 ft.
6. Bring 966 square perches to acres.
Facit 6 A. O R. 6 P.
LIQUID MEASURE.
1. Bring 91 pipes to tuns.
2. Bring 50 hogsheads to pipes.
3. Bring 945 gallons to hogsheads.
4. Bring 163 quarts to gallons.
5. Bring 87 pints to quarts.
Facit 45 T. 1 P.
Facit 25 pi.
Facit 15 nhd.
Facit 40 gal. 3 qt.
Facit 43 qt. 1 pt.

SIMPLE REDUCTION.
41
6. Bring 59 hhd. to tuns.
7. Bring 6048 pints to tuns.
DRY MEASURE.
1. Bring 308 pecks to bushels.
2. Bring 246 quarts to pecks.
3. Bring 300 pints to quarts.
Facit 14 T. 3 hhd.
Facit 3 T.
Facit 77 bu.
Facit 30 pe. 6 qt.
Facit 150 qt.
4. Bring 486 quarts to bushels. Facit 15 bu. O pe. 6 qt.
5. Bring 384 pints to bushels,
TIME.
1. Bring 675 months to years.
Facit 6 bu.
Facit 56 Y. 3 mo.
2. Bring 208 weeks to years, (supposing 52 weeks to
make a year.)
Facit 4 Y.
3. Bring 4366 days to years, (supposing 365 days to
make a year.)
4. Bring 72 days to weeks.
5. Bring 4440 hours to days.
6. Bring 726 minutes to hours.
7. Bring 360 seconds to minutes.
8. Bring 30240 minutes to weeks.
Facit 12 Y. 6 D.
Facit 10 W. 2 D.
Facit 185 D.
Facit 12 h. 6 min.
Facit 5 min.
Facit 5 W.
PROMISCUOUS EXAMPLES.
1. How many shillings are there in 20 pounds?
Ans. 400.
2. What number of pounds do 65 shillings make?
Ans. 3£ 5s.
3. How many cents are there in 65 dollars?
Ans. 6500.
Ans. 34.
4. In 3400 cents, how many dollars?
5. How many quarters of a cent are there in 96 cents?
Ans. 384.
6. How many cents are there in 480 quarters of a
Ans. 120.
cent?
7. What number of half pence do 45 pence make?
Ans. 90.
8. How many three pences are there in 10 shillings?
Ans. 40.
9. How many six pences are there in 6 shillings?
Ans. 12.
10. How many shillings are there in 18 three pences?
Ans. 4s. 6d

42
COMPOUND ADDITION.
11. How many penny weights are there in 50 grains?
Ans. 2 dwt. 2 gr.
(Troy Weight.)
12. How many ounces are there in 15 pounds? (Troy
Weight.)
Ans. 180.
13. In S6 drams, how many ounces? (Avoir. Wt.)
Ans. 5 oz. 6 dr.
14. In 5 tons, how many hundred weight? (Avoir.
Weight.)
Ans. 100.
15. How many scruples are there in 15 drams?
Ans. 5.
16. How many ounces are there in 14 pounds?
(Apoth. Weignt.)
Ans. 168.
17. How many inches are there in 12 feet? Ans. 144.
18. In 25 furlongs, how many miles? Ans. 3 m. 1 fur.
19. How many nails are there in 3 quarters of a yard?
Ans. 12.
Ans. 15.
20. How many English ells are there in 75 quarters
of a yard?
21. How many square yards are there in 37 square
feet?
Ans. 4 yd. 1 ft.
22. In 125 roods, how many square perches?
Ans. 5000.
23. In 79 pints, how many quarts? Ans. 39 qt. 1 pt.
24. How many gallons are there in three hogsheads?
Ans. 189.
Ans. 225.
25. In 900 pecks, how many bushels?
26. How many minutes are there in 360 seconds?
Ans. 6.
27. How many days are there in 12 weeks? Ans. 84.
COMPOUND ADDITION.
Compound Addition is the adding of sums or quanti-
ties which consist of several denominations.
RULE.
Place the sums or quantities so that the numbers of
the same denomination may stand directly under each
other, and form a separate column; then add up the
several columns, successively, beginning with the one
of the lowest denomination: if the amount of either of
•

COMPOUND ADDITION.
43
..
the columns be not as much as 1 of the next higher
denomination, set it down; but if it be, reduce it to
that denomination, and add the number it contains of
said denomination into the column of the same.
If a remainder occur on reducing the amount of any
column, set it under that column.
Proof: as in Simple Addition.
MONEY.
PENCE TABLE.
TABLE OF SHILLINGS.
d.
s.
d.
S.
£
s.
20 pence make
1 8
20
1 0
30
2
6
30
1 10
40
3 4
40
2 0
50
4
2
50
60
5 0 60
1
3
2 10
0
70
80
90
100
110
120
240
8
2110
- 10
20
0
8 |
0 120
130
5 10 70
6 8 80
7 6
8
9
4100
3 10
4 0
90
4 10
5
0
5 10
6
0
6 10
EXAMPLES.
£
s. .d.
10 2 3
45 10 2
36 1 1
20 3 0
120 0 5
£ S d.
175 12 6
280 10 4
362 8 3
484 13 10
40 18 11
£
456 12 52
320 12 0
8.
Sལས
400 10 14
45 6 24
1000 18 34
231 16 11
£ S. d.
1344 3 10
2223 19 03
eR
S.
d.
F
£
s. d.
75 10 2
14 0 0
12 2 0
25 0 6
13 3 0
24 10 6
10 0
0
14 6 8
5 3 0
36 4 8
15 3 6
17 10 9
4 2 1
45 0 11
4 5 0
18 0 0
6
1 8
14 0 0
2 7 6
20 0 0
E

44
COMPOUND ADDITION.
£ s. d.
174 14 101
240 18 7
£
s. d.
4900 16 10
£ S. d.
340 0 0
222 0 0
1346 18 0
3704 0 34
320 14 8
2000 0 0
642 7 21
4540 10 24
2432 2 3
438 14 10₫
3200 15 01/2
346 3 5
4620 18 4
3460 15 93
3420 10
23
11. Add the following sums: viz. 15£ 6s. 3d.-75£
10s. 2d.-65£ and 94£.
12. Add 45£ 12s. 3d. -56£ 10s.-346 18s.--and
1£ 19s. 24d.
13. Add 145£-72£ Os. 3d.-14£ 8s. 94d.-18s.
92d.—and 42£ 2s. 4 d.
14. Add 410£ 5s.—1600£ 18s.-and 4426£ 19s.
TROY WEIGHT.
lb. oz.
lb. oz. dwt.
lb. oz.dwt.
.gr.
37 11
93 11 18
17 9 11 15
62 0
6 0 1
82
200
72 10
14
4 12
12
8 16 0
13 4
72
11 3
9
4 0 0
4. Add 276. 11oz. 10dwt. 2gr.—15lb. 10oz. 2dwt.—
and 145lb. 2oz. 2dwt.
5. Add 14lb. 2oz.-2lb. Ooz. 10dwt.-15lb. 5oz.
19dwt. 14gr.-and 25lb. 10oz.
AVOIRDUPOIS WEIGHT.
T. cwt.qrs. T. cwt. qrs. lb. cwt. gr. lb. oz. dr.
40
11 3
15 10 2
18 0 1
9 12 0
3 19 3 27
2 12 1
8 3 12 15 8
12
12 0
2
9 4
10
5
0
0
8 2 1
0 3
3
2
10
91
26
8 2
4. Add 20 tons, 2 hundred weight, 2 quarters; 12
tons; 15 tons, 2 quarters; and 2 tons.
5. Add 15 hundred weight, 3 quarters, 27 pounds;

COMPOUND ADDITION.
45
17 hundred weight, 15 pounds; and 1 hundred weight,
10 pounds.
APOTHECARIES WEIGHT.
# 339
NO♡ Q
B7482
#b
39 gr.
2
10
4 0 2
4
2 1
0
2 1
12
7
4
0
14 11 7
2
0
0
4
9
6
3
1
1
1
2
29
0
4
0
7 1
2 1
4. Add 10 pounds, 7 drams, 2 scruples; 15 pounds,
11 ounces, 2 drams; 45 pounds, 4 ounces; and 36
pounds.
5. Add 14 pounds, 2 ounces; 1 pound, 3 grains; 2
ounces, 3 drams, 2 scruples; and 4 drams, 12 grains.
LONG MEASURE.
L. M. F. P
8 2 7 16
હું ૨ ૬
L. M. F
yd. ft. in.
7
2 7
2
9
6
1
4
20
2
7
1
1
1
0
12 1
0
24
1
4
9
10
1
3
14 0 3
50
0
0
0
12
0 4
4. Add 14 leagues, 2 miles, 6 furlongs; 4 leagues, 4
furlongs, 30 poles; 1 league, 2 miles, 15 poles; and 42
leagues.
5. Add 2 yards, 2 feet, 9 inches; 1 yard, 11 inches;
1 foot, 6 inches; and 10 yards, 5 inches.
CLOTH MEASURE.
yd. qr. na.
E. E. gr. na.
E. Fl. qr. na.
79 2 1
86
4 2
14 2 1
25 1 3
44
3 0
25
1 0
14 3 2
21
0 2
14
0 3
46 2 1
5 0 3
25 0 0
4. Add 15 yards, 3 quarters, 2 nails; 45 yards, 2
quarters; 1 yard, 3 nails; and 125 yards.
5. Add 14 English ells, 3 quarters; 25 English ells,
2 quarters, 3 nails; and 3 quarters, 1 nail.

46.
COMPOUND ADDITION.
LAND MEASURE.
A. R. P.
A. R. P.
75 3 2
150
3 39
yd. ft. in.
8 2 12
24 0 0
265
2
11
10
1 95
98 1 0
284
1
9
12
1 115
75 3 0
326 0
0
20
0 46
4. Add 125 acres, 2 roods; 400 acres, 3 roods, 28
perches; 56 acres, 20 perches; and 500 acres.
5. Add 15 yards, 2 feet; 2 yards, 1 foot; 14 yards,
2 feet; and 25 yards.
T. hhd. gal.
LIQUID MEASURE.
hhd. gal. qt.
gal. qt. pt.
12 2 45
2 15 3
24
2 1
10 1
17
4 14
2
14
1
0
24 0
0
10
6
2
6
3
0
20 3 0
12
U O
8 2 1
4. Add 10 tuns, 3 hogsheads, 15 gallons; 4 tuns, 2
hogsheads, 9 gallons; 2 hogsheads, 46 gallons; and 14
tuns.
5. Add 14 gallons, 3 quarts, 1 pint; 25 gallons, 2
quarts; 2 gallons, 2 pints; and 13 gallons.
DRY MEASURE.
bu. pe. qt.
bu. pe. qt.
bu. pe. qt. pt.
25 3 2
18 2 0
21 1 0
40 0 0
10 3 2
117 1 3
215 2 4
450 3 0
36 1 1 1
48
3 2 1
50 0 0 0
3
7
1
4. Add 115 bushels, 3 pecks, 7 quarts; 345 bushels,
2 pecks; 40 bushels, 4 quarts; and 375 bushels.
5. Add 2 bushels, 3 pecks, 4 quarts, 1 pint; 4 bushels,
2 pecks, 2 quarts; and 1 peck, 6 quarts, 1 pint.

COMPOUND ADDITION.
47
TIME.
Y. mo.
75
W. D. H. min.
D. H. min. sec.
5
2
1 10 40
4 20 56 54
16 10
1
6
9
20
3 19
25 22
14
11
3 5
20
12
2
8
0 3
25
2
2
3
7 56
6 O 0
4. Add 10 years, 3 months; 45 years, 6 months; 75
years, 11 months; 15 years; and 96 years.
5. Add 7 weeks, 1 day, 5 hours, 45 minutes; 2
weeks, 4 days, 22 hours; 6 days, 15 hours, 10 minutes;
and 5 hours.
APPLICATION.
1. Bought an English Reader, for 5s. 74d.; a Sequel,
for 6s. 6d.; an Arithmetic, for 35. 9d.; and a Slate for
2s. 4d. What do they all come to? Ans. 18s. 3d.
2. If a storekeeper buy cloth to the amount of 310£
7s. 6d.; linen to the amount of 37£ 5s.; and groceries
to the amount of 209£ 15s. 44d.: what sum must he
pay for the whole?
Ans. 557£ 7s. 10дd.
3. Bought a horse for 17£ 10s. 6d.; a cow for 5£ 14s.
7d.; and a quantity of hay for 6£ 12s. 6d. What is the
Ans. 29£ 17s. 7d.
4. Laid out in market, for a pair of fowls, 5s. 7½d.;
for a goose, 7s. 6d.; for a bushel of potatoes, 3s. 9d.; for
a piece of beef, 15s.; and for turnips, 4s. Sd. How much
was laid out in all?
Ans. 1 16s. 61d.
amount?
5. Bought of a silversmith, dishes, weighing 16lb.
10oz. 13dwt.; plates, weighing 35lb. 10oz. 11dwt.; table-
spoons, 61b. 11oz.; and tea-spoons, 2lb. 8oz. What was
the weight of the whole?
Ans. 62lb. 4oz. 4dwt.
6. A grocer bought four hogsheads of sugar, which
weighed as follows-No. 1, Scwt. 1qr. 261b.; No. 2,
9cwt. 3qrs. 11lb.; No. 3, 12cwt. 2qrs. 19lb.; No. 4,
12cwt. 2qrs. What did the whole weigh?
Ans. 43cwt. 2qrs.
7. Sold three boxes of spice weighing as follows-
No. 1, 1qr. 12lb. 9oz. 14dr.; No. 2, 2qrs. 13lb. 15oz.
Sdr., No. 3, 1qr. 25lb. 13oz. 12dr. How much was
the whole weight? Ans. lcwt. 1qr. 24lb. 7oz. 2dr.
I 2

48
COMPOUND ADDITION.
8. If a druggist mix several simples together, the
first, 4 ounces, 3 drams, 2 scruples; the second, 3
ounces, 1 dram, 1 scruple, 17 grains; the third, 1 pound,
7 ounces, 3 drams, 1 scruple: what will be the weight
of the mixture?
Ans. 2ft 33 03 19 17gr.
9. Admit a man travelled in one day, 27 miles, 2
furlongs in another, 32 miles, 7 furlongs, 33 perches;
in another, 19 miles, 7 furlongs, 16 perches; and in an-
other, 12 miles, 5 furlongs: how far did he travel in all?
Ans. 92m. 6fur. 9P.
10. There are three pieces of silver wire: the first
measures 10 yards, 2 feet, 6 inches; the second, 15
yards, 1 foot, 4 inches; the third, 20 yards, 11 inches:
what is the length of the whole? Ans. 46yds. Ift. 9in.
11. There are four pieces of linen: the first contains
27 yards, 2 quarters; the second, 41 yards, 3 quarters;
the third, 36 yards, 1 quarter; and the fourth, 33 yards,
2 quarters. How many yards are there in the four
pieces?
Ans. 139 yds.
12. Bought three pieces of lace containing as follows
-No. 1, 17 yards, 3 quarters, 2 nails; No. 2, 25 yards,
2 quarters, 1 nail; No. 3, 32 yards, 3 quarters, 3 nails:
How many yards were bought in all?
Ans. 76vds. 1qr. 2na.
13. A person has three farms: the first contains 120
acres, 3 roods; the second, 256 acres, 1 rood; and the
third, 300 acres. How many acres are in all? Ans. 677.
14. Sold two casks of cider, one of which contained
31 gallons, 3 quarts, and the other 36 gallons, 2 quarts,
1 pint. How much was there in the two?
Ans. 68gals. 1qt. 1pt.
15. There are three bags of wheat: the first contains
2 bushels, 3 pecks, 7 quarts; the second, 3 bushels, 3
pecks, 4 quarts, the third, 4 bushels. How much is
in the three bags ?
Ans. 10bu. 3pe. 3qt.
16. Bought 136 bushels of corn of one man; 197
bushels, 2 pecks, of another; 200 bushels, 1 peck, 6
quarts, of a third; and 764 bushels, 3 pecks, 7 quarts,
of a fourth. How much was bought in all?
Ans. 1298bu. 3pe. 5qt.
17. A person who was born in Philadelphia, resided
in that place till he was 21 years, 3 weeks old.
then went to Wilmington, spending 2 days on the road.
He

COMPOUND SUBTRACTION.
49
He resided in Wilmington 5 years, and at the end of
that time removed to Baltimore; the journey occupy-
ing 3 days. He remained in Baltimore 2 years, 3
weeks, and 3 days, and then removed to Richmond,
being 5 days in travelling thither. What was his age
at the time he arrived in Richmond?
Ans. 28 years, 7 weeks, and 6 days.
COMPOUND SUBTRACTION.
Compound Subtraction teaches to find the difference.
between any two sums or quantities, which consist of
several denominations.
RULE.
Place the sums or quantities as in Compound Addi-
tion, with the less under the greater; then, beginning
with the lowest denomination, subtract each under
number from the one above it, and set down the re-
mainder but if the number of either denomination in
the under sum be greater than the one above it, sub-
tract it from as many of that denomination as will make
one of the next higher, add the difference to the upper
number, set down the amount, and carry 1 to the under
number of the next higher denomination.
Proof as in Simple Subtraction.
EXAMPLES.
MONEY.
£ S. d.
From 5 0 6
Take 2 9 3
£
s. d.
£
S. ૧.
10 6 3
14€
18 91
5 7
6
104
12 104
Rem. 2 11 3
4 18 9
4) 5 114
£ 8. d.
L S.
£
d.
£
>
s.
d.
10 3 37 12 5
6 2 27 18 9
25 1 9
46
2
3
14
5
6
25
1
9

50
COMPOUND SUBTRACTION.
£ S. d.
45 6 32
22 4 61
£
S. d.
£
S. d.
142 10 33
45 9 24
2640 18 114
1221 19 61
11. Subtract 24 pounds, 10 shillings, and 6 pence,
from 36 pounds, 9 shillings, and 3 pence.
12. Subtract 26 pounds, from 120 pounds, 15 shil-
lings, and 9 pence.
13. Subtract 9000 pounds, from 9672 pounds, 18
shillings, and 11½ pence.
14. Subtract 45 pounds, 14 shillings, and 34 pence,
from 500 pounds.
lb. oz.
TROY WEIGHT.
lb. oz. dwt.
lb. oz. dut. gr.
From 48 2
Take 10 1
15 5 2
45 9 4 3
12 0 2
15 6 18 17
Rem.
4. Subtract 14 pounds, 9 ounces, from 65 pounds, `3
ounces, 10 penny weights.
5. Subtract 10 pounds, 6 ounces, 10 pennyweights,
11 grains, from 15 pounds, 6 grains.
AVOIRDUPOIS WEIGHT.
T. cwt. qr. T. cwt. qr. lb.
52 12 3 15
cwt.qr. lb. oz. dr.
17 0 0 0 0
From 45 11 3
Take 15 10 2
24 10 0 26
6 3 21 15 9
Rem.
4. Subtract 76 tons, 18 hundred weight, 3 quarters,
from 195 tons, 2 hundred weight, 2 quarters.
5. Subtract 14 pounds, 6 ounces, 3 drams, from 20
pounds, 2 ounces.
#b
From 48
APOTHECARIES WEIGHT.
Hb 339
ROUND
૨૦૦૨
Take 10 12
Remi.
th
22 0 1 1 0
42
19 3 2 1
•NOQ
9 gr.
0 1
25 0 0 0 3

COMPOUND SUBTRACTION.
51
4. Subtract 16 pounds, 5 ounces, 2 drams, from 24
pounds, 10 ounces, 3 drams.
5. Take 3 ounces, 3 drams, 2 scruples, from 5 pounds,
9 ounces, 2 drams, 2 grains.
LONG MEASURE.
L. M. fur.
L. M. fur. P.
yds. ft. in.
From 24 1 7
Take 18 2 4
56 1 0 19
6 2 10
10 0 7 20
3 2 7
Rem.
4. Subtract 45 miles, 5 furlongs, 20 poles, from 320
miles, 3 furlongs, 36 poles.
5. Subtract 15 yards, 2 feet, 6 inches, from 36 yards,
1 foot, 11 inches.
CLOTH MEASURE.
qrs.na.
yds.qrs.na.
From 71 3 1
E. E.
42 0
E. Fl. qrs.na.
51 2 2
Take 14 2 3
19 2 3
42 2 1
Rem.
4. Subtract 95 yards, 3 quarters, 2 nails, from 156
yards, 2 quarters, 3 nails.
5. Subtract 14 English ells, 1 quarter, 2 nails, from
52 English ells, 3 quarters, 2 nails.
LAND MEASURE.
A. R. P.
From 96 3 36
Take 25 2 39
195
A. R. P.
2 2
36 3 1
yds. ft. in.
25
2 72
14 7 10
Rem.
4. Subtract 36 acres, 2 roods, from 900 acres, 3
roods, 16 perches.
5. Subtract 72 acres, from 360 acres, 2 roods, 29
perches.

52
COMPOUND SUBTRACTION.
LIQUID MEASURE.
Tun. hhd. gal.
From 25 3 45
Take 17 2 62
Rem.
hhd. gal. qt.
gal. qt. pt.
45 13 2
75
3 1
25 2 3
22 1 0
4. Subtract 14 tuns, 2 hogsheads, 10 gallons, from
24 tuns, 1 hogshead, 9 gallons.
5. Take 22 hogsheads, 2 quarts, from 95 hogsheads,
10 gallons, 3 quarts, 1 pint.
DRY MEASURE.
bu. pe. qt.
bu. pe. qt.
pe. qt. pt.
From 95 3 2
84 2 1
3
7 0
Take 22 0 1
36 3 2
2
3 1
Rem.
4. Subtract 125 bushels, 3 pecks, 2 quarts, from 195
bushels.
5. Subtract 450 bushels, from 500 bushels, 3 pecks.
TIME.
Y. mo.
From 75 3
Take 25 4
W. D. H.
32 6 20
12 4 22
D.
H. min. sec.
36
14 30 25
15 12 25 32
Rem.
4. Subtract 125 years, 9 months, from 450 years, 11
months.
5. Take 122 days, 18 hours, 36 minutes, from 200
days, 18 hours.
APPLICATION.
1. A merchant has in his desk 375£ 10 s. If he take
out 122£ 11 s. 3 d. to pay for goods, how much will
remain ?
Ans. 252£ 18 s. 9 d.
2. A person borrowed of me 125£ 10 s. 6 d., but has
since paid me 75£ 18 s. 2 d. How much does he still
owe me?
Ans. 49£ 12 s. 4 d.

COMPOUND SUBTRACTION,
53
3. If a merchant buy a quantity of tobacco, for 1500
pounds, 16 shillings, and afterward sell it for 1595
pounds, 19 shillings, and 9 pence; how much will he
gain by the transaction?
Ans. 95£ 3 s. 9 d.
4. If a person sell goods for 136 pounds, 12 shillings,
and 6 pence, which cost him 149 pounds, 10 shillings,
and 3 pence, how much will he lose by the sale?
Ans. 12£ 17 s. 9 d.
5. A silversmith had 26 pounds, 9 ounces, 10 penny-
weights of silver, but sold 18 pounds, 16 pennyweights,
10 grains. How much had he left?
Ans. 8 lb. S oz. 13 dwt. 14 gr.
6. A grocer has 13 hundred weight, 2 quarters, 16
pounds of sugar. If he sell 9 hundred weight, 2 quar-
ters, 7 pounds, how much will remain unsold?
Ans. 4 cwt. 9 lb.
7. There is a quantity of spice, which, with the box
that contains it, weighs 34 pounds, 10 ounces, 1 dram;
the box itself weighs 10 pounds, 10 ounces, 2 drams.
What is the weight of the spice? Ans. 23lb. 15oz. 15 dr.
S. If out of 6 pounds, 10 ounces, 6 drams, 2 scruples,
of medicine, be taken 4 pounds, 5 ounces, 4 drams, 1
scruple, 17 grains; what quantity will remain?
J
+
Ans. 2b 53 23 09 3 grs.
9. A certain rope is 365 yards, 1 foot, 6 inches long.
If 84 yards, 2 feet, 4 inches, be cut off from it, how
long will the remainder be? Ans. 280 yds. 2 ft. 2 in.
10. The distance from Philadelphia to Trenton is
about 30 miles, 3 furlongs, 16 poles. A person, going
from one place to the other, stopped at an inn, when
he had travelled 18 miles, 3 furlongs, 26 poles. How
much further had he still to go?
Ans. 11 M. 7 fur. 30 P.
11. Bought 145 yards, 3 quarters. of cloth, and sold
thereof 95 yards, 2 quarters, 3 nails. How much re-
mains?
Ans. 50 yds. 1 na.
12. If from a piece of cambrick, containing 25 yards,
3 quarters, 3 nails, there be taken 16 yards, 2 quarters,
how much will be left?
Ans. 9 yds. 1 qr. 3 na.
13. A farmer had 450 acres, 3 roods of land, but
gave
his son 150 acres, 3 roods, 25 perches. How much
ad he remaining?
Ans. 299 A. 3 R. 15 P.
14. Bought several casks of cider, containing in all,

54
COMPOUND SUBTRACTION.
120 gallons, 3 quarts; and disposed of one cask which
contained 31 gallons, 2 quarts, 1 pint. How much is
there in the other casks?
Ans. 89 gal. 1 pt.
15. From a barrel of beer containing 31 gallons, 2
qts., there has been drawn 15 gallons, 2 quarts, 1 pint.
How much remains in the barrel? Ans. 15 gal. 3 qt. 1 pt.
1
16. Out of a granary which contained 500 bushels of
wheat, there has been taken 374 bushels, 2 pecks, 7
quarts. What quantity remains? Ans. 125bu. I pe. 1 qt.
17. Charles was bound as an apprentice for 7 years.
He has served 2 years, and 5 months. How long has
he still to serve?
Ans. 4 Y. 7 mo.
18. James is 13 years, 2 months old, and John 9
years, 3 months. How much older is James than John?
Ans. 3 Y. 11 mo.
Note. The interval or space of time between two
given dates is thus found :---Set the prior date under the
subsequent date; and when the lower number of days
is greater than the upper, take it from as many days as
are in the month of the prior date, add the difference to
the upper number, and set down the amount; then carry
one to the months of the prior date, and subtract as in
the foregoing examples.
19. Henry was born on the 20th of the 8th month,
1789, and Charles on the 18th of the 9th month, 1808.
What is the difference in their ages?
Ans.
Y. mo. da.
1808 9 18 subsequent date.
1789 8 20 prior date.
19 0 29
20. A person was born on the 18th of the 5th month,
(May,) 1781. What was his age on the 12th of the 7th
month, (July,) 1808.
Ans. 27 Y. 1 mo. 25 D.
21. A bond was given the 21st of the 11th month,
(November,) 1798, and was taken up the 12th of the
9th month, (September,) 1811. What time elapsed fror
the day the bond was given till the day it was tak
up?
Ans. 12 Y. 9 mo. 21

COMPOUND MULTIPLICATION.
55
COMPOUND MULTIPLICATION.
Compound Multiplication is the multiplying of any
sum or quantity which consists of divers denominations.
When the multiplier does not exceed 12, work by
RULE 1.
Multiply the several denominations of the given sum
or quantity, one after another, beginning with the
lowest: if the product of either of them be not equal
to one or more of the next higher denomination, set it
down: but if it be, reduce it to that denomination, and
add the number it contains thereof to the product of the
same; and so proceed. If, on reducing the product of
any denomination, there be a remainder, it must be
placed under that denomination.
PROOF.
Double the multiplicand, and multiply by half the
multiplier.
#
EXAMPLES.
MONEY.
£ s. d.
5
42
༢༠ ༤
£ s.
10 15 6
3
d.
£
s. d.
21 9 24
£
s. d.
15 0 9
4
5
10 8 4
32 6 6
85 16 9
75 4 04
£ s. d.
4 2 1
3
£
s. d.
12 3 9
6
બે વર્ષ
£
s. d.
£
s. d.
25
4 1
96 4 9
7
S
9. Multiply 2
10.
11.
الله
عالم
£
S.
Q.
6
4 by 5 Product 11
16
34 by 6
1
2
64 by 9
L
S.
d.
11
S
4 17
10
2
1 3
2 by 10
12
91 by 11
r 2 14 by 12
11 11
7
13 5 3
F

56
COMPOUND MULTIPLICATION.
WEIGHTS AND MEASURES.
lb. oz.dwt.gr. T.cwt.qrs. lb. oz. dr. t 3 3 9 gr.
17 5 12 6 6 17 3 13 2 15 4 10 7 2 13
3
4
5
L. M. fur. P.
yds. ft. in.
yds. qrs.na
15 2 7 30
14 2 11
16 3 3
6
7
8
E.E.qrs.na.
4 1
42
A. R. P.
47 3 15
9
2
T.hhd. gal. qt. pt.
2 3 40 3 1
10
bu. pe. qt.
bu. pe. qt.
6
37
5
14 3 2
6
W. D. H. min. sec.
4 5 20
20 32 10
7.
When the multiplier exceeds 12, and is the product
of two factors in the multiplication table, work by
RULE 2.
Multiply the given sum by one of said factors, and
then multiply the product by the other factor.
Proof: Change the factors.
F
s. d.
EXAMPLES,
1. Multiply 3 2 6 by 14.
£
3
S.
GS OR
2
d.
6
2
F
£ 5. d.
Product 43
15 7
S. d.
3 2
6
7
21 17 9
2
Product 43 15 7
Proof 43 15 7
6 5 1
7

COMPOUND MULTIPLICATION
57
£ S. d.
£
s. d.
2. Multiply 1
12
3 by 15
Product 24 3 9
3.
2
14
1½ by 54
146
2 9
4.
11
14 by
96
53
6 0
5.
12
3 by 35
21
9 5
ext
6.
7 6 by 120
45 0 0
When the multiplier is not the exact product of any
two factors in the multiplication table, work by
RULE 3.
Use those two factors whose product is the least short
of the multiplier; then multiply the given sum by the
number which supplies the deficiency, and add its pro-
duct to the sum produced by the two factors.
EXAMPLES.
1. Multiply 2£ 1s. 3d. by 68.
Product 140£ 5s.
£
S.
d.
£ s.
d.
2
3x2
2
1
3×2
11
6
22 13
}
9
12
7 6
11
136 2
6
4 2
6
136
2
6
4
2
6
Prod. 140 5 0
LS. d.
2. Multiply 3 13 4 by 31
3.
4.
5.
6.
1 18 10 by 68
1 11 6 by 23
16 6 by 47
16 8 by 112
Proof 140 5 0
£ S.
Product 113 13
132
When the multiplier is greater tha
any two factors in
plication

58
COMPOUND MULTIPLICATION.
ten by the tens figure,-the product of the second ten
(if any) by the hundreds figure, &c.; then add the pro-
ducts of these several figures together, and their amount
will be the product required.
s. d.
EXAMPLES.
s. d.
1. Multiply 1 7 by 276. 2. Multiply 2 6 by 3452
S.
d.
1
7X6
16
10
3X7
10
8 2 6
2
16
5
0
9
9
9
5 13
Prod. 22 8 6
Multiply
F
s. d.
2
6x2
10
1 5 0x5
10
12 10 0x4
10
125 0 0
3
譬
​375
OK
0
5 0
6
5
50 0 0
Prod. 431 10 0
s.
d.
£ .S.
d
1
2 by 195
Product 11
1
73
02.02.09
3
9
3 by 435
3 by 407
4 by 820
3 6 by 165
62 by 276
114 by 2123
APPLIC
27 3 9
66 2
95 13
193 17 6
7 9 6
99 10 34

COMPOUND DIVISION.
4. Bought 8 yards of linen for 3 L.
was the price per yard?
11 s. 8 d. Wha
Ans. 8 s. 11 d
5. Sold 132 yards of cloth for 221 L. 18 s. 6 d. How
much was it per yard?
Ans. 1 L. 13 s. 7 d.
6. What is the price of a bushel of wheat, when 42
bushels are sold for 17 L. 13 s. 6 d.?
Ans. 8 s. 5 d.
PROMISCUOUS QUESTIONS.
1. Bought 2 pieces of linen, one of which contained
30 yards, and the other 25 yards; the price was 7s. 6 d
per yard: what was the cost of the two pieces?
Ans. 20 L. 12 s. 6
2. Sold one piece of cloth, containing 41 yards, a
2 L. 18 s. per yard; and another piece containing 3
yards, at 2 L. 6 s. 6 d. per yard: what is the amount o
the whole?
Ans. 202 L. 12:
3. A person has 500 L. 18 s. 9 d. He owes to on
man 25 L 10 s.; to another, 76 L. 18 s. 9 d.; to anoth
175 L. 10 s.; and to another, 100 L. What sum
he have left after paying these debts? Ans. 12
4. A grocer has 10 bags of coffee, weighing each
pounds, and 2 bags, weighing each 160 pounds. I
sell 560 pounds, what quantity will remain?
Ans. 900
5. Bought 4 pieces of linen, containing 25 yards,
quarters, each, and 3 pieces containing 32 yard
quarters, each; from which was afterwards cold
yards: what number of yards was then remaining?
Ans. 75 yds.
6. A farmer has three tracts of land, the first coro
125 acres, 3 roods; the second, 200 acres, 2 rood
perches; the third, 175 acres, 10 perches. He in
dividing this land equally between his two sons: wh
will be the share of each son? Ans. 250 A. 2 R. 341
7. A person, at his decease, left propety to il
nt of 2425 L. 19 s. His will directed that
should be given to the poor, and that
r should be divided, equally, amongst
What is the portion of each daughter?
Ans. 741 L. 199

COMPOUND DIVISION,
8. Bought 10 yards of muslin, at 3 s. per yard; 6
yards of tape, at 3 d. per yard; and 7 yards of linen, at
7 s. 6 d. per yard: how much did the whole amount to?
Ans. 4 L. 4 s.
9. Sold 19 bushels of wheat, at $2.37% per bushel; 15
bushels of rye, at 75 cents per bushel; and 95 bushels
of Indian corn, at 873 cents per bushel: how much did
the whole sale amount to?
Ans. $139.50.
10. If I buy 15 pounds of sugar, at 103 cents per lb.,
and 17 pounds of rice, at 54 cents per pound, and 19
pounds of candles, at 174 cents per pound; how much
must I pay for the whole?
Ans. $5.78%
11. What is the a:nount of the following 'bill?
JAMES JOHNSON,
"
Philadelphia.
Bought of Samuel Williams,
7 yards of coating at 17 s. 6 d. a yard.
of broad cloth at 45
18
9
ditto
at 48
9
23
of cassimere
at 18
4
37
ditto
at 21
6
97
drugget
at 9
6
Ans. £180.5 101
ל:
A TABLE
Foreign Coins, &c., with their value in Federal
honey, as established by a late Act of Congress.
and Sterling,
nd of Ireland,
da of India,
al of China,
-
lill-ree of Portugal,
able of Russia,
e of Bengal, -
ilder of the United Netherlands,
Banco of Hamburgh,
e Tournois of France,
D. c.
m.
4, 44
4
4, 10
0
1, 94
0
1, 48
0
1, 24
0, 66
0, 55
0,
Plate of Spain,

A TABLE OF COINS
Which pass current in the United States of North America, with their Sterling and Federal Value.
NAMES OF COINS.
Standard
weight.
Sterling Mo-
Britain.
ney of Great shire, Massa-
New-Hamp-
chusetts,
Rhode Island,
Connecticut,
and Virginia.
New-York and
New-Jersey,
North Caro-
lina.
Pennsylvania,
South
lina
Delaware, and
Caro-
and
Georgia.
Federal value.
Maryland.
Dollars
Cents
Mills
65
GOLD.
A Johannes,.
...
An half Johannes,.
dwt. gr.
L. s. d.
L. S.
d.
L. s. d.
L. 8. d.
L. s. d.
D.
C. M.
18
3 12
0
4 16 0
6
8 0
6 0 0
4
0 0
16
00 0
9 0
1 16 0
2 8
0
3 4 4
0
3 0
0
2
0
0
8 00 0
1
A Doubloon,..
16 21
3
6
0
4
8
5 16 0
5.12
6
3 10
0
14
93 3
4
1 Moidore,..
6 18
1
7
0
1 16
0
2 8
0
2 5
0
1
8
0
6
00 0
Zoomo
An English Guinea,..
5 6
1 1 0
1
8
French Guinea,.
5
5
1
0
1
A Spanish Pistole,.
4 6
0 16 6
1
A French Pistole,..
4 4
0 16
0
1
∞ 72 2
0
1 17
0
1 15 0
1
1
9
4
66 7
6
1 16
0
1 14 6
1
1
5
4
60 0
2 0
1
9
0
1
8 0
0 18 0
3
77 3
0
1
8
0
1
7
7.6
0 17 6
3
66 7
SILVER.
An English or French
Crown,.
18 0
0 5 0
0 6 8
8 9
0 5 0
1 10 0
The Dollar of Spain, Swe-
den, or Denmark,.
17
6
0
4 6
0 6 0
0
8
0
0
7 6
0 4 8
1.00
0
An English Shilling,
3 18
0
10
0
1 4
0
1
9
0
1 8
0
1 1
22 2
A Pistareen,.
3 11
0
0 103
1 2
0
1
7
0
1 6
0
0 11
0
20 0
* All other Gold Coins of equal fineness, at 89 cents per dwt., and Silver at 111 cents per oz.

66
COMPOUND REDUCTION.
COMPOUND REDUCTION.
Compound Reduction teaches to change any sum or
quantity which consists of several denominations, to a
given denomination; and to change a sum of one kind
of money to a given denomination of another kind.
When a sum or quantity, consisting of several de-
nominations, is to be changed to a given denomination,
work by the following
RULE.
Reduce the highest denomination to the next lower
one, and this again to the next lower, and so on; ob-
serving to add to the amount of each denomination the
number there is of that denomination in the given sum
or quantity.
EXAMPLES.
MONEY.
Reduce 25L. 10s. 64d. to farthings.
L. S. d.
Or thus,*
25 10
6
L. S.
d.
20
25 10
64
20
500
10
510
qrs.
4)24507 •
12
510
12)6126+3
12
6126
4
6120
6
210)51|8+6
24507 Proof 252.10s. 62d
6126
4
24504
3
24507 farthings.
* The frimer of these two operations is given merely to render the
application of the rule more intelligible: the method of adding in the
denominations of the sum, as in the latter operation, should be explained
to the scholar.

COMPOUND REDUCTION.
67
2. Reduce 36 L.
3. Reduce 32 L.
4. Bring 102 L.
15 s. to shillings.
12 s. 4 d. to pence.
19 s. 74 d. to farthings.
Facit 735 s.
Result 7828 d.
Result 98861 qrs.
5. Bring 21 L. 10 s. 63 d. to farthings.
Result 20666 qrs.
6. Reduce 137 L. 15 s. 63 d. to farthings.
Result 132267 qrs.
7. Bring 45 L. 3 s. 14 d. to halfpence.
Result 21675 halfpence.
8. Bring 10 L. 10s to halfpence. Result 5040 halfp.
9. Bring 5 L. 6 s. to farthings.
10. Bring 2 L. 0 s. 64 d. to farthings.
11. Bring 6 s. 64 d. to farthings.
Noe 1.-A sum of Federal Money,
Result 5088 qrs.
Facit 1945 qrs.
Facit 313 qrs.
which consists
Facit 2550 cts.
of dollars and cents, is reduced to cents, by simply re-
moving the separating point.
12. Reduce $25.50 to cents.
13. Bring $456.05 to cents.
14. Bring $967.10 to cents.
Result 45605 cts..
Result 96710 cts.
Note 2.-To reduce a sum which consists of dollars
and cents, to fourths, thirds, or halves of a cent, &c.,
reduce it first to cents as in the foregoing examples,
then reduce those cents to fourths, thirds, or halves,
&c., as under rule 1, note 2, Simple Reduction.
15. Reduce $5.25 to fourths or quarters of a cent.
Facit 2100 fourths.
16. Bring $10.183 to fourths of a cent
Result 4075 fourths.
17. Bring $95.12 to fourths of a cent.
Result 38050 fourths.
18. Reduce $17.333 to thirds of a cent,
Result 5200 thirds.
19 Reduce $56.66% to thirds of a cent.
Result 17000 thirds.
Facit 84020 halves.
20. Bring $420.10 to half cents.
21. Bring $375.12 to half cents. Result 75025 halves.
Note 3-To reduce pence, Pennsylvania currency,*
to cents, annex a cypher to their number, and divide by
To reduce pence to mills, annex two cyphers, and
divide hy 9.
9.
* The same rule that applies to Pennsylvania currency applies also
the currencies of New Jersey, Delaware, and Maryland.

68
COMPOUND REDUCTION.
22. Reduce 1575 pence to cents.
Facit 1750 cts.
9)15750
1750 cents.
Result 5250 cts.
23. Bring 4725 pence to cents.
24. Bring 3150 pence to cents.
25. Reduce 1575 pence to mills.
Result 3500 cts.
Facit 17500 mills.
Note 4.-To change pounds, Pennsylvania currency,
to Federal Money, annex two cyphers to their number,
then multiply by 8, and divide the product by 3; the
quotient will be cents, which reduce to dollars.
26. Reduce 25 pounds to dollars.
2500
8
3)20000
$66.662
27. Bring 150 pounds to dollars.
28. Bring 756 pounds to dollars.
29. Bring 17 pounds to dollars.
Facit $66.663
Result $400.00.
Result $2016.00.
Result $45.33}.
If there are shillings, or shillings and pence, with the
pounds, reduce the whole to pence; then reduce those
pence to cents, and the cents to dollars.
30. Reduce 156 pounds, 6 shillings, to dollars.
Facit $416.80.
31. Bring 29 pounds, 12 shillings, to dollars.
Result $78.93§.
32. Bring 100 pounds, 12 shillings, and 6 pence, to
dollars.
Result $268.33.
Note 5.-To reduce cents to pence, Pennsylvania
currency, multiply by 9, and separate one figure from
the right of the product.
33. Reduce 359 cents to pence.
359
Result 323 pence
+
9
323 1

COMPOUND REDUCTION.
69
34. Bring 89 cents to pence.
35. Bring 350 cents to pence.
Facit 80 pence.
Facit 315 pence.
Note 6.-To reduce dollars, or dollars and cents, to
pounds, Pennsylvania currency, reduce them first to
cents, then reduce those cents to pence, and then re-
duce those pence to pounds.
36. Reduce $68.30 to pounds. 37. Bring $450 to pounds.
6830
9
45000
9
12)6147|0
2|0)51|2 3
Result 25 L. 12 s. 3 d.
12)40500 0
2|0)337|5
Result 168 L. 15 s.
38. Reduce 125 dollars to pounds.
39. Bring $246.29 to pounds.
40. Bring 728 dollars to pounds.
41. Bring $79.60 to pounds.
Facit 46 L. 17 s. 6 d.
Result 92 L. 7 s. 2 d.
Result 273 L.
Result 29 L. 17 s.
Note 7.-—To reduce pounds sterling to Federal Mo-
ney, bring them to sixpences, or to pence, and to these
annex two cyphers; then, if sixpences, divide by 9, but
pence, divide by 54, and the quotient will be cents,
which reduce to dollars.
if
42. Reduce 230 L. 15s. 6d. sterling to Federal Money.
230 L. 15 s. 6 d.
20
4615
2
9)923100
Result $1025.66+
43. Reduce 218 L. 19 s. 6 d. sterling to Federal Mo-
ev
Facit $973.22+
44. Bring 25 L. sterling to Federal Money.
Result $111.11+
45. Bring 437 L. 18 s. sterling to Federal Money.
Facit 1946 dols. 22 cts.

70
COMPOUND REDUCTION.
Note 8.-A general rule to change the currency of
each of the states to Federal Money.
Reduce the given sum to shillings, or to sixpences.
or to pence, and to these annex two cyphers; then di-
vide by the number of shillings, sixpences, or pence
in a dollar, as it passes in each state: the quotient will
be cents. (For the value of a dollar, see the table at
page 65.)
46. Reduce 63 L. 15 s., New England or Virginia
currency, to Federal Money, a dollar being 6 s.
Facit $212.50
47. Reduce 112 L. 16 s., New York or North Caro
lina currency, to Federal Money.
Result $282.00.
48. Reduce 161 L. 14 s., South Carolina or Georgia
currency, to Federal Money.
Result $693.00.
WEIGHTS AND MEASURES.
1. Reduce 47 pounds, 10 ounces, 15 pennyweights.
to pennyweights.
Facit 11495 awt
2. Reduce 5 lb. 6 oz. 4 dwt. 20 gr. to grains.
Result 31796 gr.
3. Bring 2 tons, 15 cwt. 2 quarters, to quarters.
Result 222 qrs.
Result 6745 lb.
4. Bring 3 tons, 25 lb. to pounds.
5. Reduce 7 cwt. 3 qrs. 10 lb. to ounces.
Facit 14048 oz.
6. Bring 27 73 23 19 2 grs. to grains.
Result 150022 grs.
7. Bring 3 leagues, 2 miles. 7 furlongs, to furlongs,
8. Bring 57 miles, 2 furlongs, to poles.
Result 95 fur.
Result 18320 P.
9. Reduce 15 yards, 2 feet, to inches. Result 564 in.
10. Bring 42 English ells, 3 quarters, to quarters.
Result 213 qrs,
11. Bring 17 yards, 2 quarters, 2 nails, to na
Result 282 d.
12. Reduce 11 acres, 2 roods, 19 perches, to porches.
Result 1859 P.
13. Bring 17 acres, 3 roods, to perches. Result 2840 P.

THE SINGLE RULE OF THREE.
75
2. If 10 shillings will pay for 20 pounds of beef, how
many pounds will 5 s. pay for?
S.
S.
lb.
10 : 5:
20
5
10)100
Ans. 10 lb.
§.
S.
lb.
5 : 10 :: 10
10
5)109
Proof 20 lb.
3. If1lb. of sugar cost 9d., what will 2ewt. 2 qrs. 10lb. cost?
lb. cwt.qrs. lb.
d. cwt.qrs. lb. lb.
L. s. d.
1 : 10:
2 2 10 :: 9
2 2 10 : 1 :: 10 17 6
4
4
20
10
10
217
28
28
12
80
80
21
21
290)2610(9d. Pr.
2610
290 lb.
9
290
12)2610 pence.
2|0)2117 6
Ans. 10 L. 17 s. 6 d.
4. Sold 125 bushels of wheat, at 11 s. 3 d. a bushel;
what did it come to?
bu.
bu.
Ans. 70 lb. 6 s. 2 d.
L. s. d.
125 : 1 :: 70 6 3
bu. bu.
s. d.
1 125 11 3
12
135
125
20
1406
12
675
125)16875(12135
270
125
135
Pr. 11s. 3d
437
12)16875 pence.
375
2)140 6 3
625
625
70 L. 6 s. 3 d.
7
+

76
THE SINGLE RULE OF THREE.
5. If 3 pounds of sugar cost 4 shillings, what will 6
pounds cost?
Ans. 8 s.
6. If 8 yards of muslin cost 24 shillings, what will
96 yards come to?
Ans. 14 L. 8 s.
7. If 12 bushels of wheat be worth 16 dollars, how
much are 48 bushels worth?
Ans. 64 dols.
8. If 1 pound of butter bring 16 pence, what will 56
pounds bring?
Ans. 3 L. 14 s. 8 d.
9. Sold 12 yards of cloth for 72 dollars: how much
was it per yard?
Ans. 6 dols.
10. If 12 yards of cloth cost 19 L. 16 s., what will 192
yards come to?
Ans. 316 L. 16 s.
11. If 96 pounds of sugar cost 3 L. 12 s., what is it
per pound?
Ans. 9 d.
12. What will 421 bushels of wheat come to, at 1
dollar 35 cents per bushel?
Ans. $568.35.
13. What will 128 pounds of pork come to, at 8
cents a pound?
Ans. 10 dols. 24 cts.
14. How much will 75 pounds of almonds come to,
at 37 cents a pound?
Ans. $28.12}.
15. If 3 yards of cloth cost 5 L. 12 s. 6 d., how much
will 225 yards cost?
Ans. 421 L. 17 s. 6 d.
16. If 1 pound of rice cost 4½ d., what will 48 pounds
cost?
17. Bought 230 bushels of coal for
how much was it per bushel ?
18. Bought 120 bushels of corn for
much is that a bushel?
Ans. 18 s.
26 L. 16 s. 8 d.,
Ans. 2 s. 4 d.
58 dollars: how
Ans. 48 cts
19. If 891 gallons of molasses cost 176 L. 6 s. 10 d.
what is it per gallon?
Ans. 3 s. 11½ d
20. What must be paid for 45 bushels, 3 pecks of
potatoes, at 2 s. 8 d. a bushel ?
Ans. 6 L. 2 s.
21. If 1 dozen of penknives cost 2 dollars 0 cent
how much will 4 dozen come to?
Ans. $0.0
22. If the price of one acre of land be 18 dolla s
cents, what will 50 acres, 2 roods, 20 perches, con
Ans. $923.00
23. If 1 hundred weight of sugar cost 2 L. 11 s. 4
what is the price of,1 pound?
Ans 54d
24. If i hundred weight of iron be worth 11. 63
hat is the value of 33 cwt. 1 qr. 22 lb.?
Ans. 46 L
old hundred weight of tobacco, a

THE SINGLE RULE OF THREE.
79
54. If 8 yards of velvet cost $3.20, what will 96
Ans. $38.40.
yards come to?
55. If 9 lb. of sugar cost 9 s. 4 d., what is the value of
27 lb.?
Ans. 1 L. 8 s.
56. How much will 60 bushels of apples come to, if
4 bushels cost 1 L. 4 s.?
Ans. 18 L.
INVERSE PROPORTION.
Questions in Inverse Proportion may be solved pre-
cisely in the same manner as the foregoing examples.
EXAMPLES.
1. If 12 men can build a house in 48 ys, in what
time could 36 men build it?
Ans. 16 days.
2. If 48 men can build a wall in 24 days, how many
men can do it in 192 days?
Ans. 6 men.
3. If 100 men can finish a piece of work in 12 days,
how many can do it in 3 days?
Ans. 400 men.
4. How many labourers must be employed to finish
a piece of work in 15 days, which 5 can do in 24 days?
Ans. 8.
5. If 6 reapers can reap a field of wheat in 12 days,
in what time could 24 do it?
Ans. 3 days.
6. If 100 dollars in 12 months bring 6 dollars inte-
rest, what sum will bring the same in 8 months?
Ans. $150.
7. If a footman perform a journey in 3 days, when
the days are 16 hours long, how many days will he re-
qure of 12 hours long, to perform the same in? Ans. 4.
8. How many yards of matting, 2 feet 6 inches broad,
will cover a floor that is 27 feet long, and 20 feet broad?
Ans. 72 yards.
9. What quantity of shalloon, that is 3 quarters of a
yard wide, will line 74 yards of cloth, that is 14 yards
wide?
Ans. 15 yards.
10. How many yards of carpeting, that is 3 quarters
of a yard wide. are sufficient to cover a floor that is 18
feet wide, and 60 feet long?
Ans. 160 yds.
11. If a board be 9 inches broad, how long must it
be to measure 12 square feet?
Ans. 16 feet.
12. How much in length that is 4½ inches broad will
anke a square foot?
H
Ans. 32 inches.

80
THE DOUBLE RULE OF THREE.
PROMISCUOUS EXAMPLES.
1. Calculate the value of 26 yards of linen, at 5 s.
6 d. a yard?
Facit 7 L. 5 s. 9 d.
2. Purchased 156 lb. of soap for 15 dollars 60 cents;
what was the price per pound?
Ans. 10 cts.
3. How many yards of cloth, 3 quarters of a yard
wide, are equal in measure to 30 yards, of 5 quarters
wide?
Ans. 50 yards.
4. Bought 274 yards of muslin, at 6 s. 9½ 1. per yard.
what does it amount to?
Ans. 9 L. 5 s. 04 d. +2
5. In what time will 600 dollars gain the interest
which 80 dollars would gain in 15 years? Ans. 2 years.
6. What quantity of wine, at 6 s. per gallon, may be
bought with 18 L. 18 s.?
Ans. 63 gals.
7. If 1 hundred weight of sugar cost 13 dollars 50
cents, what must be paid for 17 cwt. 3 qrs. 14 lb.?
Ans. $241.314 cents.
8. A cistern has a pipe which will empty it in 10
hours how many pipes of the same capacity will
empty it in 30 minutes?
Ans. 20.
9. How many yards of paper, 2 feet wide, will be
required to cover a wall which is 12 feet long, and 9
feet high?
Ans. 14 yds. 1 ft. 2 in
10. If 1½ oz. of spice cost 64 d., what will 34 oz. cosí
at the same rate?
Ans. 1 s. 1 d. +
11. What is the value of a piece of cloth containing
52 English ells, 3 quarters, at one dollar 76 cents per
yard?
Ans. $115.72.
COMPOUND PROPORTION,
OR
THE DOUBLE RULE OF THREE.
Compound Proportion is compounded of two or more
ranks of proportionals; five, seven, nine, &c. terms being
given, to find a sixth, eighth, &c.
RULE.
Work by two or more statings in Simple Propor-
tion; or,
Set that term which is like the term sought, in the

THE DOUBLE RULE OF THREE.
81
third place, and consider each pair of similar terms and
this third one, as the terms of a stating in Simple Pro-
portion, and set them severally, in the first and second.
places, agreeably to the directions under that rule.
When the question is thus stated, reduce the similar
terms to like denominations, and then multiply all the
terms in the second and third places together, and di-
vide the product by the product of those in the first
place: the quotient will be the answer, or term sought.
The above rule is preferred for reasons similar to those which have
been given for adopting the new rule for Simple Proportion: the one for-
merly used is, however, subjoined.
RULE FOR STATING.
Set the two terms of supposition which are of the same name or kind
as those of the demand, one under the other, in the first place; that of
the same kind as the answer in the second, and those of the demand in
the third, with the two corresponding terms of the supposition and de-
mand opposite to each other, and of the same denomination.
When a question is stated, consider the two upper terms with the mid-
dle one, as a stating in the Single Rule of Three, and also the two under
terms, with the middle one, as a stating in the same rule; if, in both in-
stances, the proportion be direct, the question is in direct proportion;
but if in either of them the proportion be inverse, the question is in in-
verse proportion.
RULE FOR DIRECT PROPORTION.
Multiply the two terms in the third place together, and multiply the
product by the middle term; divide the last product by the product of
the terms in the first place, and the quotient will be the answer, in the
same denomination as the middle term.
EXAMPLE.
If 6 men in 8 days eat 10 lb. of bread, how much
will 12 men eat in 24 days?
6 men
10 lb.
8 days S
S12 men
{24 days
।
48
Ans. 60.
Contracted.
$10 542 2
224 3
{
24
288
10
48)2880(60 lb.
288
0
6
10
60 lb.

82
THE DOUBLE RULE OF THREE.
PROOF.
By two statings in the Single Rule of Three.
Note.-If either of the two first terms, or both, will
divide, or can be divided by any of the three last, or if
any other number will divide one of the first and one
of the last, without a remainder, the operation may be
contracted by using their quotients in their stead.
EXAMPLES.
1. If 6 men in 8 days eat 10 lb. of bread, how much
will 12 men eat in 24 days?
Ans. 60.
men 6: 127
days 8 245
:: 10 lb.
288
Contracted.
0:42 22
8:24 35
:: 10 lb.
10
6
48)2880(60 Ans.
288
10
60 Ans.
0
2. If 3 men in 4 days eat 5 lb. of bread, how much
will suffice 6 men for 12 days?
Ans. 30 lb.
3. Suppose 4 men in 12 days mow 48 acres, how
many acres can 8 men mow in 16 days?
RULE FOR INVERSE PROPORTION.
Ans. 128 A.
Transpose the inverse extremes; that is, set that which is in the first
place under the third; and that which is in the third place under the
first; then work as in Direct Proportion.
EXAMPLE.
If 7 men reap 84 acres of wheat in 12 days, how many
men can reap 100 acres in 5 days?
$ inverse
84 A.
12 D.
5
}
7m.
{
S100 direct
12
420
1200
7
420)8400(20
840
0
Ans. 20.
[App(20 Ans.
Contracted.
)$4
ÅØ
7m.
$
12

THE DOUBLE RULE OF THREE.
83
4. If 10 bushels of oats be sufficient for 18 horses 20
days, how many bushels will serve 60 horses 36 days,
at that rate?
Ans. 60 bu.
5. If 7 quarters of malt are sufficient for a family of
7 persons 4 months, how many quarters will 46 per-
sons use in 10 months?
Ans. 115.
6. Suppose the wages of 6 persons for 21 weeks be
288 dollars, what must 14 persons receive for 46
weeks?
Ans. 1472 dols.
7. If 8 reapers have 3 L. 4 s. for 4 days work, how
much will 48 men have for 16 days work? Ans. 76 L. 16s.
8. If 100 L. in 12 months gain 6 L. interest, how much
will 75 L. gain in 9 months?
Ans. 3L. 7 s. 6 d.
9. If 100 L. in 52 weeks gain 6 L. interest, how much
will 200 L. gain in 26 weeks?
Ans. 6 L.
10. If the carriage of 8 cwt. 128 miles cost $12.80,
what must be paid for the carriage of 4 cwt. 32 miles?
Ans. $1.60.
11. If 16 L. 18 s. be the wages of 16 men for 8 days,
what sum will 32 men earn in 24 days? Ans. 101 L. 8s.
12. If 350 L. in half a year gain 10 L. 10 s. interest,
what will be the interest of 400 L. for 4 years? Ans. 96 L.
INVERSE PROPORTION.
1. If 7 men reap 84 acres of wheat in 12 days, how
many men can reap 100 acres in 5 days? Ans. 20 men.
2. If 4 dollars be the hire of 8 men for 3 days, how
many days must 20 men work for 40 dollars? Ans. 12.
3. If 4 men have $3.20 for 3 days work, how many
men will earn $12.80 in 16 days?
Ans. 3 men.
4. If 4 reapers have 12 dollars for 3 days work, how
many will earn 48 dollars in 16 days?
Ans. 3.
5. If 100 L. in 12 months gain 6 L. interest, what
m will gain 3 L. 7 s. 6 d. in 9 months? Ans. 75 L.
6. If a footman travel 240 miles in 12 days, when
the days are 12 hours long; how many days will he
require to travel 720 miles, when the days are 16 hours.
long?
7. If 100 L. in 12 months gain 8 L.
sum will gain SL. 12 s. in 5 months?
8. If 200 lb. be carried 40 miles for 40
may 20200 lb. be carried for $60.60?
Ans. 27 days.
interest, what
Ans. 258 L.
cents, how far
Ans. 60 miles.
R2

84
PRACTICE.
PROMISCUOUS EXAMPLES.
1. If 4 men in 5 days eat 7 lb. of bread, how much
will suffice 16 men 15 days?
Ans. 84 lb.
2. If 100 dols. gain $3.50 interest in one year, what
sum will gain $38.50 in 1 year and three months?
Ans. 880 dols.
3. If it take 5 men to make 150 pair of shoes in 20
days, how many men can make 1350 pair in 60 days?
Ans. 15.
4. If the wages of 6 men for 21 weeks be 120 L.,
what will be the wages of 14 men for 46 weeks?
Ans. 613 L. 6 s. 8 d.
5. If 333 L. 6 s. 8 d gain 15 L. interest in 9 months,
what sum will gain 6 L. in 12 months? Ans. 100 L.
6. A wall which is to be built to the height of 27
feet, has been raised 9 feet in 6 days, by 12 men: how
many men must be employed to finish the work in 4
days?
Ans. 36 men.
PRACTICE.
Practice is a short method of ascertaining the value of
any number of articles, or of pounds, yards, &c. by the
given price of one article, one pound, or one yard, &c.
Practice may be proved by Compound Multiplica-
tion, or by the Single Rule of Three Direct.
qr.
1
2
d.
3
6
At H
H@ Alex H:0 HIN
~୯
of ld.
of a shilling.
TABLES OF ALIQUOT PARts.
s. d.
10
J 8
20
26
3
4
4 0
5 0
68
10 0
20
I
12
1
6
~OHVAT A
of a pound.
cts.
50=/
25
20
12
10
of a dollar.
*
lb.
7=1/6
8
со н
14
16
10
28
64 16
56
ठ
-FAT AS
of an cwt
* An aliquot part of a number is any number that will divide it with-
out a remainder; thus 4 is an aliquot part of 20, and 8 of 56. A sum or
quantity is an aliquot part of a greater sum or quantity, when a certain
number thereof will make the greater; thus a shilling is an aliquot part
of a pound, because 20 shillings make one pound.

PRACTICE.
85
When the price is less than a penny, work by
RULE 1.
If the price be a farthing, or a halfpenny, set down
the value of the given number at a penny, and take
such part of that sum as the price is of a penny, for the
answer in pence.
*
If the price be three farthings, find the value of the
given number at a halfpenny, and afterwards at a far-
thing; then add the two results together, and their
amount will be the answer.
** If the learner be unable to tell the denomination
of a quotient, or how to proceed with remainders, it
would be useful to refer him to examples 14, 15, and
16, under Rule 1, and 7, 8, under Rule 3, Compound
Division.
EXAMPLES.
1. What is the value of 4528 quilis, at each?
2. What is the value of 4528 quills, at 4 each?
(1) d.
| 4 | | 4528 value at 1 d.
12)1132 Ans. in pence.
2|0)9|4 4
Ans. redu. 4 L. 14 s. 4 d.
42 44
(2) d.
4528 value at 1 d.
2264 value at .
1132 value at 4.
12)3396 Ans.
210)28|3
Ans. reduced 14 L. 3 s.
3.
64 at 1
7612 at
5. 2345 at
L. S.
ď.
Answer
1 4
7 18 7
4 17 S
* The value of any number of articles at a penny, each, is that num-
ber of pence: thus, the value of two things at a penny, each, is two
pence of three things, three pence; of twenty things, twenty pence, &c.;
and, is a farthing is the fourth part of a penny, the value at a farthing
must be a fourth part of the value at a penny; and as two farthings are
the half of a penny, the value at two farthings must be half of the value
at a penny, &c.
This explanation of the rule, with a little variation, will apply to most
of the other rules of Practice.

36
PRACTICE.
L.
s. d.
6. 6812 at
Answer 14 3 10
7. 1487 at
4 12 11
8.
4712 at
14 14 6
When the price is not less than a penny, but less than
a shilling, and is an aliquot part of a shilling, work by
RULE 2.
Set down the value of the given number at a shilling,
and take such part of it as the price is of a shilling, for
the answer.
EXAMPLES.
1. What is the value of 7612 lb. of rosin, at 1 d. per
lb. and also at 1 d. per lb.?
1
S.
S.
| 1 d. | ½ | 7612 value at 1 s. | 1½ d. | | | 7612 value at 1 s.
2
210)63|4 4
2|0)951 6
Ans. redu. 31L. 14 s. 4d. Ans. reduced 47 L. 11 s. 6 d.
L. S. d.
d.
2.
24 at 1
3.
3806 at 12
4.
1769 at 2
5. 7649 at 3
Answer
2 0
23 15 9
14 14 10
6. S120 at 4
7.
2764 at 6
95 12 3
1 5 6 8
¡9 2 0
When the price is not less than a penny, but less
than a shilling, and is no aliquot part of a shilling,
work by
RULE 3.
Separate the price into parts, one of which shall be
an aliquot part of a shilling, and the rest either aliquot
parts of a shilling or of one of the other parts. Find
the value at each of the parts, agreeably to the tenor of
the preceding rules, and add the several results toge-
ther, for the answer.
EXAMPLES.
1. What is the value of 6192 yards of tape, at 24 d.
per yard?
2. What is the value of 3711 lb. of sugar, at 73 d.
per lb.?

PRACTICE.
87
S.
(1.) | 2d. | | 6192 value at 1 s.
PH
1032 value at 2 d.
129 value at
2|0)116 1 Answer in shillings.
Answer reduced 58 L. 1 s.
(2.) | 4d. | 3711 value at 1 s.
3 d.
+++
More
1237
at 4 d.
927 9 at 3 d.
154
7 at
77 34 at
2|0)239 6 8 Ans. in shillings, &c.
Answer reduced 119 L. 16 s. Så d.
Note.-In working the former of these examples,
we find the value of the given number at 2 d. by Rule
2, and divide the result by 8 to find the value at 4; for
as is an eighth part of 2 d., the value at must be an
eighth part of the value at 2 d. The latter example is
wrought in a similar manner.
d.
3. 3596 at 24
L. S. d.
3355
4.
1861 at
14
5.
7000 at
44
6. 7181 at 5
7.
3762 at 7
8.
3747 at 7
9.
4697 at 8
10.
7924 at
9
11. 7796 at 10
12.
3064 at 11
Answer 33 14 3
9 13 104
123 19 2
149 12 1
109 14 6
*
117 1 10
156 11
4
313
13
341
1
140
S 8
∞ a ro
2
6
When the price is not less than a shilling, but less
than two shillings, work by
RULE 4.
Set down the value of the given number at a shil-
ling, and to this add the value at the rest of the price,
found by the preceding rules.

88
PRACTICE.
EXAMPLES.
1. What is the value of 725 yards of muslin, at 13 d.
per yard?
1725 s.
11 히
​value at 1 s.
90 7
value at 13 d.
2|0)81|5 7 Ans. în shillings, &c.
Ans. reduced 40 L. 15 s. 7 d.
d.
R
2.
15 at 13
3.
360 at 14
4. 1479 at 15
}
5. 7121 at 164
6.
2340 at 17
7.
7890 at 182
8.
8900 at 19
9.
7120 at 20+
10.
1376 at 21
11.
6812 at 224
12.
9999 at 23
L. S.
d.
Answer
16 10/1/1
21
0 0
92 8 9
482 3 04
170 12 6
616 8 1
704 11 8
600 15 0
120 8 0
645 14 5
989 9 81
When the price is any number of shillings under 20,
work by
RULE 5.
Set down the value of the given number at a shilling,
and multiply that sum by the number of shillings in the
price: the product will be the answer. Or,
*
If the price be an aliquot part of a pound, set down the
value of the given number at a pound, and take such
part of that sum as the price is of a pound, for the answer.
EXAMPLES.
1. What is the value of 528 bu. of apples, at 3 s. per bu.?
2. What is the value of 750 yards of linen, at 5 s. per yd.?
S.
528alue at 1 s.
3
2|0)15814 Ans. in shillings.
Ans. 79 L. 4 s.
L.
5s. || 750 value at 1L..
Ans. 187 L. 10 s.
* As two shillings are twice one shilling, the value of any number of
articles, at two shillings, each, must be twice their value at one shilling;
and as three shillings are three times une shilling, the value at three
shillings must be three times the value at one shilling, &c.

PRACTICE.
89
S.
3. 264 at 3.
334
4.
486 at
2.
5.
121 at
5.
6. 1286 at 4.
7.
860 at 7.
8.
242 at 11.
9.
2798 at 13.
Answer
L. S. d.
39 12
48 12
800
30 5 0
257 4 0
301 0 0
133 2 0
1818 14 0
3127 3 0
10.
3679 at 17.
Note. When the price is an even number of shil-
lings, the answer may be found thus:-Multiply the
given number by half the price, doubling the right
hand figure of the product for shillings; the rest of the
product will be pounds.
11. 473 at 4 s.
Ans. 94 L. 12 s.
473
2
Ans. 94 L. 12 s.
S.
L. S. d.
12.
946 at 4.
Answer 189 4 0
13.
713 at 6.
213 18 0
14.
916 at 8.
366 8 0
15.
739 at 12.
443 8 0
16. 171 at 16.
136 16 0
"When the price is shillings and pence, or shillings,
pence, and farthings, work by
RULE 6.
If the price be an aliquot part of a pound, set down
the value of the given number at a pound, and take such
part of that value as the price is of a pound, for the
answer: but,
If the price be not an aliquot part of a pound, find
the value at the shillings, by rule 5; and to this add
the value at the rest of the price, found by the pre-
ceding rules.
EXAMPLES.
1. 764 yards, at 2 s. 6 d.
L.
Ans. 95 L. 10 s.
2s. 6d. | 764 value at 1 L.
Ans. 95 L. 10 s.

90
PRACTICE.
Ans. 123 L. I s.
A 428 lb. at 5 s. 9 d.
6 d.
差
​H
3 d.
S.
428 value at 1 s.
5
2140 value at 5 shillings.
214
at 6 pence.
107
•
at 3 pence.
210)246|1 Answer in shillings.
Ans. reduced 123 L. 1 s.
S. d.
L. S.
d.
3.
378 at 1 8
Answer
31 10 0
4.
324 at
2
6
40 10 0
5.
126 at
3 4
6.
716 at 6
8
7.
673 at 5
10
21 0 0
238 13 4
197 13 10/
8.
2547 at
7
31
928 11 10/
9. 3715 at 9
4
1741 8 1/
10.
2572 at 13
73
1752 3 6
11.
7251 at 14
843
5324 19 02
12.
1924 at 19
6
1875 18 0
13.
2710 at 19
21
2602 14 7
When the price is pounds, or pounds, shillings, &c..
work by
RULE 7
Set down the value of the given number at a pound.
and multiply that sum by the number of pounds in the
price the product will be the value at the pounds, to
which add the value at the remainder of the price (if
any) found agreeably to the tenor of the preceding
rules: or,
Reduce the pounds and shillings of the price to shil-
Jings, and find the answer by Rule 6.

PRACTICE.
91
EXAMPLES.
1. 428 tons, at 3 L. 4 s. 61 d. per ton.
48.
L.
428 value at 1 L.
Ans. 1381 L. 3 s. 10 d.
Or thus:
6 d.
3
1284
6 d.
18
85 12
C
10 14
12
17 10
11/20
12
Answer 1381 L. 3s. 10 d.
S.
428
64
1712
2568
27392
214
17 10.
L. s. d.
2.
47 at
3
3 4
3.
17 at 2
6
8
4.
17 at 11
14
0
5.
20 at 4
13
4
6.
71 at 6 13
4
7. 156 at 3 6
84
2102762|3 10
Ans. 1381 L. 3 s. 10 d.
L. S. d.
Answer 148 16 8
39 13 4
198 18 0
93 6 68
473
6 8
520 3 3
8. 457 at 14 17 92
6804 10 9½
When the given quantity consists of several denomi-
nations, and the price relates to the highest of those
denominations, work by
RULE S.
Multiply the price by the number of the highest
denomination in the given quantity, and the product
will be the value thereof; to which add the value of
the remaining denominations, found by taking parts of
the price or,
Find the value of the number of the highest deno-
mination by one of the preceding rules, to which add
the value of the remaining denominations, found as
before.
EXAMPLES.
1. What is the value of 171 cwt. 1 qr. 7 lb. of sugar,
at 3 L. 6 s. 8 d. per cwt.?
Ans. 571 L. 0 s. 10 d.

92
PRACTICE.
qr.
L. s.
d.
Or thus:
1 | 4
3 6
8x1
s. d.
L.
10
68171
1
qr.
3
33 6 8x7
10
513
57
333
6
8
value of
7 lb. 4
0
16 8
3
6
8
lb.
233 6
8
8f171cwt.
49
4 2
74
16
8 value of 1 qr.
Ans. 571 L. 0 s. 10 d.
4 2 value of 7 lb.
Ans. 571 L. Os. 10 d.
cwt.grs. lb.
L. s. d.
L. s. d.
2.
12 2 14 at 3
14 0 per cwt.
Ans. 46 14 3
3. 17 3
19 at 2
26
38 1 6
4.
10 0
12 at 1
19 6
19 19 2
5.
9 2
26 at 4
10 4
43 19 6
6.
5
1
0 at 2
17 0
14 19 3
410
7. 7 0 19 at 3 16 0
8.
lb. oz. dwt. L. s. d.
27 10 O`at 1 4 per lb.
9. 73 5 15 at 3 9 0
27
1 17 14
253 10 0
yds.qrs. S. d.
10.
67 2 at 12
2 per yard.
41 1 3
11. 68 1 at 8
1
27 11 8
12. 419 3 at 12 6
262 6 10
A. R. P. L. s. d.
13. 476 3 28 at 3 7 11 per acre
1619 11 12
14. 238 1 34 at 6 15 10
1619 11 12
EXAMPLES IN FEDERAL MONEY.
Note.—When the given price of an article is in
Federal money, the question may generally be answer-
ed by Multiplication, or by the Rule of Three, more
readily than by Practice. It is useful, however, to be
acquainted with the method of working by Practice,
as it afords a means of proving the correctness of one-
rations performed by those other rules.
The examples that are given in this place are chiefly
confined to cases in which the price is an aliquot

PRACTICE.
93
part of a dollar for working which the following is a
GENERAL RULE.
Set down the value of the given number, at a dollar,
and take such part of that sum as the price is of a dol-
lar, for the answer.
1. What is the value of 800 loaves of bread, at 64
cents each, and also at 12 cents each?
dols.
dols.
| 6 || 800(50 dols. Ans. |12|| 800 value at 1 dol.
2.
6.
80
0
Ans. 100 dollars.
720 lb. of pork, at 64 cts. per lb.
3. 1446 lb. of beef, at 64 cts. per lb.
4. 680 lb. of sugar, at 10 cts. per lb.
5. 2128 lb. of cheese, at 10 cts. per lb.
336 lb. of sugar, at 124 cts. per lb.
7. 1364 lb. of ham, at 12 cts. per lb.
8. 160 yds. of muslin, at 20 cts. per yd.
9. 1462 yds. of check, at 20 cts. per yd.
240 lb. of coffee, at 25 cts. per lb.
726 yds. of muslin, at 25 cts. per yd.
324 yds. of linen, at 50 cts. per yd.
75 bush. of potatoes, at 50 cts. per bu.
10.
11.
12.
13.
Dols.cts.
Ans. 45.00
90.37
68.00
212.80
42.00
170.50
32.00
292.40
60.00
181.50
162.00
37.50
Note. When the given quantity consists of several
denominations, proceed as directed in Rule 8.
14. 2cwt. 3 qrs. 14lb. at $7.00 per cwt.
Or thus:
2 qrs.
Ans. $20.12
dols.
2 qrs.
PN
7
12qrs.
7.00
2
14
14.00
1qr.
3.50
1 qr.
}
3.50
14 lb.
1.75
14 lb.
1.75
87/
87/
Ans. $20.12
Ans. $20.12
15. 37 cwt. 2 qrs. 14 lb. at $20.10
per cwt.
Ans. $756.264.
16. 7 cwt. Oqrs. 16 lb. at $6.20 per cwt.
Ans. $44.281.

94
PRACTICE.
17. 4 cwt. 1 qr. 16 lb. at $14.43 per cwt.
18. 47 lb. 10 oz. (Troy weight,) at
Ans. $63.38+
$1.25 per lb.
Ans. $59.79+
Ans. $145.68+
19. 64 yds. 3 qrs. at $2.25 per yd.
20. 240 A. 1R. 10 P. at $15.25 per acre.
Ans. $3664.76
APPLICATION.
1. What is the value of 120lb. of rice, at 3d. per lb.
Ans. 1 L. 10 s.
2. Bought 640 lb. of pork, at 4 d. per lb.; what is the
amount?
Ans. 10 L. 13 s. 4 d.
3. How much will 3906 lb. of beef come to, at 7½ d.
per lb.?
Ans. 122 L. 1 s. 3 d.
4. What is the amount of 2004 lb. of sugar, at 10 d.
per lb.?
Ans. 87 L. 13 s. 6 d.
5. How much will 121 lb. of cheese come to, at 1 s.
per lb.?
Ans. 6 L. 1 s.
6. What is the value of 1234 yards of muslin, at 1 s.
114 d. per yard?
Ans. 122 L. 2 s. 33 d.
7. If one yard of linen cost 4 s., how much will 987
yards cost?
S. If 1 gallon of wine sell for
gallons bring?
9. How much will 800 bushels
at 13 s. 4 d. per bushel?
Ans. 197 L. 8 s.
11 s., what will 543
Ans. 298 L. 13 s.
of wheat amount to,
Ans. 533 L. 6s. 8 d.
10. How much will 47 tons of hay amount to, at 6 L.
6 s. 8 d. per ton?
11. If 1 yard of cloth cost 1 L.
will 1677 yards come to?
Ans. 297 L. 13s. 4d.
19s. 4 d., how much
Ans. 3298 L. 2s.
cents per lb.; what
Ans. 488 dols. 25 cts.
12. Sold 3906 lb. of sugar, at 124
is the amount?
13. Bought 324 yards of calico, at 25 cents per yard;
what is the amount?
Ans. 81 dols.
14. What will 16 cwt. 2qrs. 17 lb. of sugar amount to,
Ans. 93 L. 2 s. 24 d.
at 5L. 11s. 10d. per cwt.?
15. Sold 83 yds. 2qrs. of superfine cloth, at 10 dols.
50 cts. per yard: how much does it amount to?
Ans. $876.75.
16. If 1 acre of land be worth 11 L. 15s., what is the
Ans. 6800L. 6s. 3d.
value of 578 acres 3 roods?

TARE AND TRET.
95
•
TARE AND TRET.
Tare and Tret are allowances made by the seller to
the buyer, on some particular commodities.
Tare is an allowance made for the weight of the barrel,
box, bag, or whatever contains the commodity.
Tret is an allowance of 4 lb. in every 104 lb. for
waste, dust, &c.
Gross weight is the weight of the goods, together with
the barrel, box, bag, or whatever contains them.
Neat weight is the weight of the goods after all
allowances are deducted.
CASE 1.
To find the neat weight when the tare is so much in
the whole gross weight.
RULE
Subtract the tare from the gross weight, and the re-
mainder will be the neat weight.
EXAMPLES.
1. The gross weight of a certain hogshead of sugar is
7 cwt. 3 qrs. 16 lb.; the tare is 3 qrs. 10 lb.; what is the
neat weight?
Ans. 7 cwt. O qrs. 6 lb.
2. What is the neat weight of 12 hogsheads of sugar,
the gross weight of each hhd. being 6 cwt. 2 qrs. 171b.;
the tare in the whole 8 cwt. 3 qrs. 14 lb.?
(1)
cwt.qrs. lb.
cwt. 93
16 gross.
3 10 tare.
Âns. 70 cwt. 3 qrs. 22 lb.
(2)
cwt.qrs. lb.
6 2 17 gross, each
12
Ans. 7 0 6 neat.
79 3
8
3
8 gross in all
14 tare in all.
Ans. 70 3 22 neat weight.
3. The gross weight of a certain hogshead of sugar is
8 cwt. 3 qrs. 17 lb.; the tare is 3 qrs. 16 lb.; what is the
neat weight?
Ans. S cwt. O qrs. 1 lb.
4. What is the neat weight of 456 cwt. 1 qr. 19 lb. of
tobacco, tare in the whole 15 cwt. 2 qrs. 13 lb.:
Ans. 440 cwt. 3 qrs. 6 lb.
12

96
TARE AND TRET.
5. What is the neat weight of 4 casks of indigo, the
gross weight of each cask being 4 cwt. 2 qrs. 14 lb.; the
tare in the whole 1 cwt. 0 qrs. 26 lh.?
Ans. 17 cwt. 1 qr. 2 lb.
6. What is the neat weight of 5 casks of sugar, the
gross weight and tare as follows?
Cut. qrs. lb.
qrs. lb.
No. 1. Gross 4 2 14 Tare 1 5
2.
3
0
17
1 1
3.
5
3
10
2
11
4.
6
1
16
2
27
5.
3
2
18
1 3
Ans. 21 cwt. 2 qrs.
CASE 2.
To find the neat weight when the tare is so much
per barrel, box, &c.
RULE.
Multiply the tare per barrel, box, &c. by the num-
ber of barrels, boxes, &c., and the product will be the
whole tare: subtract the whole tare from the whole
gross weight, and the remainder will be the neat
weight.
EXAMPLES.
1. What is the neat weight of 15 casks of raisins,
each weighing 2 cwt. 3 qrs. 12 lb. gross-tare 21 lb.
cask?
cirt. qrs. lb.
2 3 12
15 casks.
21 tare per cask.
per
5
15
14 1
4
30
3
28)315lb. (4)11 qrs.
42
3
2 3
12 gross.
7 tare.
28
2 cwt. 3 qrs. 7lb.
35
40 0
5 neat.
23
77

TARE AND TRET.
97
2. What is the neat weight of 4 hogsheads of tobacco,
each weighing 10 cwt. 3 qrs. 10 lb. gross ;-tare 100 lb.
per hhd.?
Ans. 39 cwt. 3 qrs. 4 lb.
3. What is the neat weight of 6 casks of raisins, each
weighing 3 cwt. 2 qrs. 10 lb. gross; tare 20lb. per cask?
Ans. 20 cwt. 1qr. 24 lb.
4. What is the neat weight of 35 bales of silk, each
weighing 317 lb. gross; tare 16 lb. per bale?
CASE 3.
Ans. 10535 lb.
To find the neat weight when the tare is so much
per hundred weight.
RULE.
Subtract from the gross such aliquot part or parts of it,
as the tare is of a cwt.; the remainder will be the neat.
Or, multiply the pounds gross by the tare per cwt.,
then divide the product by 112, and the quotient will
be the tare. Subtract the tare from the pounds gross,
and the remainder will be the neat weight.
EXAMPLES.
1. What is the neat weight of 40 kegs of figs, gross
weight 75 cwt. 3 qrs. 12 lb.-tare per cwt. 14 lb.
Cwt. grs. lb.
| 14 | 4 | 75 3
12 gross.
9 1 26 tare.
66 1 14 neat.
2. What is the neat weight of 35 kegs of raisins,
gross weight 37 cwt. 1 qr. 20 lb.;-tare per cwt. 14 lb.?
Ans. 32 cwt. 3 qrs.
3. What is the neat weight of 6 hogsheads of sugar,
each weighing 8 cwt. 2 qrs. 14 lb. gross; tare 16 lb. per
cwt.?
Ans. 44 cwt. 1 qr. 12 lb.
4. What is the neat weight of 9 hogsheads of tobacco,
each weighing 6 cwt. 2qrs. 12 lb. gross;-tare 17 lb. per
cwt.?
Ans. 50 cwt. 1 qr. 22 lb.
CASE 4.
To find the neat weight when tret is allowed with
tare.
RULE.
Subtract the tare from the gross weight as before: the
1

98
TARE AND TRET.
remainder is called suttle.
the quotient will be tret.
Divide the suttle by 26, and
Subtract the tret from the
suttle, and the remainder will be the neat weight.
EXAMPLES.
1. What is the neat weight of 8 cwt. 3 qrs. 20 lb.
gross;―tare 38 lb.-tret 4 lb. per 104 lb.
Cwt. grs. lb.
8 3 20
4
35
Suttle 962
26)962 37 tret.
78
925 lb. neat.
28
300
70
182
182
1000 lb. gross.
38 lb. tare.
962 lb. suttle.
2. What is the neat weight of 17 chests of sugar,
weighing 120 cwt. 2 qrs. gross;-tare 176 lb.-tret 4 lb.
per 104 lb.?
Ans. 12808 lb. or 114 cwt. 1 qr. 12 lb.
3. What is the neat weight of 5 hogsheads of sugar,
each 10 cwt. 1 qr. 20 lb. gross;-tare 3 qrs. 25 lb.
hhd.—tret 4 lb. per 104 lb.? Ans. 45 cwt. 1 qr. 24lb.
}
APPLICATION,
per
1. There are 24 hogsheads of tobacco: each hogshead,
weighs 6 cwt. 2 qrs. 17lb. gross; tare in all, 17 cwt. 3 qrs.
27 lb. How much will the tobacco amount to, at 1˚L.
10's. 6 d. per cwt.
Ans. 216 L. 0 s. 4 d.
which weighed
How much did
Ans. $116.25.
of raisins; each
2. Bought 5 bags of coffee, each of
95 lb. gross; tare in the whole, 10 lb.
it amount to, at 25 cents per pound?
3. What is the amount of 30 casks
cask weighing 2 cwt. 3 qrs. 12 lb. gross; tare 21 lb. per
cask; price, $7.35 per cwt.?
Ans. $588.608
4. What is the value of 10 casks of alum; the whole
weighing 33 cwt. 2qrs. 15lb. gross; tare 15lb. per cask
price, 23 s. 4 d. per cwt.?
Ans. 37 L. 13 s. 61 d
3. Sold 12 butts of currants; each butt weighed 7cwt.

SIMPLE INTEREST.
99
1 qr. 10 lb. gross; tare 16 lb. per cwt.
amount at $9.20 per cwt.?
What was the
Ans. $694.514.
6. What is the value of 8 hogsheads of sugar, each
weighing 8 cwt. 3 qrs. 7 lb.; tare 12 lb. per cwt.; price
72 s. 6 d. per cwt.?
Ans. 228 L. 3 s. 7 d.
SIMPLE INTEREST.
Interest is a consideration allowed for the use of
money; relative to which are four particulars, viz., the
principal, time, rate per cent., and amount.
The principal is the money for which interest is to
be received.
The rate per cent. per annum is the interest of 100
pounds or dollars for one year.
The time is the number of years or months, &c. for
which interest is to be calculated.
The amount is the sum of the principal and in-
terest.
CASE 1.
}
To find the interest when the time is one year, and
the rate per cent. is pounds or dollars only.
RULE.
*
Multiply the principal by the rate per cent., and
divide the product by 100: the quotient will be the
interest for 1 year.
PROOF.
By the Single Rule of Three.
EXAMPLES.
1. What is the interest of 525 L. for 1 year, at 6 L.
per cent. per annum?
Ans. 31 L. 10 s.
2. What is the interest of 650 L. 15 s. for 1 year, at
& L. per cent. per annum?
* This rule agrees with the Single Rule of Three except that the
stating required by that rule is omitted in this.
4.
#

100
SIMPLE INTEREST.
(1)
525
6
L. 31 50
20
s. 10|00
(2)
L.
S.
650
15
6
39104 10
20
0|90
12
10|80
3|20
3. What is the interest of 500 L. for one year, at 6 L.
per cent. per annum?
Ans. 30 L.
4. What is the interest of 1000 L. for one year, at 7 L.
per cent. per annum?
Ans. 70 L.
5. What is the interest of 350 L. 17 s. 8 d. for one
year, at 6 L. per cent. per annum? Ans. 21 L. 1 s. 0 d.
6. What is the interest of 220 L. for one year, at 4 L.
per cent. per annum?
7. What is the interest of 76 L. for
per cent. per annum ?
year, at 5 L.
Ans. 8 L. 16 s.
one year, at 5 L.
Ans. 3 L. 16 s.
8. What is the interest of 270 L. 10 s. 6 d. for one
per cent. per annum?
9. What is the interest of 542
at 6 dollars per cent. per annum ?
Ans. 13 L. 10s. 64 d.
dollars for one year,
Ans. $3252.
542
6
$32.52
10. What is the interest of 756 dollars for one year,
at 7 dollars per cent. per annum?
Ans. $52.92.
11. What is the interest of 600 dollars for one year,
Ans. $30.00.
12. What is the interest of $438.25 for one year, at
6 dollars per cent. per annum?
at 5 dollars per cent. per annum ?
Ans. 26 dols. 29 cts. 5 m. or $26.291.

SIMPLE INTEREST.
101
D. cts.
438 25
Or thus:
D.
cts.
6
438 25
6
26|29 50
$26.29.5|0
100
29|50
10
5|00
13. What is the interest of $322.71 for one year, at
5 dollars per cent. per annum?
Ans. $16.134.
14. What is the interest of $75.95 for one year, at 7
dellars per cent. per annum?
Ans. $5.31+
Note.-When the amount is required, add the prin-
cipal to the interest.
15. What is the amount of 173 L. 17 s. 83 d. for one
year, at 7 L. per cent. per annum?
Ans. 186 L. 1 s. 1 d.
16. What is the amount of a bond for 756 dollars,
for one year, at 6 dollars per cent. per annum?
CASE 2.
Ans. $801.36.
When there is a fraction, as 4, 1, 4, &c. in the rate.
per cent.
RULE.
Multiply the principal by the pounds or dollars of
the rate per cent; to the product add 4, 4, or 4, &c. of
said principal, and divide the result by 100, as in the
foregoing case.
EXAMPLES.
1. What is the interest of 432 L. 10 s. for one year,
at 5 L. per cent. per annum?
Ans. 23 L. 15 s. 9 d.

102
SIMPLE INTEREST.
2. What is the interest of 428 dollars for one year,
at 6 dollars per cent. per annum?
Ans. $26.75.
(1)
L.
(2)
S.
Dols.
| 432
10
428
51
64
2162 10
2568
216
5
107
23 78 15
$26.75
20
15175
12
9100
3. What is the interest of 216 L. 5 s. for one year, at
5 L. per cent. per annum?
Ans. 11 L. 17 s. 10 d.
4. What is the interest of 500 L.
64 L. per cent. per annum?
5. What is the interest of 855 L.
year, at 54 L. per cent. per annum ?
for one year, at
Ans. 31 L. 5 s.
17 s. 6 d. for one
Ans. 49 L. 4 s. 3 d.
6. What is the interest of 300 dollars for one year,
at 64 dollars per cent. per annum ?
CASE 3.
Ans. $18.75.
To find the interest when the time is two or more
years.
RULE.
Find the interest of the given sum for 1 year: then
multiply the interest for 1 year by the number of years
given.
PROOF.
By the Double Rule of Three.

SIMPLE INTEREST.
103
EXAMPLES.
1. What is the interest of 700 L. 16s. 8d. for 5 years,
at 6 L. per cent. per annum?
d.
Ans. 210 L. 5 s.
L. E.
700 16 8
L. S.
Interest for 1 year 42
1
6
5
42|05 0 0
Interest for 5 years 210
5
20
•
100
2. What is the interest of 750 L. for 3 years, at 6 L.
per cent. per annum ?
Ans. 135 L.
3. What is the interest of 375 L. 10 s. 6 d. for 4 years,
at 7 L. per cent. per annum? Ans. 105 L. 2 s. 11 d.
4. What is the interest of 353 L. 6 s. 3 d. for 9
years,
at 5 L. per cent. per annum ? Ans. 158 L. 19 s. 9 d.
5. What is the interest of $438.25 for 5 years, at 6
dollars per cent. per annum?
Ans. $131.471.
6. What is the interest of 1000 L. for 4 years, at
64 L. per cent. per annum?
Ans. 250 L.
7. What is the interest of 1711 L. 15 s. for 2 years,
at 5 L. per cent. per annum?
Ans. 196 L. 17 s.
5 dollars per cent. per annum?
per cent. per annum ?
8. What is the interest of 320 dollars for 6 years, at
Ans. $105.60.
9. What is the amount of 720 L. for 3 years, at 6 L.
Ans. 849 L. 12 s.
10. On a mortgage for 1256 dollars ther is 4 years
interest due, at 6 dollars per cent. per annum, which
is to be paid with the principal; what sum will dis-
charge the debt?
Ans. $1557.44.
CASE 4.
To find the interest when the given time is months,
weeks, or days, less or more than a year.
RULE.
Find the interest of the given sum for one year, then,
As one year
Is to the given time,
So is the interest of the giver sum for one year,
To the interest required.
Or, take parts of the yearly interest for the aliquot
K

104
SIMPLE INTEREST.
parts of a year that are in the given time, and add the
interest for the odd days (if any) found by the Rule of
Three.
EXAMPLES.
1. What is the interest of 350 L. for 3 years and 10
months, at 6 L. per cent. per annum?
Ans. 80 L. 10s.
L.
Y. Y. M.
L.
350
1 : 3 10 :
::
21
6
12
12
46
L. 21/00
12
46
126
84
12)966
80 L. 10 s
Or thus:
M.
6 |
4
L.
| 21 interest for one year.
3
63 interest for three years.
10 10-
for six months.
for four months.
7 00-
80 L. 10 s.
2. What is the interest of 150 L. 19 s. for 3 years
and 4 months, at 6 L. per cent. per annum?
Ans. 30 L. 3 s. 9 d.
3. What is the interest of 57 L. 17s. 8d. for 3 months,
Ans. 17 s. 44 d.
4. What is the interest of 7500 dollars for 4 months,
at 7 dollars per cent. per annum? Ans. $175.00.
5. What is the interest of 400 L. for 1 week, at 5 L.
per cent. per annum?
at 6 L. per cent. per annum?
Ans. 7 s. 84 d. +
6. What is the interest of 126 L. 12 s. for 16 weeks,
at 4 per cent. per annum?
Ans. 1 L. 15 s. 0 d. +
per cent. per annum?
7. What is the interest of 250 L. for 73 days, at 7 L.
Ans. 3 L. 10 s.
days, at 6 L.
Ans. 12 L.
d. for 1 year,
8. What is the interest of 500 L. for 146
per cent. per annum ?
9. What is the interest of 71 L. 3s. 11

SIMPLE INTEREST.
105
5 months, and 25 days, at 6 L. per cent. per annum?
Ans. 6 L. 6 s. 10 d.
10. What is the amount of a bond for 967 dollars,
for 2 years and 4 months, at 6 dollars per cent. per
annum ?
Ans. 1102.38.
11. What is the amount of 100 L. for 18 months, at
8 per cent. per annum ?
Ans. 12 L.
Note. The answers to the following questions may
be found by the concise method at the bottom of the
page: but it is thought best that the learner should first
obtain them by the preceding general rule.
12. What is the interest of 900 L. for 8 months, at
6 L. per cent. per annum ?
Ans. 36 L.
13. What is the interest of 450 L. for 4 months, at
7 L. per cent. per annum?
14. What is the interest of 148 L.
months, at 6 per cent. per annum?
12 s. 6
Ans. 10 L. 10 s.
d. for 11
Ans. 8 L. 3 s. 5 d.
15. What is the interest of 1260 dollars for 4 months,
at 6 dollars per cent. per annum : and also at 7 per
cent. per annum?
Ans. {$25.20 interest at 6 per cent.
interest at 7 per cent.
* The interest of any sum for any number of months, may be con-
cisely found by the following method.
Multiply the principal by half the number of months, and divide the
product by 100; or multiply the principal by the whole number of
months, divide the product by 2, and divide the quotient by 100: the re-
sult of either operation will be the interest at 6 per cent. per annum.
For the interest at any other rate per cent. take aliquot parts of the
interest at 6 per cent., and add or subtract as the case requires.
EXAMPLE.
What is the interest of 450 L. for 8 months, at 6 L. per cent. per annum;
and also at 7 L. per cent. per annum ?
L.
450
Or thus:
L.
450
4
8
| + |
18100
2)3600
18|00
Le
18 interest at 6 per cent.
3 interest at 1 per cent.
21 interest at 7 per cent.

106
SIMPLE INTEREST.
16. What is the interest of 630 dollars for 8 months,
at 6 dollars per cent. per annum ?
Ans. $25.20.
17. What is the interest of 7342 dollars for 16 months,
at 6 dollars per cent. per annum? Ans. $587.36.
18. What is the interest of 750 dollars for 9 months,
at 7 dollars per cent. per annum ?
Ans. $39.371.
19. What is the interest of 375 dollars for 5 months,
at 6 dollars per cent. per annum? Ans. $10.314.
* 20. What is the interest of $460.50 for 4 months,
at 6 dollars per cent. per annum?
Ans. $9.21.
21. What is the interest of $230.25 for 8 months, at
Ans. $10.74).
7 dollars per cent. per annum?
22., What is the interest of $764.50 for 3 years and
10 months, at 6 dollars per cent. per annum ?
Ans. $175.831.
To find the interest of any given sum as computed
at the banks, at 6 per cent.
RULE.
1. Multiply the dollars by the number of days, and
divide the product by 6; the quotient will be the in-
terest in mills. Or,
2. If the principal be any number of dollars, the in-
terest for 60 days, at 6 per cent., will be exactly that
When the principal consists of dollars and cents, the interest may
be found thus:
Reduce the principal to cents, by removing the separating point: then
multiply and divide as directed in the last note, and separate one figure
from the right of the result as a remainder or fraction; the figures on the
left of this will be the interest in mills.
EXAMPLE.
What is the interest of $425.98 for 3 months, at 6 dollars per cent. per
annum? and also at 7 dollars per cent. per annum?
D.cts.m.
cts.
42598
3
2)127794
6.38.9 int. at 6 p. ct.
1.06.4+int. at 1 p. ct.
Interest in mills 6389|7
$7.45.3+int. at 7 p. ct.

SIMPLE INTEREST.
107
number of cents: and for the time more or less than 60
take aliquot parts.*
EXAMPLES
1. What is the interest of 1542 dollars for 90 days,
at 6 per cent. per annum? also, for 60,† for 30, and for
20 days, at the same rate?
1542 int. for 60 days in cents.
Dols.
30
1542
90
20
771 int. for 30 days in cents.
6)138780
23130 mills.
514 int. for 20 days in cents.
D. cts
Or, int. for 60 days 15. 42.
Or, $23.13 int. for 90 days.
for 30
for 20
7. 71.
5. 14.
2. What is the interest of 771 dollars for 90 days, at
6 per cent. per annum ?
Ans. $11.562.
3. What is the interest of 3084 dollars for 30 days,
at 6 per cent. per annum?
Ans. $15.42.
4. What is the interest of 2324 dollars for 54 days,
at 6 per cent. per annum?
Ans. $20.911 -
5. What is the interest of 3942 dollars for 50 days,
at 6 per cent. per annum?
Ans. $32.85.
CASE 5.
To find the principal, when the amount, time, and
rate per cent. are given.
RULE.
Find the amount of 100 pounds, or dollars, at the
rate and time given: then,
As the amount of 100 pounds, or dollars.
Is to the amount given,
So are 100 pounds, or dollars,
To the principal required.
*This is calculating after the rate of 360 instead of 365 days to the
year, which will always make the interest rather too much.
If the interest found by this rule be divided by 73, the quotient will
show by how much it exceeds the true interest.
+ When a note is drawn for 60 days, the interest is mostly calculated
for 63, on account of three days called days of grace, which are commonly
allowed the payer, on all notes, after the time expires for which they are
drawn. The interest is here only computed for the given time.
K 2

108
SIMPLE INTEREST.
EXAMPLES.
1. What principal at interest for five years, at 6 per
cent. per annum, will amount to 650 L. ?
Ans. 500 1.
L.
6
5 years
L.
*•
L.
130
650 :: 100
100
L.
1:30)65000(500
650
00
30 int. of 100 L. for 5 years.
100
130 amt. of 100 L. for 5 years.
2. What principal at interest for 10 years, at 6 per
cent. per annum, will amount to 1300 L.
Ans. 812 L. 10 s.
3. What principal at interest for 4 years, at 5 per
cent. per annum, will amount to $571.20?
CASE 6.
Ans. 476 dollars.
To find the rate per cent. when the amount, time, and
principal are given.
RULE.
Subtract the principal from the amount, and the re-
mainder will be the interest for the given time: then,
As the principal,
Is to one hundred pounds or dollars,
So is the interest of the principal, for the given time,
To the interest of 100 pounds, or dollars, for the
same time:
Again,
As the given time,
Is to one year,
So is the interest last found,
To the rate per cent. required.
EXAMPLES.
1. At what rate per cent. per annum, will 500 L.
amount to 650 L. in 5 years?
L.
L.
650 Amount.
500 : 100
500 Principal.
again,
Y.
150 Int. for the given time.
Ans. 6 L. per cent
L.
L.
:: 150 00 30
Y.
L. L.
5 : 1 :: 30: 6

INSURANCE, COMMISSION, AND BROKAGE. 109
2. At what rate per cent. per annum, will 500 L.
amount to 725 L. in 9 years? Ans. 5 L. per cent.
3. At what rate per cent. per annum, will 600 dollars
amount to 856 dols. 50 cts. in 9 years and 6 months?
Ans. 41 per cent.
CASE 7.
To find the time, when the principal, amount, and
rate per cent. are given.
RULE.
Find the interest of the principal for one year.
Find the interest for the time required, by subtract-
ing the principal from the amount then,
As the interest of the principal for one year,
Is to the interest for the time required,
So is one year,
To the time required.
EXAMPLES.
1. In what time will 500 L. amount to 725 L., at 5
fer cent. per annum ?
Ans. 9 years.
L. L.
Y. Y.
25: 225 :: 1:9
L.
L.
500
725
5
500
225 Interest for time required.
4
L. 25 00
2. In what time will 540 L. amount to 734 L. 8 s., at
per cent. per annum ?
Ans. 9 years.
3. In what time will 600 dollars amount to 798 dol-
Ans. 54 years.
lars, at 6 per cent. per annum?
INSURANCE, COMMISSION, AND
· BROKAGE.
Insurance, Commission, and Brokage, are allowances.
made to insurers, factors, and brokers, at a stipulated
rate per cent.
RULE.
Work as if to find the interest of the given sum for
one year, at the proposed rate: or, if the rate be less

110
INSURANCE, COMMISSION, AND BROKAGE.
than 1 per cent., take such aliquot part or parts of the
interest at 1 per cent. as the rate is of a pound, or dol-
lar.
EXAMPLES.
1. What is the commission on 596 L. 18 s. 4 d., at 6
per cent.?
Ans. 35 L. 16 s. 31 d.
L. S.
d.
596
18
4
6
10 0
20
L. 35 81
s. 16 30
12
d. 3|60
4
3½ L. per cent.?
qr. 2/40
2. What is the commission on 1371 L. 9 s. 5 d., at 5
Ans. 68 L. 11 s. 5 d.
per cent.?
3. What is the commission on 526 L. 11 s. 5 d., at
Ans. 18 L. 8 s. 7 d.
4. What is the commission on 1974 dollars, at 5 dol-
lars per cent.?
Ans. $98.70.
5. A factor has sold goods for a merchant to the
amount of 930 L. 10 s., and is to receive 34 L. per cent.
commission: what sum is due to him?
Ans. 30 L. 4 s. 9 d.
6. What is the insurance of 924 L., at 7 L. per cent.?
Ans. 64 L. 13 s. 7 d.
7. What is the insurance of 1250 dollars, at 7½ dol-
lars per cent.?
Ans. $93.75.
s. What is the insurance of an East India ship and
cargo, valued at 14813 L. 15 s., at 15 L. per cent.?
Ans. 2333 L. 3 s. 3 d.
9. What is the brokage on 1321 L. 11 s. 4 d., at 18 L.
Ans. 14 L. 17 s. 4 d.
per cent.?
10. What is the brokage on 874 L. 15 s. 3 d., at 5 s.
or 4 L. per cent.?
Ans. 2 L. 3 s. 84 d.

COMPOUND INTEREST.
111
11. If a broker buy goods for me to the amount of
$1853, and I allow him dollars per cent. for his ser-
vice, what sum must 1 pay him?
Ans. $13.893.
COMPOUND INTEREST.
Compound Interest is that which arises from a prin-
cipal increased by its interest, as the interest becomes
due.
RULE.
Find the amount of the given principal for the first
year, by simple interest; this amount will be the prin-
cipal for the second year, and the amount of this prin-
cipal, found as before, will be the principal for the third
year, and so on.
From the last amount, subtract the given principal,
and the remainder will be the compound interest.
EXAMPLES.
1. What is the compound interest of 500 L. for 3
Ans. 78 L. 16 s. 3 d.
years, at 5 per cent.?
Principal
L.
500
Interest for 1st year 25
Amount 1st year
525
Interest 2d year
26 5
Amount 2d year
551 5
Interest 3d year
27 11 3
Amount 3d year
Principai
578 16 3
500
L. 78 16 s. 3 d.
2. What is the compound interest of 450 L. for 3
years, at 5 per cent. per annum? Ans. 70L. 18s. 7½ d.
3. What is the con.pound interest of 760 L. 10 s. for
4 years, at 6 per cent. per annum?
Ans. 199 L. 12 s. 2 d.

112
DISCOUNT.
4. What is the compound interest of 500 dollars for
4 years, at 6 per cent. per annum? Ans. $131.234.
5. How much will 400 L. amount to in 4 years, at 6
per cent. per annum?
Ans. 504 L. 19 s. 94 d.
DISCOUNT.
Discount is an allowance made for the payment of a
sum of money before it becomes due, according to a
certain rate per cent. agreed on between the parties
concerned.
The present worth of any debt, not yet due, is so
much money as, being put to interest, at a given rate
per cent. till the debt become payable, will amount to
a sum equal to the dent.
RULE.
Find the amount of 100 pounds, or dollars, at the
rate and time given. then,
As the amount of 100 pounds, or dollars,
Is to the given sum, or debt,
So is 100 pounds, or dollars,
To the present worth.
Subtract the present worth from the debt, and the
remainder will be the discount.
PROCF.
Find the amount of the present worth for the time
and rate proposed, which must
debt.
EXAMPLES.
the given sum or
1. What is the present worth of 540 dollars, due in
3 years, discount at 6 per cent. per irrum?
$6 rate.
3 years.
18
100
118 amt. of $100.
Ans. 500 dollars.
$
$
GB
118 :
599
100
100
118)59000į 500 dols.
590
00

DISCOUNT
113
2. What is the discount of 795 L. 11 s. 2 d. for 11
months, at 6 per cent. per annum ?
Ans. 41 L. 9 s. 6 d.
M. M. L. L. s.
L. s.
d.
12: 11 :: 6 : 5 10
795 11
2 whole debt.
100 0
754 1
8 present worth.
Amt. of 100L. 105 10
41 9
6 discount.
L. s.
L. s. d.
L. L.
s. d.
105 10: 795 11 2 :: 100: 754 1 8 present worth.
3. What is the present worth of 672 L. due in 2
years; discount at 6 per cent. per annum? Ans. 600 L.
4. What is the present worth of 308 L. 15 s. due in
18 months; discount at 8 per cent. per annum ?
Ans. 275 L. 13 s. 4 d.
5. What is the present worth of $430.67, due in 19
months; discount at 5 per cent. per annum?
Ans. $399.07.
6. What is the discount of 112 L. 12 s. due in 20
months, at 7 per cent. per annum? Ans. 11 L. 15s. 3 d.
7. What is the present worth of 100 L., one half due
in 4 months, and the other half in 8 months; discount
at 5 per cent. per annum?
Ans. 97 L. 11 s. 4 d.
8. Bought goods amounting to $615.75, at 6 months.
credit; how much ready money must be paid, if a dis-
count of 4 per cent. per annum be allowed?
Ans. $602.20.
9. What is the difference between the interest of
1204 dollars, at 5 per cent. per annum for 8 years; and
the discount of the same sum for the same time and
rate per cent.?
Ans. $137.60.
Note.-Discount for present payment is often made
without regard to time; it is then found precisely as
the interest of the given sum for 1 year.
EXAMPLES.
1. How much is the discount of 853 dollars, at 2 per
cent.?
Ans. $17.06.
$53
2
17.06
2. How much is the discount of 750 dollars, at 3 per
cent.?
Ans. $22.50.

114
EQUATION.
3. How much is the discount of 650 L., at 4. per
cent.?
Ans. 26 L.
4. Bought goods on credit, amounting to 1656 dol-
lars; how much ready money must be paid for them, if
a discount of 5 per cent. be allowed? Ans. $1573.20.
5. A holds B's note for 175 L. 10 s.; he agrees to al-
low B a discount of 3 per cent. for present payment:
what sum must B pay?
Ans. 170 L. 4 s. 84 d.
EQUATION.
Equation is a method of reducing several stated
times, at which money is payable, to one mean or
equated time.
RULE.
Multiply each payment by its time, add the several
products together, and divide the sum by the whole
debt; the quotient will be the equated time.
PROOF.
The interest of the sum payable at the equated time,
at any given rate, will equal the interest of the several
payments, for their respective times, at the same rate.
EXAMPLES.
1. Cowes D 100 dollars, of which 50 dollars is to be
paid at 2 months, and 50 at 4 months; but they agree
that the whole shall be paid at one time; when must it
be paid?
Ans. 3 months.
50×2=100
50X4=200
1|00)3100
3 months.
2. A owes B 380 L., of which 100 L. is to be paid at
6 months, 120 L. at 7 months, and 160 L. at 10 months,
but they agree that the whole shall be paid at one time:
when must it be paid?
Ans. at 8 months.
3. A merchant has owing to him 300 L. to be paid as

BARTER.
115
follows: 50 L. at 2 months, 100 L. at 5 months, and 150 L.
at 8 months; it is agreed to make one payment of the
whole: at what time must it be paid? Ans. 6 months.
4. F owes H 2400 dollars, of which 480 dollars are
to be paid at present, 960 dollars at 5 months, and the
rest at 10 months, but they agree to make one payment
of the whole, and wish to know the time.
·
Ans. 6 months.
5. A merchant has purchased goods to the amount of
2000 dollars, of which sum 400 dollars are to be paid at
present, 800 dollars at 6 months, and the rest at 9
months; but it is agreed to make one payment of the
whole what is the equated time? Ans. 6 months.
6. G owes K 420 L. which will be due 6 months
hence: it is agreed that 60 L. shall be paid now, and that
the rest remain unpaid a longer time than 6 months
when must it be paid?
Ans. in 7 months.
BARTER.
Barter is the exchanging of one commodity for an-
other, according to the price or value agreed upon by
the parties concerned.
Questions relating to barter are solved either by the
Rule of Three or by Practice.
Note.-When a given quantity of any commodity at
a given price is to be bartered for another commodity
at a given price, find the value, in money, of that com-
modity whose quantity is given; then find what quan-
tity of the other may be had for that value.
EXAMPLES.
1. Hov much sugar, at 11 d. per lb., must be given in
barter for 100 lb. of rice, at 31 d. per lb.? Ans. 350 lb.
lb.
lb.
d.
d.
11 : 3850
1 : 350
1100
31/2
3300
550
3850 d. the value of the rice.
L

116
BARTER.
2. How much sugar, at 9 d. per lb., must be given in
barter for 492 lb. of rice, at 3 d. per lb.? Ans. 164 lb.
3. How much tea, at 64 cents per lb., must be given
in barter for 448 lb. of coffee, at 20 cents
4. What quantity of tea, at 10s. per
given for 720 lb. of chocolate, at 4 s. 2 d.
per lb.?
Ans. 140 lb
lh., must be
per Ih.?
Ans. 300 lb.
per bushel, is
5. How much wheat, at 1 dol. 25 cts.
equal in value to 50 bushels of rye, at 70 cents per
bushel ?
6. B has 75 yards of muslin, at 1
which he is to give to H for linen, at 5
much linen will he receive?
Ans. 28 bushels.
s. 4 d. per yard,
s. per yard; how
Ans. 20 yards.
7. A has sugar at 9 d. per lb., for a quantity of which
F is to give him 225 lb. of tea, at 6 s. per lb.; how much
Ans. 1800 lb.
sugar must F receive for his tea?
8. How much sugar, at 8 d. per lb., must be given in
barter for 20 cwt. of tobacco, at 3 L. per cwt.?
Ans. 16 cwt. 0 qrs. 8 lb.
9. A merchant has 1000 yards of canvass, at 91 d. per
yard, which he is to barter for serge, at 104 d. per yard;
how many yards of serge should he receive?
Ans. 92634 yards.
10. A grocer bartered 5 cwt. of sugar, at 6 d. per lb.,
for cinnamon, at 10 s. 8 d. per lb.; how much cinnamon
did he receive?
Ans. 26 lb. 4 oz.
11. A has 41 cwt. of hops, at 30 s. per cwt., for which
B is to give him 20 L. in money, and the rest in prunes,
at 5 d. per lb.: what quantity of prunes must A receive?
Ans. 1992 lb.
12. A and B barter: A has 320 lb. of chocolate, at
4 s. 6 d. per lb., for which B is to give him 30 L. in
money, and the rest in cotton, at 8 d. per lb. How
much cotton is B to give A?
Ans. 1260 lb.
13. L has 41 cwt. of hops, at 4 dols. 50 cts. per cwt.,
for which M is to give him 28 dols. 50 cts. in money,
and the rest in salt, at 80 cts. per bushel; what quantity
of salt is M to give L?
Ans. 195 bushels.
14. G has 28 lb. of tea, at 11 s. 6 d. per lb., for which
B is to give him 40 yards of linen, at 7 s. 4 d. per yard,
and the rest in money; how much money must
ceive?
re-
Ans. 1L. 149. 5d.
j.
அ

LOSS AND GAIN.
117
15. R gave 189 yards of linen, at 6 s. 8 d. per yard,
to C, for 42 yards of cloth; what was the cloth per
yard?
Ans. 30 s.
16. A has 608 yards of cloth, at 14 s. per yard, for
which B is to give him 125 L. 12 s. in money, and 85
cwt. 2 qrs. 24 lb. of bees wax. At how much is the
Ans. 3 L. 10 s.
bees wax valued per cwt.?
17. C has wheat at $1.25 cents per bushel, ready
money; but in barter he will have $1.50 per bushel
D has cotton at 20 cents per lb. ready money: what
price must the cotton be in barter, and how much cot-
ton must be given for 100 bushels of wheat?
Ans. {
The cotton must be 24 cts. per lb., and 625 lb.
must be given for 100 bushels of wheat.
LOSS AND GAIN.
Loss and gain instructs merchants and traders, so to
estimate their goods in buying and selling, as to know
what they gain or lose in dealing.
Questions in Loss and Gain are solved by the Rule
of Three, or by Practice.
EXAMPLES.
1. A storekeeper sold 100 yards of silk, at $1.50 per
yard, which cost him $1.25 per yard; how much did
he gain by the sale?
$1.50
$1.25
25 gain per yard.
2. If a grocer buy 265 lb. of
afterwards sell the whole at 7 s.
he gain by the transaction?
265
7
2|0)185 5
92 L. 15 s.
yd. yds. cts.
1 : 100 :: 25
100
Whole gain $25.00
tea for 79 L. 10 s., and
per lb., how much will
L.
S.
Sold for 92
15
Cost 79
10
Gain L. 13
5
3. A shopkeeper bought 53 yards of silk, at 12 s. per
yard, and afterwards sold it at 14 s. per yard; how much
did he gain by the sale?
Ans. 5 L. 6 s.

118
LOSS AND GAIN.
4. G bought 650 lb. of sugar, at 10 cents per lb., and
sold it at 12 cents per lb.; how much did he gain?
Ans. $13.00.
5. If I buy 765 yards of baize, at 3 s. 4 d. per yard,
and sell it at 3 s. 9 d. per yard, how much do I gain?
Ans. 14 L. 6 s. 10 d.
6. Bought 2016 lb. of rice, at 3 d. per lb., and sold it
at 3 d. per lb.; how much was gained by the trans-
action?
Ans. 4 L. 4 s.
7. If I lay out 1000 dollars in hats, at 4 dollars each,
and sell them afterwards at 4 dols. 50 cts. each, how
much will I gain?
Ans. 125 dols.
8. A merchant bought 1300 lb. of coffee, at 22 cts.
per lb., and was afterwards obliged to sell it at 20 cts.
per lb.; how much did he lose?
Ans. $26.00.
9. B laid out 250 L. in cloth, at 30 s. per yard, and,
afterwards, finding it was damaged, sold it at 26 s. 3 d.
per yard; how much did he lose? Ans. 31 L. 5 s.
10. A shopkeeper bought 42 yards of muslin for 4 L.
14 s. 8 d., and sold it at 2 s. 6 d. per yard; whether did he
gain or lose, and how much? Ans. He gained 10s. 4 d.
11. A draper bought 100 yards of cloth for 56 dollars;
how must he sell it per yard, to gain 19 dollars in the
whole?
Ans. 75 cents.
cent.?
12. If a grocer buy a quantity of tea for 125 L., and
sell it again for 150 L., how much will he gain per
Ans. 20 per cent.
13. If a yard of mantua be purchased for $1.20, and
sold again for $1.50, what is the gain per cent.?
Ans. 25 per cent.
14. If a yard of velvet be bought for 16 s., and sold
again for 12 s., what is the loss per cent.?
Ans. 25 per cent.
15. Bought a chest of tea, weighing 490 lb. for 326
dollars, and sold it for $370.10, what was the profit on
each lb.?
Ans. 9 cents.
16. If I buy 100 yards of cambric for 56 L., at how
much must I sell it per yard, to gain 15 per cent.?
Ans. 12 s. 10 d.
17. Bought 12 pieces of white cloth, for 6 L. 10 s.
per piece, and paid 20 s. 10d. per piece for dying it;
how much must each piece be sold for, to gain 20 per
Ans. 9 L. 1 s.
cent.?

FELLOWSHIP.
119
18. If a trader gain 14 d. per shilling on his goods,
how much does he gain per cent.? Ans. 12 per ct.
19. If I buy 28 pieces of stuffs at 4 L. per piece, and
sell 10 of the pieces at 6 L. per piece, and 8 at 5 L. per
piece; at what rate per piece must I sell the rest, to gain
20 per cent. by the whole?
Ans. 3 L. S s. 9 d.
20. Having bought a parcel of goods for 18 L., and
sold the same immediately for 25 L. with 4 months
credit, what is gained per cent. per annum?
Ans. 116 L. 13 s. 3 d.
FELLOWSHIP.
Fellowship is a rule, by which merchants, &c. trad-
ing in company with a joint stock, are enabled to ascer-
tain each person's particular share of the gain or loss,
in proportion to his share in the joint stock.
By this rule, also, legacies are adjusted, and the ef-
fects of bankrupts divided, &c.
CASE 1.
When the several stocks in company are considered
without regard to time.
RULE.
As the whole sum, or stock,
Is to either person's share in stock, &c.
So is the whole gain or loss,
To that person's share of the gain or loss,
PROOF.
The sum of the several shares must equal the whole
gain or loss.
EXAMPLES.
1. Three merchants, trading together, gained 800 dol-
lars; A's stock was 1200 dols., B's 4800 dols., and C's
2000 dols.: what was each man's share of the gain?
A's stock 1200 dols.
B's stock 4800 dols.
C's stock 2000 dols.
Whole stock 8000 dollars.
BARZEZ TERSISKYRIUATRERINGENA

120
FELLOWSHIP.
As 8000
1200 :: 800: 120 A's share of gain.
As 8000 4800 :: 800: 480 B's share of gain.
As 8000: 2000 :: 800: 200 C's share of gain.
2. D, E, and F, trading together, gained 120 L.; D's
stock was 140 L., E's was 300 L., and F's was 160 L.:
what was each man's share of the gain?
Ans. D's share was 28 L., E's 60 L., F's 32 L.
3. Three merchants, trading together, lost goods to
the value of 1920 dols.; now suppose A's stock was
2880 dols., B's 11520 dols., and C's 4800 dols.: what
share of the loss must each man sustain ?
JA's share 288 dols.
Ans. B's
C's
1152 dols.
489 dols.
4. A, B, and C freighted a ship with 108 tuns of
wine, of which A had 48 tuns, B 36, and C 24, but by
reason of stormy weather were obliged to cast 45 tuns
overboard; how much must each man sustain of the
loss?
Ans. A 20 tuns, B 15, and C 10.
5. If the money and effects of a bankrupt amount to
1400 L. 14 s. 6 d., and he is indebted to M 742 L. 12 s.,
to B 641 L. 19 s. 8 d., and to C 987 L. 19 s. 9 d.; how
must the property be divided among them?
M must have 438 L. 8 s. 44 d.
Ans. B
C
379 L. 0 s. 32 d.
583 L. 5 s. 93 d.
6. Suppose a person is indebted to S 70 L., to T
400 L., and to V 140 L. 12 s. 6 d., but upon his decease
his property is found to be worth only 409 L. 14 s.;
how must it be divided among his creditors?
{
S
Ans. T
must have 46 L. 19 s. 33 d.
V -
268 L. 7 s. 74 d.
94 L. 7 s. 04 d.
7. Three graziers pay among them 120 dols. for a
grass enclosure, into which they put 300 oxen, whereof
L had 80, N 100, and C 120; how much should each
person pay ? Ans. L 32 dols., N 40 dols., and C. 48 dols.
CASE 2.
When the respective stocks in company are consi
dered with time.
RULE.
Multiply each man's stock by its time; then,

FELLOWSHIP.
121
As the sum of the products,
Is to either particular product;
So is the whole gain or loss,
To the gain or loss of the stock from which that pro-
duct is obtained.
EXAMPLES.
1. Three merchants traded together; A put in 120 L.
for 9 months, B 100 L. for 16 months, and C 100 L. for
14 months, and they gained 100 L.; what is each man's
share?
L. m.
A's stock 120 x 9
1080
B's stock 100 x 16
1600
C's stock 100 x 14 = 1400
Sum 4080
Sum
Prod. L.
L. s. d.
As 4080
1080 :: 100: 26 9 42 + A's share.
1600 :: 100 39 4 34+ B's share.
As 4080 1400 :: 100: 34 6 34+ C's share.
As 4080
2. B, C, and D traded together; B put in 50 dollars
for 4 months, C 100 dollars for 6 months, and D 150
dollars for 8 months; they gained $126.80: what is
each man's share of the gain?
Ans. B's share is $12.68, C's $38.4, D´s $76.8.
3. B and C trade in company; B put in 950 L. for 5
months, and C 785 L. for 6 months, and by trading they
gain 275 L. 18 s. 4 d.;_ what is each man's share of the
gain? Ans. B's 138 L. 10 s. 10 d., C's 137 L. 7 s. 6 d.
4. Three merchants trade in company; A put in
400 L. for 9 months, B 680 L. for 5 months, and C 120 L.
for 12 months; but by misfortunes lost goods to the va-
lue of 500 L.: what must each man sustain of the loss?
A must lose 213 L. 5s. 43 d. +
201 L. Ss. 5d. +
85 L. 6s. 14 d. +
Į
Ans. B
C
5. A, B, and C made a stock for 12 months; A put
in at first $873.60, and 4 months after he put in $96.00
more; B put in at first $979.20, and at the end of 7
months he took out $206.40; C put in at first $355.20,
and 3 months after he put in $206.40, and 5 months
after that he put in $240.00 more. At the end of 12

122
EXCHANGE.
months their gain is found to be $3446.40; what is
SA's share is $1334.821
each man's share of the gain?
Ans. B's
C's
$1271.614+
$839.96
EXCHANGE.
Exchange is the reduction of the money of one state
or country to that of another.
Par is equality in value;
but the course of exchange
is frequently above or below par.
Agio is a term used to signify the difference, in some
countries, between bank and current money.
DOMESTIC EXCHANGE.
To change the currency of either state into that of
any other, work by the Rule of Three; or by the theo-
rems in the following page.
EXAMPLES.
1. What is the value of 750 L. Massachusetts cur
rency in New York ?
S. L. S.
6: 750: 8
20
15000
Ans. 1000 L.
Or thus:
L.
3)750
250
1000 L.
6)120000
2|0)2000|0
1000 L.
*Note.-A Spanish dollar is valued at 4 s. 6 d. ster-
ling, and at 7 s. 6 d. Pennsylvania currency: 4 s. 6 d.
sterling is therefore equal to 7 s. 6 d. Pennsylvania cur-
rency, and 100 L. of the former is equal to 1663 L. of
the latter. When exchange between England and
Pennsylvania is at this rate, it is said to be at par.

EXCHANGE.
123
A TABLE,
Exhibiting the value of a dollar in each of the United States; and practical theorems
for exchanging the currency of either into that of any other.
To exchange
from
to
* New England States
and Virginia.
(New England Pennsylvania,
Pennsylvania, N. Jersey,
Delaware, and Maryland.
New York and North
Carolina.
South Carolina and
Georgia.
and Maryland, N. Carolina.
New York
States and
Firginia.
Jersey, Delaw.
and
S. Carolina
and
Georgia.
Dollar
6s. Od.
Add
one 4th.
Add
Subtract
one 3d.
twice.
Subtract
one fifth.
Add
one 15th.
Subtract
one fourth.
Subtract
one 16th.
Dollar
8s. Od.
Add
two 7ths.
Add, that
½
and that.
Multiply by 12
and divide by 7
Dollar
7s. 6d.
5
Multiply by 7
and divide by 12.
Dollar
4s. 8d.
*The New England States are, New Hampshire, Vermont, Massachusetts, Rhode
Island, and Connecticut.
Note.
either opposite to that state, or under it, in the table.
The value of a dollar in any state is found,

124
EXCHANGE.
Ans. 2000 L.
2. What is the value of 1500 L., Massachusetts cur-
rency, in New York?
3. What is the value of 240 L., Pennsylvania cur-
rency, in New York?
Ans. 256 L.
4. What is the value of 933 L. 6 s. 8 d., South Caro-
lina currency, in Pennsylvania?
Ans. 1500 L.
5. What sum in Pennsylvania currency is equal to
120 L. 10 s. in New York? Ans. 112 L. 19 s. 4 d.
6. What sum in Pennsylvania currency is equal to
234 L. 4 s. in New England or Virginia?
Ans. 292 L. 15 s.
7. What is the value of 173 L. 16 s., New Jersey cur
rency in New York?
Ans. 185 L. 7 s. 84 d.
S. What is the value of 900 L., New England or Vir-
ginia currency, in South Carolina?
Ans. 700 L.
9. Change 792 L. 19 s. 7 d. of North Carolina into
Pennsylvania currency.
Facit 743 L. 8 s. 44 d.+
10. What sum of Maryland currency is equal to 6307 L.
13 s. 5 d. of New York? Ans. 5913 L. 8 s. 94 d. +
11. What is the value of a bill of 750 L., Pennsyl-
vania currency, in New York or North Carolina cur-
rency?
Ans. 800 L.
12. A merchant in Virginia consigns to his agent in
New York a quantity of tobacco; which, when sold,
and the charges deducted, amounts to 625 L. 6 s., what
is the value thereof in Virginia currency; also in Fe-
deral Money?
Ans. (468 L. 19s. 6 d., Virginia currency.
1563 dols. 25 cts.
FOREIGN EXCHANGE.
Accounts are kept in England, Ireland, and the
English West India Islands, in pounds, shillings, pence,
and farthings; though their intrinsic values in these
places are different.
A TABLE OF DIFFERENT MONEYS.
12
20
Deniers
Sols
3
Livres
France.
1 Sol,
1 Livre,
1 Crown.

EXCHANGE.
125
Spain.
4 Marvadies Vellon,or 1 Quarta,
2 Marvadies of plate
8 Quartas, or
34 Marvadies Vellon
16 Quartas, or
34 Marvadies of plate
8 Rials of plate
10 Rials of plate
5 Piastres
12
Deniers
20
Sols -
5 Livres
Italy.
1 Rial Vellor,
1 Rial of plate,
1 Piastre,
1 Dollar,
1 Spanish Pistole.
1 Sol,
1 Livre,
1 Piece of Eight at Genoa,
6 Livres
6 Solidi
24 Grosses
400 Reas
1000 Reas
1 do.
1 Gross,
1 Ducat.
Portugal.
1 Crusadoe,
1 Millrea.
at Leghorn,
Holland.
8 Pennings
2 Groats
6 Stivers
20 Stivers
2 Florins
6 Florins
5 Guilders*
16 Shillings
6 Marks
-
32 Rustics
6 Copper Dollars
1 Groat,
1 Stiver, 2d.
1 Shilling,
1 Florin or Guilder,
1 Rix Dollar,
1 L. Flemish.
1 Ducat.
Denmark.
1 Mark,
1 Rix Dollar,
1 Copper Dollar,
1 Rix Dollar.
* A stiver is estimated at 2 cents; and a florin or guilder at 40 cents,
nearly.

126
EXCHANGE.
Russia.
1 Gros,
18 Pennings
30
Gros
3
Florins
2 Rix Dollars
1 Florin,
1 Rix Dollar,
1 Gold Ducat.
All the operations in Exchange may be performed by
the Rule of Three or by Practice.
Note. The par of exchange between the United
States and most other trading countries, may be ascer-
tained by the table at page 65.
EXAMPLES.
1. A of Philadelphia is indebted to B of London
1474 L. 16 s. currency; how much sterling must be re-
mitted, when the exchange is at 64 per cent.?
Ans. 899 L. 5 s. 44 d.
L.
L. S.
L. L. s. d.
As 164 1474 16 :: 100: 899 5 44
2. B of London is indebted to C of Philadelphia
943 L. 17 s. 54 d. sterling; for how much currency
may C draw on B, exchange being at 64 per cent.?
L.
L. S. d.
50943
10
H4C* Hills Hikus mko
17 54
471
18 8/3/2
94 7 82
18 7 6
18 7 6
Answer L. 1547 18 11 currency.
3. C of Philadelphia is indebted to D of London
750 L. 2 s. 4½ d. sterling; how much Pennsylvania cur-
rency will discharge the debt, exchange being at 78
per cent.?
Ans. 1335 L. 4 s. 24 d.
4. How much sterling is equal to 1341 L. 9 s. 4 d.,
Pennsylvania currency, exchange being at 67% per
cent.?
Ans. 800 L. 17 s. 63 d.

EXCHANGE.
127
5.
Philadelphia,
Exchange for 452 Ľ. 10 s. 6 d.
Thirty days after sight of this my first of exchange,
second and third of like tenor and date not paid, pay to
Samuel Sims, or order, four hundred and fifty-two
pounds, ten shillings, and sixpence sterling, value re-
ceived, which place to account of
Peter Simpson.
To Samuel Jones, merchant,
London.
What is the value of this bill in Pennsylvania cur-
rency, exchange at 773 per cent.?
Ans. 803 L. 4 s. 7½ d.
6. M of Philadelphia owes P of London 1474 dol-
lars 80 cents; how much sterling must be remitted,
when exchange is at par?
cts. dols.cts. s. d. L. s. d.
As 100 1474.80 :: 4 6: 331 16 7. Ans.
7. What sum sterling is equal to 260 L. 8 s. 6 d. Vir-
ginia currency; exchange 44 per cent.?
Ans. 180 L. 17 s.
8. M of Dublin draws upon M of London for 740 L.
14 s. 6 d. Irish; exchange at 12 per cent.: how much
sterling will discharge this bill? Ans. 661 L. 7 s. 24 d.
9. P of London remits to G of Ireland 651 L. 14 s.
114 d. sterling; with how much Irish must P be credit-
ed, exchange being at 12 per cent.?
Ans. 729 L. 19 s. 2 d.
10. Purchased in Ireland goods to the value of 400 L.
17 s. 9 d. Irish; what sum Pennsylvania currency will
discharge the debt, exchange being at 514 per cent.?
Ans. 607 L. 6 s. 10 d.
11. B of Jamaica is indebted to C of London 1470 L.
12 s. 8 d. sterling; with how much currency will C be
credited at Jamaica, when exchange is at 36 per cent.?
Ans. 2007 L. 8 s. 34 d.
12. D of Jamaica is indebted to E of London 806 L.
5 s. sterling; with how much currency must E be credit-
ed in Jamaica, when the exchange is at 35 per cent.?
Ans. 1088 L. 8 s. 9 d.
13. P of Philadelphia received of A of Amsterdam,
an invoice of goods amounting to 10235 florins, 17 sti-
veis, 8 pennings; what sum of Fennsylvania currency
will discharge the bill, at 354 d. per florin? and what is
M

128
EXCHANGE.
the sum in sterling,
L. sterling?
exchange at 38 s. 6 d. Flemish per
1503 L. 7 s. 10 d. currency.
386 L. 4 s. 54 d. sterling.
Ans.
{
14. A merchant in Rotterdam has a bill drawn on
him for 673 L. 16 s. 8 d. sterling, exchange at 33 s. 4 d.
Flem. per pound sterling; how much Flemish must he
pay?
Ans. 1123 L. 1 s. 14 d.
15. A Connecticut merchant imported goods from
France, amounting, per invoice, to 49008 livres; how
much currency of that state, at 15 d. per livre, will they
amount to?
Ans. 3063 L. currency.
16.
Philadelphia,
Exchange for 4226 livres, 12 sols, & deniers.
Thirty days after sight of this my second of exchange,
(first of the same tenor and date not paid) pay to
Thomas Broker, or order, four thousand two hundred
and twenty-six livres, twelve sols, and eight deniers,
value received; which place to account of
To Thomas Lamot, Merchant,
Bordeaux.
Ans.
Silas Stroud.
How much sterling is the above bill, at 10½ d. per
livre; and how much Pennsylvania currency, at 17 d.
per livre ?
184 L. 18s. 34 d. sterling.
308 L. 3s. 10d. currency.
17. What sum Pennsylvania currency is equal to
2524 piastres, 7 rials, 33 marv. at 7 s. 6 d. per piastre?
Ans. 946 L. 17 s. 5½ d.
18. A merchant of North Carolina shipped a quantity
of flour, which, when disposed of, amounted to 1186
millreas, 500 reas; and received in return 17 pipes of
wine; how much was the wine per pipe, a millrea
reckoned at 7 s. 6 d.
Ans. 26 L. 3 s. 54 d.
19. A Virginia merchant sent goods to Norway,
worth 1743 L. 16s. Virginia currency; how many rix
dollars, at 6s. each, must he receive?
Ans. 5812 dols. 4 s.
Note.-To change bank into current money say: As
100 bank, is to 100 with the agio added; so is the bank
given, to the current required.
To change current money into bank say: As 100
with the agio added, is to 100; so is the current given,
to the bank required.

VULGAR FRACTIONS.
129
20. Change 794 guilders, 15 stivers, current money,
into bank florins, agio 4 per cent.
Result 761 guild. 8 stiv. 11 pennings.
21. Change 761 guilders, 9 stivers bank, into current
money, agio 4 per cent.
Result 794 guild. 15 stiv. 4 pennings.
22. A merchant in Holland wishes to change 4376
florins currency into bank, the agio at 4 per cent.; how
many pounds Flemish bank must he receive?
Ans. 701 L. 1 flo. 13 stiv. 13 pen.
23. In 290 L. 11 s. 10 d., sterling, how many pounds
Flemish; exchange at 33 s. 10d. Flemish bank per pound
sterling, and agio at 4 per cent.? Ans. 513L. 14s. 1 d.
24. A merchant in Philadelphia receives from Lon-
don a parcel of goods, charged in the invoice at 450 L.
10 s. sterling, which he immediately sells at an advance
of 78 per cent.; what is the amount in Pennsylvania
currency; also in Federal money?
Ans. S801 L. 17 s. 9 d.
2138 dols. 37½ cts.
25. Amsterdam changes on London 34 s. 3 d. per L.
sterling, and on Lisbon, at 52 d. Flemish for 400 reas;
how then ought the exchange to go between London
and Lisbon ?
Ans. 754 d. +sterling per millrea.
VULGAR FRACTIONS.
A vulgar fraction is a part, or parts of a unit or inte-
ger expressed by two numbers, placed one above the
other, with a line drawn between them; as 4 one fourth,
two thirds.
The number above the line is called the numerator,
and that below the line the denominator.
The denominator denotes the part, and the numera-
tor informs how many of that part are designed to be
expressed.
Vulgar fractions are either proper, improper, com-
pound, or mixed.
A proper fraction is that of which the numerator is
less than the denominator; as 1, 4, 3, &c.

130
VULGAR FRACTIONS.
An improper fraction is that of which the numera-
tor is equal to, or greater than the denominator; as 4,
1, 11, &c.
7
12
A compound fraction is a fraction of a fraction; as
½ of, or 2 of 3 of, &c.
A mixed number consists of a whole number and a
fraction; as 44, 7%, &c.
REDUCTION OF VULGAR FRACTIONS.
CASE 1.
To reduce a fraction to its lowest terms.
RULE.
Divide the greater term by the less, and that divisor
by the remainder, till nothing be left; the last divisor
will be the common measure. Divide both terms by
the common measure, and the quotients will be the nu-
merator and denominator of the fraction required: or,
Divide the terms by any number that will divide
them both without a remainder, and divide the quo-
tients in the same manner, and so on, till no number
greater than 1 will divide them; the fraction is then at
its lowest terms.
Note. If the common measure be 1, the fraction is
already at its lowest terms. Ciphers on the right hand
of both terms may be rejected; thus 400-4.
EXAMPLES.
1. Reduce to its lowest terms.
72)96(1
72
Or thus:
12) 2)
72
== Result.
V.
Com. measure 24)72(3
72
24)72=Result.
2. Reduce to its lowest terms.
27
3. Reduce 48 to its lowest terms.
56
4. Reduce 2 to its lowest terms.
94
5. Reduce to its lowest terms.
125
6. Reduce to its lowest terms.
170
7. Reduce 1344 to its lowest terms
1536
Result.
Result 4.
Result
Result 1.
Result 43
Result ठ्ठ .

VULGAR FRACTIONS.
131
CASE 2.
To reduce several fractions to others of the same
value, and having a common denominator.
RULE.
Multiply each numerator into all the denominators
but its own, for its respective numerator; and all the
denominators into each other, for a common denomi-
nator.
EXAMPLES.
1. Reduce 4, §, and § to a common denominator.
3X5X6= 90
4X4X6 96 Numerators.
5X4×5=100
4x5×6=120 Common denominator.
Result, and 198.
2. Reduce 2, 4, and § to a common denominator.
7
100.
40
Result 3, 44, 44.
64 64
3. Reduce ½, 1, §, and to a common denominator.
4. Reduce, , and
ड
8
5. Reduce,,, and 4
Result 눈​, 눈​, 음​,
1,28,
144 192 252
음​.
288 2889 28 37 28 8
to a common denominator.
Result 735, 560 504, 720
to a common denominator.
Result 12, 140, 240, 240
8409 840) 840, 840`
CASE 3.
2409
200 60
To reduce a mixed number to an improper fraction.
RULE.
Multiply the whole number by the denominator of
the fraction, and add the numerator to the product for
a new numerator, under which place the given denomi-
nator.
EXAMPLES.
1. Reduce 12 to an improper fraction. Result 112.
12×9+4=112
new numerator. 112
denominator.
9
2. Reduce 1912 to an improper fraction.
3. Reduce 12 to an improper fraction.
4. Reduce 10019 to an improper fraction.
18
Result 35.
Resuit.
.
Result 512
59
M 2

132
VULGAR FRACTIONS.
5. Reduce 514 to an improper fraction.
Result 2222.
6. Reduce 478141 to an improper fraction.
CASE 4.
Result 37841.
8406
To reduce an improper fraction to a whole or mixed
number.
RULE.
Divide the upper term by the lower.
Note. This case and case 3 prove each other.
EXAMPLES.
1. Reduce 219 to its proper terms.
17) 219(121
17
Result 121.
49
34
15
17
317
19
17
2. Reduce to its proper terms.
3. Reduce to its proper terms.
4. Reduce 126 to its proper terms.
5. Reduce 3848 to its proper terms.
21
6. Reduce 1245 to its proper terms.
22
CASE 5.
Result 161
Result 87.
Result 21
Result 183
Result 561
21
To reduce a compound fraction to a single one.
RULE,
Multiply the numerators together for a new numera
tor, and all the denominators for a new denominator.
Note.-Like figures in the numerators and denomi
nators may be cancelled, and frequently others con
tracted, by taking their aliquot parts.

VULGAR FRACTIONS.
133
EXAMPLES.
1 Reduce of of to a single fraction.
2×3×4=24
3x4x5=60
2
B
ड
Q
Result .
Or, 2 of 3 of 24=3.
A
of of -- as before.
Or cancelled, of – of
5
2. Reduce of of to a single fraction.
ΤΟ
Result 135
135 = 10.
3. Reduce of of to a single fraction.
40 6.
Result 44077S.
4. Reduce of of to a single fraction.
Result 112
5. Reduce of 12 of 2 to a single fraction,
29
304
Result 3003
143
232
6. Reduce 12 of ½ of to a single fraction.
je
4872
CASE 6.
Result f
5
168 14
To reduce the fraction of one denomination to the
fraction of another, but greater, retaining the same
value.
RULE.
Make the fraction a compound one, by comparing it
with all the denominations between it and that to
which it is to be reduced; which fraction reduce to a
single one
EXAMPLES.
1 Reduce & of a penny to the fraction of a pound.
je
5
of of 10 of a pound.
12
20 1440
28 8
2. Reduce of a penny to the fraction of a pound.
Result L.
1
300
3. Reduce of a farthing to the fraction of a shilling.
Results.
4. Reduce of a cent to the fraction of a dollar.
Result dol.
1
180
5. Reduce of an oz., troy, to the fraction of a pound.
៖
Result lb.
27

134
VULGAR FRACTIONS.
6. Reduce of a lb. avoirdupois to the fraction of a
cwt.
T3
Result 3 cwt.
392
7. Reduce of a pint of wine to the fraction of a
hhd.
1
Result hhd.
728
8. Reduce 1 of a minute to the fraction of a day.
CASE 7.
1
Result day.
T384
To reduce the fraction of one denomination to the
fraction of another, but less, retaining the same value.
RULE.
Multiply the given numerator by the parts of the
denomination between it and that to which it is to be
reduced, for a new numerator, and place it over the
given denominator; which reduce to its lowest terms.
Note. This case and case 6 prove each other.
5
EXAMPLES.
1. Reduce of a pound to the fraction of a penny.
1140
5×20×12=1200 d. Result.
4
14 4
2. Reduce of a pound to the fraction of a penny
3. Reduce
1200
Result 4 d.
of a shilling to the fraction of a farthing.
4. Reduce of a dollar to the fraction of a cent.
Result qr.
Result & ct.
5. Reduce of a lb. troy to the fraction of an oz.
27
Result oz.
6. Reduce of a cwt. to the fraction of a lb. avoir-
dupois.
7. Reduce
392
728
1
Result & lb.
017
of a hhd. to the fraction of a pint.
13
Result pt.
8. Reduce of a day to the fraction of a minute.
1584
CASE 8.
Result 19.
To reduce a fraction to its proper value or quantity
in integers.
RULE.
Multiply the numerator by the known parts of the
integer, and divide by the denominator.

VULGAR FRACTIONS.
135
EXAMPLES.
1. Reduce of a pound to its proper value.
2 thirds of a pound.
20
3)40 thirds of a shilling.
13s.+1 third of a shilling
12
3)12 thirds of a penny.
4 d.
Or thus:
L. L.
of 20-40-13 s. 4 d.
Result 13s. 4 d.
2. Reduce of a pound to its proper value.
3. Reduce
4 3
Result 6 s. 8 d.
Result 5 d..
of a shilling to its proper value.
4. Reduce of 5 L. 9 s. to its proper value.
Result 4 L. 13 s. 5 d. 4.
5. Reduce of a dollar to its proper value.
៖
Result 60 cents.
6. Reduce 12 of a lb. troy to its proper quantity.
6
Result 9 oz.
7. Reduce of a lb. avoirdupois to its proper quan-
tity.
Result 8 oz. 14 dr.
8. Reduce of a ton to its proper quantity.
죽음
​Result 3 cwt. 8 lb. 9 oz. 13 dr. 43.
9. Reduce of a mile to its proper quantity.
10
10. Reduce
11. Reduce
12. Reduce
44
13. Reduce
tity.
14. Reduce
Result 4 fur. 125 yds. 2 ft. 1 in. §.
of a yard to its proper quantity.
Result 2 ft. 8 in. 2.
of an acre to its proper quantity.
Result 1 rood, 30 perches.
of a tun of wine to its proper quantity.
Result i hhd. 49 gal.
of a yard of cloth to its proper quan-
Result 3 qrs. 2 nails.
of a year to its proper quantity.
Result 328 days, 12 hours.

136
VULGAR FRACTIONS.
CASE 9.
To reduce any given value, or quantity, to a fraction
of any greater denomination of the same kind.
RULE.
Reduce the given quantity to the lowest denomina-
tion mentioned for a numerator, and the integer into
the same denomination for a denominator.
Note.-If a fraction be given, multiply both parts by
the denominator thereof, and to the numerator add the
numerator of the given fraction.
EXAMPLES.
1. Reduce 13 s. 4 d. to the fraction of a pound.
13 s. 4 d. 160 d.
1 L.=20s.=240 d.
=3 L.
2. Reduce 10 s. 6 d. to the fraction of a pound.
Result 21 L.
3. Reduce 4 d. to the fraction of a shilling. Results.
4. Reduce 5 d. to the fraction of a shilling.
Results.
5. Reduce 9 oz. troy to the fraction of a lb. Result 1h.
6. Reduce 9 oz. 2 dr. & avoirdupois to the fraction
of a pound.
Result 4.
7. Reduce 3 cwt. 8 lb. 9 oz. 13 dr., to the fraction
of a ton.
Result 2 ton.
8. Reduce 3 qr. 2 na. to the fraction of a yard.
T3
7
Řesult yd.
9. Reduce 6 furlongs, 16 poles, to the fraction of a
mile.
Result .
10. Reduce 2 roods, 20 perches, to the fraction of
an acre.
Result
ADDITION OF VULGAR FRACTIONS.
GENERAL RULE.
Reduce the given fractions, if necessary, to single
ones, and to a common denominator; then add all the
numerators together, and place the sum over the com-
mon denominator.

VULGAR FRACTIONS.
137
If fractions be of different denominations, find their
value separately, and add as in Compound Áddition.
EXAMPLES.
Note 1.-When the given fractions have a common
denominator, add their numerators together, and
place the sum over the common denominator.
Result.
1. Add 4, 2, and
together.
4+3+4=4
sum of the numerators.
common denominator.
2. Add 3, 3, and
2
Result .
Result .
Result 1.
Result 21.
together.
8
3. Add 13,, and 7 together.
4. Add,, and together.
5. Add 1, 2, 4, 5, and together.
15
7
Note 2.-When the given fractions have not a
common denominator, reduce them to such as have,
by rule in Case 2 of Reduction; then add, as in the
foregoing examples.
6. Add and together.
Result 17.
1x5= 51
17
numerators.
3x4=12
4x5-20 com. denom.
17 sum.
4
Result 128=0.
Result 1.
91
Result 211
180*
7. Add 1, 3, and together.
7
12
8. Add and together.
9. Add, and together.
Note 3.-When mixed numbers are given, add
the fractions as under note 1 or 2; then, if their
sum be an improper fraction, reduce it to a mixed
number, and add its integers to the given integers;
but if it be not an improper fraction, annex it to
the sum of the given integers.
10. Add 5%, 6%, and 44 together.
Fractions.
号​++季​=2.
24
5
Integers 6
4
Result 174
Sum of fractions 24.
173.

138
VULGAR FRACTIONS.
11. Add 24 and 34 together.
Result 64.
12. Add 73 and 5 together.
Result 1255.
Result 18.
Result 2011
Result 1359
110
13 Add 17½ and together.
14. Add 4, 6, 9, 4, and 5 together.
15. Add 5, 71, 3, and together.
IT
Note 4.-When compound fractions are given,
reduce them to single fractions, and proceed as be-
fore.
16. Add of, and of together.
& of 9=45
6
12 of 2 = 28
7
Result 13.
4
28 +18=113
Result 31
Result 11.
48
Result 16,73
17. Add of and of 18 together.
3 4 10
20
18. Add 13, 4 of 3, and 93 together.
of, and 7 together.
19. Add 1, 63,
120.
Note 5.-When the given fractions are of several
denominations, reduce them to their proper values
or quantities, and add as in the following example.
20. Add 3 of a pound to of a shilling.
10
Result 15 s. 10,4 d.
s. d.
3 of a L.-15 63
of a s. 0 3
S. d.
d. d. d.
3+3=1 /
15
6
0
3
4
11/8
15 105
of a shilling.
of a pound.
21. Add of a pound to
22. Add ₫ of a penny to
23. Add ½ lb. troy to
24. Add of a mile to
25. Add of a yard to
26. Add § of a day to
1/2
Result 18 s. 3 d.
Result 2 s. 3 d. 1 qr. 3.
of an ounce.
Result 6 oz. 11 dwt. 16 grs.
of a furlong.
Result 6 furlongs, 28 poles.
of a foot. Result 2 ft. 2 in.
of an hour.
Result 8 h. 30 min.
27. Add § of a week, 4 of a day, and of an hour
Result 2 days, 14 hours, 30 min.
together.

VULGAR FRACTIONS.
139
SUBTRACTION OF VULGAR FRACTIONS.
GENERAL RULE.
Prepare the given fractions as in Addition, then sub-
tract the less numerator from the greater, and place the
difference over the common denominator.
1. From
EXAMPLES.
take 3.
2. From take 4
ΤΣ
3. From take
To
12
10.
Result .
Result
Result 2.
Note 1.-When the given fractions have not a
common denominator, reduce them to such as have,
ana Izen subtract as before.
4. From take 3.
Result
1
3x5=152
2x7=145
new numerators.
5x7 35 com. denom.
1
35
5. From
6. From
7. From
8. From
take 4.
ΙΣ
1 2
take 3.
take 1.
15
take 2
9
Result .
Result 4.
Result .
Result
Note 2.-When mixed numbers are given, reduce
them to improper fractions, and reduce these im-
proper fractions to such as have a common denomi-
nator; then subtract as before.
9. From 8 take 6,3.
10. From 9 take 4.
11. From 73 take 33.
Result 194
Result 40
Result 4
99
9
5
24•
Note 3.-When compound fractions are given,
reduce them to single ones, and reduce these single
fractions to such as have a common denominator,
then subtract as before.
12. From of take
10
12
of
13. From & of 11 take 1 of
14. From take 3 of §.
N
काळ नाल
180
Result
Result 133
240.
Result 24

140
VULGAR FRACTIONS.
Note 4.-When a fraction or a mixed number is to be sub-
tracted from a whole number, subtract the numerator of the
fraction from its denominator, and under the remainder set
the denominator; then carry 1 to be subtracted from the inte
gers.
15. From 6 take §.
16. From 3 take.
17. From 100 take 99,9%%
!
Result 51
Result 248.
Result T
Note 5. When the given fractions are of different de
nominations, reduce them to their proper values, or quanti-
ties, and subtract as in the following example.
18. From of a pound, take
f of a L.-15s. 6d.
of a s. = 3 d.
15s. 3d. Result.
of a shilling.
d. d.
19. From of a L. take of a shilling.
Result 14s. 3d.
20. From
21. From of a yard take of an inch.
of a lb. troy, take of an ounce.
Result 8oz. 16dwt. 16grs.
43
22. From & of a L. take of of a shilling.
Result 5in. }.
43
Result 10s. 7d. 1qr. t.
MULTIPLICATION OF VULGAR FRACTIONS.
GENERAL RULE.
Reduce compound fractions to single ones, and mix-
ed numbers to improper fractions; then multiply the
numeraters together for a new numerator, and the de-
nominators for a new denominator.
1. Multiply by .
2. Multiply
by .
EXAMPLES.
3. Multiply by 18.
Result
Result
Result 23
ठ उ

VULGAR FRACTIONS.
141
123-3
4. Multiply 123 by 73.
123 and 7-23; then 3×23=1449963.
Result 96%.
5. Multiply 74 by Št.
Result 61%.
6. Multiply 4 by .
Result 16
7. Multiply by 13%.
Result 1213.
8. Multiply of by 7% of H.
Result
77
9. Multiply 4 by 2 of 2.
10. Multiply of 7 by 3.
Result 23.
Result 14.
11. Multiply 23 by 14, and multiply the product by
of of 4.
Result.
DIVISION OF VULGAR FRACTIONS.
GENERAL RULE.
Prepare the given fractions, if necessary, then invert
the divisor, and proceed as in Multiplication.
EXAMPLES,
1. Divide by 1.
8×4=32
Result.
7x9=63
2. Divide by 4.
Result .
Result 124.
3. Divide by 3.
4. Divide 1 by 410
5 Divide 31 by 93.
6. Divide by 4.
7. Divide 4 by 3.
8. Divide of by 3 of 4.
9 Divide of 19 by of .
10. Divide 45 by 3 of 4.
11. Divide of by 4 of 73.
12. Divide 52054 by ÷ of 91.
Result
Result.
Result
उठ्ठ.
Result 44.
Result .
Result 7.
Result 2
Result
Result 711.

142
VULGAR FRACTIONS.
THE SINGLE RULE OF THREE,
IN VULGAR FRACTIONS.
RULE.
Prepare the given terms, if necessary, and state
them as in whole numbers; multiply the second and
third terms together, and divide the product by the
first. Or,
Invert the dividing term, and multiply the three
terms together, as in Multiplication.
EXAMPLES.
1. If of a yard cost of a shilling, what will } of a
yard come to?
Ans. 2 s. 4 d.
5
yd. yd. s.
ว
8
S.
If 4 : ¦ : & : s. 2 s. 4 d.
Inverted xx}=-2s 4d.
15
14
2. If of a yard cost of a pound, what will of
a yard come to?
13
Ans. 3 s. 4 d.
3. If of a lb. of sugar cost of a shilling, what
cost of a lb.?
T5
1657
Ans. 4 d. 3 qrs. 1983.
4. It of a yard of lawn cost 7 s. 3 d. what will 10
yards cost?
Ans. 4 L. 19 s. 10 d. 2 qrs. §.
5. If 14 yard cost 9 s., what is the value of 164
yards?
Ans. 5 L. 17 s.
6. What is the value of 100 yards of cloth, at 1}
shillings per yard?
7. If I ounce of silver cost 5
of 1611 oz.?
Ans. 6 L.
s., what is the value
Ans. 4 L. 12 s. 13 qrs.
7
105
8. How much will 45 lb. of cheese come to, at 12
cents per lb.?
Ans. 55 cents.
9. What will & of a pound come to, if of a lb. cost
of a shilling?
Ans 4,22 d.
10. If one yard of cloth cost 15 s., what will 4 pieces,
each containing 27 yards, cost? Ans. 85L. 10s. 114 d.
11. A person having of a sloop, sells of his share
for 319 L., what is the value of the whole vessel at that
rate?
Ans. 598 L. 2 s. 6 d.

VULGAR FRACTIONS.
143
12. A merchant had 5g cwt. of sugar, at
which he bartered for tea, at 8 s. per lb.
tea did he receive for the sugar?
8½
64 d. per lb.,
How much
Ans. 43 b.
ठठ्ठ
INVERSE PROPORTION.
1. How much shalloon, of a yard wide, will line
4 yards of cloth, 1 yard wide?
4/2/2
1/1/
Q'ama
As 29
yd. yd. yd. yd.
&
2. What quantity of shalloon,
7 yards of cloth, 1 yards wide?
Ans. 9 yards.
Or thus:
2×3×24089.
12
yard wide, will line
Ans. 15 yards.
3. If 16 men finish a piece of work in 28 days, how
long will 12 men require to do the same work?
Ans. 373 days.
4. If 3 men can do a piece of work in 4 hours, in
how many hours will 10 men do the same?
Ans. 1.
5. How many pieces of cloth, at 20 dollars per piece,
are equal in value to 240
piece?
equal
6. A merchant bartered 5
lb., for tea, at 8§ s. per lb.
ceive?
pieces, at 12 dollars per
Ans. 149177 pieces.
cwt. of sugar, at 63 d. per
How much tea did he re-
Ans. 43 lb.
6 9
THE DOUBLE RULE OF THREE,
IN VULGAR FRACTIONS.
RULE.
Prepare the given terms, when necessary, by reduc-
tion, then proceed as directed in whole numbers. Or,
Invert the dividing terms, and multiply the upper
figures continually for the numerator, and those below
for the denominator of the fractional answer.
N 2

144
DECIMAL FRACTIONS.
EXAMPLES.
1. If yard of cloth, yard wide, cost & L., what is
the value of yard, 12 yards wide, of the same qua
lity?
yd.
H
B
yd.
yd.
lyd. S
:: } L. :
21
32
Sax T = 1
21 × 1 × 1 = 128=17.
70
160
D
21
то
2 =
24
Answer.
7 ÷ } } } } = L. 13 s. 4 d.
33
2. If 24 yards of cloth, 13 yd. wide, cost 33 L., what
is the value of 384 yds. 2 yds. wide?
Ans. 76 L. 10 s.
days labour, how
3. If 3 men receive 8 L. for 19
much must 20 men have for 100 days?
39
Ans. 305 L. 0 s. 8, d.
4. If 50 L. in 5 months gain 237 L. interest, in what
time will 13 L. gain 1½ L.?
Ans. in 9 months.
5. If the carriage of 60 cwt. 20 miles cost 144 dol-
lars, what weight can I have carried 30 miles for 57
dollars?
Ans. 15 cwt.
DECIMAL FRACTIONS.
A Decimal Fraction is a fraction whose denominator
is 1, with as many cyphers annexed as there are places
in the numerator, and is usually expressed by writing
the numerator only with a point prefixed to it: thus
%, 75%, 25%, are decimal fractions, and are expressed
by .5, .75, .625.
5
10009
A mixed number, consisting of a whole number and
a decimal, as 25,5%, is written thus, 25.5.
As in numeration of whole numbers the values of the
figures increase in a tenfold proportion, from the right
hand to the left; so in decimals, their values decrease
in the same proportion, from the left hand to the right,
which is exemplified in the following

DECIMAL FRACTIONS.
145
TABLE.
Hundred million
Ten million.
Million.
Hundred thousand.
Ten thousand.
Thousand.
Hundred.
Ten.
Unit
Hundred thousandth.
Millionth.
Hundredth.
Thousandth.
Ten thousandth.
Hundred millionth.
Thousand millionth.
Ten millionth.
Tenth.
1 1 1 1 1 1 1 1 1.
Whole numbers.
1 1 1 1 1 1 1 1 1
Decimals.
Note.-Ciphers annexed to decimals, neither in-
crease nor decrease their value; thus, .5, .50, .500, be-
ing 10, 100, 1000, are of the same value: but ciphers
fo
prefixed to decimals, decrease them in a tenfold pro-
portion; thus .5, .05, .005; being, 150, 1000, are
of different values.
ADDITION OF DECIMALS.
RULE.
Place the given numbers according to their values,
viz. units under units, tenths under tenths, &c., and
add as in addition of whole numbers; observing to set
the point in the sum exactly under those of the given
numbers.
EXAMPLES.
.12
2.16
.14
.1
.15
.134
3.45
.24
4.12
.75
.21
40.02
.122
15.4
.92
.743
35.4
.36
76.36
63.25
.345
36.1
.141
120.16
25.
.002
125.32
.567
425.04
4.
1.554
242.45
6. Add .5, .75, .125, .496, and .750 together.
7. Add .15, 126.5, 650.17, 940.113, and 722.2560
together.
8. Add 420., 372.45, .270, 965.02, and 1.1756 to-
gether.

146
DECIMAL FRACTIONS.
SUBTRACTION OF DECIMALS.
RULE.
Place the numbers as in addition, with the less under
the greater, and subtract as in whole numbers; setting
the point in the remainder under those in the given
numbers.
EXAMPLES.
.4562
56.12
4314
.316
1.242
.312
.1402
54.878
5672.1
32.456
321.12
1.33
6. From 100.17 take 1.146.
7. From 146.265 take 45.3278.
8. From 4560. take .720.
MULTIPLICATION OF DECIMALS.
RULE
Multiply as in whole numbers: then observe how
many decimal figures there are in both factors, and point
off that many figures, for decimals, in the product.
If there are not so many figures in the product as
there are decimal figures in both factors, prefix ciphers
to supply the deficiency.
1. Multiply .612 by 4.12
EXAMPLES.
2. Multiply 1.007 by .041
.612
4.12
1224
612
2448
1.007
.041
1007
4028
.041287
2.52141
3. Multip) 37.9 by 46.5
Product
1762.35
4.
36.5 by 7.27
265.355
5.
29.831 by .952
28.399112
6.
3.92 by 196.
7.
.285 by .003
4.001 by .004
768.32
.000855
.016004
.00071 by .121
.00008591

DECIMAL FRACTIONS.
147
Note.-Multiplication of decimals may be contract-
ed thus:
Write the units place of the multiplier under that
figure of the multiplicand whose place you would re-
serve in the product; and dispose of the rest of the
figures in a contrary order to what they are usually
placed in. In multiplying, reject all the figures that
are to the right hand of the multiplying digit, and set
down the products, so that their right hand figures may
fall in a straight line below each other; observing to in-
crease the first figure of every line with what would
arise by carrying 1 from 5 to 15, 2 from 15 to 25, &c.
from the preceding figures when you begin to multi-
ply, and the sum is the product required.
EXAMPLES.
7
1. Multiply 27.14986 by 92.41035, so as to retain
only four decimal places in the product.
Contracted.
27.14986
53014.29
24434874
Common way.
27.14986
92.41035
13574930
542997
108599
2715
81
14
8144958
2714986
10859944
5429972
24434874 ·
2508.9280650510
2508.9280
2. Multiply 245.378263 by 72.4385, reserving 5
decimal places in the product. Prod. 17774.83330.
3. Multiply .243264 by .725234, reserving 6 deci-
mal places in the product.
Prod. .180049.
DIVISION OF DECIMALS.
RULE.
Divide as in whole numbers; then observe how many
more decimal figures there are in the dividend than in
the divisor, and point off that many figures, for deci-
mals, in the quotient.
If there are not so many figures in the quotient as the

148
DECIMAL FRACTIONS.
rule directs to be pointed off, prefix ciphers to supply
the defect.
If, after dividing, there be a remainder, ciphers may
be affixed to the dividend, as decimal figures, and the
quotient carried on to greater exactness.
If there are more decimal figures in the divisor than
there are in the dividend, the number of decimal figures
in the dividend must be increased by affixing ciphers.
EXAMPLES.
1. Divide .863972 by .92
2. Divide 4.13 by 572.4
572.4)4.130000(.00721+
40068
.92).863972(.9391
628
359
276
837
828
92
12320
11448
8720
5724
2996
92
3. Divide 19.25 by 38.5
Quotient
.5*
4.
234.70525 by 64.25
3.653
5.
1.0012 by .075
13.34-
6.
..1606 by .44
7.
.1606 by 4.4
.365
.0365
S.
.1606 by 44.
9.
9. by .9
10.
.9 by 9.
11.
186.9 by 7.476
.00365
10.
.1
25.
Note 1.-When a whole number is to be divided by
a greater whole number, ciphers must be affixed to the
dividend, as decimal figures.
12. Divide 3 by 4
13.
275 by 3842
14.
210 by 240
Quotient
.75
.071577+
.875
Note 2.-When any whole number is divided by
another, if there be a remainder, ciphers may be affixed
to the dividend, and the quotient continued.

DECIMAL FRACTIONS.
149
15. Divide 382 by 25
16.
17.
13689 by 75
315 by 124
Quotient 15.28
182.52
2.5403+
Division of Decimals may be contracted thus:
Take as many of the left hand figures of the divisor
as will be equal to the number of integers and decimals
in the quotient, and find how many times they may be
had in the first figures of the dividend as usual. Let
each remainder be a new dividend; and for every such
dividend, leave out one figure to the right hand of the
divisor, remembering to carry for the increase of the
figures cut off, as in contracted multiplication.
Note.-When there are not so many figures in the
divisor as are required to be in the quotient, begin the
operation with all the figures, as usual, and continue it
till the number of figures in the divisor, and those re-
maining to be found in the quotient be equal, after
which use the contraction.
EXAMPLES.
1. Divide 2508.928065051 by 92.41035, so as to
nave 4 decimal places in the quotient.
Contracted way.
92.41035)2508.928065051 (27.1498
1848207
Common way.
92.41035)2508.928065051(27.1498
660721
18482070
646872
66072106
13849
64687245
9241
13848615
4608
9241035
3696
912
46075800
36964140
832
91116605
80
72
00
83169315
79472901
73928280
5544621

150
DECIMAL FRACTIONS.
2. Divide 721.17562 by 2.257432, and let there be
only 3 places of decimals in the quotient.
3. Divide 12.169825 by 3.14159, so
places of decimals in the quotient.
Quo. 319.467
as to have 5
Quo. 3.87377
REDUCTION OF DECIMALS.
CASE 1.
To reduce a vulgar fraction to a decimal.
RULE.
Divide the numerator by the denominator.
mals.
See examples under note 1, Division of Deci-
When a compound fraction is given, first reduce it
to a single one, and then to a decimal.
EXAMPLES.
1. Reduce to a decimal.
Result .25
4)1.00
.25
4.
મેં જે મ
2. Reduce
to a decimal.
Result
.5
3.
₫ to a decimal.
.75
1
25
6.
to a decimal.
26
7.
8.
9.
5.
to a decimal.
2 to a decimal.
.375
.04
.1923076+
.333+
.6043956+
.071577+
of 2 to a decimal.
11 of 10 to a decimal.
14
13
275 to a decimal.
3842
CASE 2.
To reduce any sum, or quantity, to the decimal of
any given denomination.
RULE.
Divide the given sum or quantity, reduced to the
lowest denomination mentioned, by the proposed inte-
ger, reduced to the same denomination, and the quo-
tient will be the decimal required. Or,
Write the given numbers from the least to the great-
est in a perpendicular column, and divide each of them
by such a number as will reduce it to the next denomi-

DECIMAL FRACTIONS.
151
nation, annexing the quotient to the succeeding number;
the last quotient will be the decimal required.
EXAMPLES.
1. Reduce 17 s. 6 d. to the decimal of a pound.
S.
d. d.
17
6-210
1 L. =240
Or thus,
12 6.0
2|0|17.500
.875 decimal.
240.)210.000(.875 decimals.
1920
1800
1680
1200
1200
2. Reduce 7 s. 6 d. to the decimal of a pound.
3. Reduce 9 d. to the decimal of a pound.
Result .375
Result .0375
4. Reduce 10 s. 94 d. to the decimal of a pound.
Result 5385416+
5. Reduce 12 grains to the decimal of a lb. troy.
Result .002083+
6. Reduce 12 drachms to the decimal of a pound
avoirdupois.
Result .046875
7. Reduce 2 qrs. 14 lb. to the decimal of a cwt.
Result .625
8. Reduce 2 furlongs to the decimal of a league.
Result .0833
9. Reduce 3 qrs. 2 na. to the decimal of a yard.
Result .875
10. Reduce 4 perches to the decimal of an acre.
Result .025
11. Reduce 1 pint to the decimal of a gallon.
Result .125
12. Reduce 7 minutes to the decimal of a day.
13. Reduce 72 days to the
puting the year at 365 days.
14. Reduce 52 days to the
puting the year at 365 days.
Result .00486+
decimal of a year, com-
Result .1972602+
decimal of a year, com-
Result .142368+
15. Reduce & d. to the decimal of a shilling.
Result .0625
!

152
DECIMAL FRACTIONS.
CASE 3.
To reduce a decimal fraction to its value.
RULE.
Multiply it by the known parts of the integer, and
separate to the right of the product as many places as
there are places in the given number.
Note. To find the value of any decimal of a pound
by inspection, double the first figure after the point for
shillings, adding one thereto if the second be five or
more.
Prefix the second figure, if less than five, or its ex-
cess above five to the third, and call them so many far-
things, abating one when above twelve and two when
above thirty-six.
EXAMPLES.
1. What is the value of .S7615 of a L.?
Ans. 17 s. 64 d.
By inspection.
.87615
.S7615
20
17.52300
17 6 1
12
That is,
6.27600
8x2+1 17 s.
I s. d.
4
and 26-1-25 qrs.+64
17 64
1.10400
2. What is the value of .7854166 of a L."
3. What is the value of .76 of a L.?
Ans. 15 s. 8 d.
Ans. 15 s. 2 d. 1.6 qrs.
4. What is the value of .625 of a shilling? Ans. 71 d.
5. What is the value of .461 of a dollar?
Ans. 46 cts. 1 mill.
6. What is the value of .461 of a shilling?
Ans. 5 d. 2.128 qrs.
7. What is the value of .86 of a cwt.?
Ans. 3 qrs. 12 lb. 5 oz. 1.92 dr.
8. What is the value of .7 of a lb. troy ?
Ans. 8 oz. 8 dwt.

DECIMAL FRACTIONS.
153
9. What is the value of .71 of 4 oz. troy?
Ans. 2 oz. 16 dwt. 19.2 grs.
10. What is the value of .761 of a day?
Ans. 18 h. 15 min. 50.4 sec.
11. What is the value of .67 of a league?
Ans. 2 miles, 0 fur. 3 poles, 1 yd. 3.6 in.
12. What is the value of .712 of a furlong?
Ans. 28 poles, 2 yds. 1 ft. 11.04 in.
13. What is the value of .6875 of a yard?
Ans. 2 qrs. 3 na.
14. What is the value of .3375 of an acre?
Ans. 1 rood, 14 perches.
15. What is the value of .3 of a year?
Ans. 109 days, 12 hours.
16. What is the value of .07 of a barrel of 32 gal-
lons?
Ans. 2 gals. 1.92 pts.
17. What is the sum of .48 of a pound, and .16 of a
shilling?
Ans. 9 s. 9.12 d.
18. What is the sum of .17 of a lb. troy, and .84 of
an ounce ?
Ans. 2 oz. 17 dwt. 14.4 grs.
19. What is the difference between .17 L. and .7 s.?
Ans. 2 s. 8 d. 1.6 qr.
20. What is the difference between .41 day and .Î6
Ans. 9 h. 40 min. 48 sec.
hour?
THE SINGLE RULE OF THREE,
IN DECIMALS.
The operation both in direct and inverse proportion
is the same as in whole numbers, regard being had to
the right placing of the points.
EXAMPLES.
1. If 2.75 yards of cloth cost 4 L. 13.5 s., what are
12.25 yards worth?
Ans. 20 L. 16 s. 6 d.
yds. yds.
L. S.
L. s. d.
As 2.75 12.25 :: 4 13.5: 20 16 6
2. If 1.4 lb. of mace cost 14.5 s., what cost 75.31 lb.?
Ans. 38 L. 19 s. 11 d. 3.52 qrs.
3. If 1.5 oz. of silver be worth 7.8 s., what is the value
of 9.7 lb.
Ans. 30 L. 5 s. 3 d. 1.44 qrs.

154
DECIMAL FRACTIONS.
4. If 1.47 cwt. of sugar be worth 4.5 L., how much
is 1.7 lb. worth?
Ans. 11.1 d.
5. If 1.6 cwt. of sugar sell for 3 L. 12.76 s., what is
the value of 3 hogsheads, each 11 cwt. 3 qrs. 10.12 lb.?
Ans. 80 L. 15 s. 3 d. 3.36 qrs.
6. What is the value of 3 pieces of cloth, each con-
taining 21.5 yds., at 12.3 s. per yard?
Ans. 39 L. 13 s. 4.2 d.
7. If 1 pint of wine cost 1.2 s., what cost 12.5 hogs-
heads?
Ans. 378 L.
8. If 19 yards of linen cost 25.75 dols., what will 435.5
yards come to?
Ans. 590.217 dols. +Or,
590 dols. 21 cts. 7 m.+
9. What will a merchant gain by buying 436 yards
of linen, at 8.5 s. per yard, and selling it at 10.75 s. per
yard?
Ans. 49 L. 1 s.
10. A grocer bought 7.6 cwt. of sugar, at 40.1 s. per
cwt. and retailed it out at 4.5 d. per lb. Whether did
he gain or lose, and how much?
ib.,
Ans. He gained 14 s. 5 d. 1 12 qrs.
11. A bought 3 cwt. 1.5 qr. of cloves, at 2.75 s. per
which he afterwards sold for 60 L. 11 s. 6d. How much
did he gain by the transaction?
Ans. 8 L. 12 s.
12. If 1 yard of ribbon sell for 4.5 cents, how much
will 345 yards bring?
Ans. 15.525 dols. Or, $15.524.
INVERSE PROPORTION.
1. How long will 3 men be in performing a piece
of work which will occupy 5 men 40.5 days?
Ans. 67.5.
2. How many men can do as much work in .4 of a
month, as 16 can do in 1.5 month?
Ans. 60.
3. How much silk .75 of a yard wide, will line 25.5
yards of cloth that is 5 qrs. wide? Ans. 42.5 yds.
4. If a board be .75 of a foot broad, what length must
it be to measure 12 square feet?
Ans. 16 feet.
5. A had 40.7 yards of linen, for which B gave him
25.6 ells of Holland, at 4.5 s. per ell. How much was
the linen per yard?
Ans. 2 s. 9 d. 3.84 qrs.

INVOLUTION.
155
THE DOUBLE RULE OF THREE,
IN DECIMALS.
Questions in this rule are wrought as in whole num-
bers, placing the points agreeably to former directions.
EXAMPLES.
1. If 3 men receive 8.9 L. for 19.5 days' labour, how
much must 20 men have for 100.25 days?
3 : 20
Ans. 305 L. 0 s. 8.2 d.
:: 89 L.: 305 L. 0s. 8.2 d.
days 19.5: 100.25 days
2. If 2 persons receive
how much should 4 persons have for 10.5 days?
4.625 s. for 1 day's labour,
Ans. 4 L. 17 s. 1 d.
3. If the interest of 76.5 L. for 9.5 months be 15.24 L.,
what sum will gain 6 L. in 12.75 months?
Ans. 22 L. 8 s. 94 d.
4. How many men will reap 417.6 acres in 12 days,
if 5 men reap 52.2 acres in 6 days? Ans. 20 men.
5. If a cellar 22.5 feet long, 17.3 feet wide, and 10.25
feet deep, be dug in 2.5 days, by 6 men, working 12.3
hours a day; how many days of 8.2 hours, should 9 men
take to dig another, measuring 45 feet long, 34.6 wide,
and 12.3 deep?
Ans. 12 days.
INVOLUTION,
OR THE RAISING OF POWERS.
A power is the product arising from multiplying any
given number into itself continually a certain number
of times; thus,
2×24 the second power or square of 2.
2×2×28 the third power or cube of 2.
2×2×2×2=16 the fourth power of 2, &c.
The number denoting the power is called the index
or exponent of that power.
If two or more powers of the same number are mul-
tiplied together, their product is that power whose in-
dex is the sum of the exponents of the factors; thus,
2×2=4 the square of 2; 4×4-16=4th power of 2 ;
and 16×16-256-8th power of 2, &c.
0 2

156
INVOLUTION.
9th power.
8th power.
A TABLE OF THE FIRST NINE POWERS.
7th power.
6th power.
1 1 1
1
1
]
1
1
1
2 4
80
16 32
64
128
256
512
3 9 27 81 243
729 2187
6561
19683
65536 262144
5th power.
4th power.
Roots.
Squares.
Cubes.
416 64 256| 1024 4096 16384
390625 1953125
5 25 125 625 3125| 15625 78125
6 36 216 1296 7776 46656 279936 1679616) 10077696
7 49 343 2401 16807 117649 823543 5764801 40353607
8 64 512 4096 32768 262144 2097152 16777216134217728
981 729 6561 59049 531441 4782969 43046721 387420489|
EXAMPLES.
1. What is the square of 22?
Ans. 484.
2. What is the cube or third power of 4? Ans. 64.
4×4×4-64.
3. What is the fifth power of 7?
Ans. 16807.
4. What is the cube or third power of 35?
Ans. 42875.
5. What is the fourth power of ?
6. What is the cube or third power of .13?
7. What is the sixth power of 5.03?
Ans. 1
256.
Ans. .002197.
Ans. 16196.005304479729
EVOLUTION,
OR THE EXTRACTING OF ROOTS.
The root of a number, or power, is such a number, as
being multiplied into itself a certain number of times,

THE SQUARE ROOT.
157
will produce that power. Thus 2 is the square root
of 4, because 2×2-4; and 4 is the cube root of 64,
because 4x4x4-64, and so on.
THE SQUARE ROOT.
The square of a number is the product arising from
that number multiplied into itself.
Extraction of the square root is the finding of such a
number as being multiplied by itself will produce the
number proposed.
RULE.
1. Separate the given number into periods of two
figures, each, beginning at the units place.
2. Find the greatest square contained in the left hand
period, and set its root on the right of the given num-
ber: subtract said square from the left hand period,
and to the remainder bring down the next period for a
dividual.
3. Double the root for a divisor, and try how often
this divisor (with the figure used in the trial thereto
annexed) is contained in the dividual set the number
of times in the root; then, multiply and subtract as in
division, and bring down the next period to the re-
mainder for a new dividual.
4. Double the ascertained root for a new divisor, and
proceed as before, till all the periods are brought down.
Note. If, when all the periods are brought down,
there be a remainder, annex cyphers to the given num-
ber, for decimals, and proceed till the root is obtained
with a sufficient degree of exactness.
Observe that the decimal periods are to be pointed
off from the decimal point toward the right hand; and
that there must be as many whole number figures in the
root, as there are periods of whole numbers, and as many
decimal figures as there are periods of decimals
PROOF.
Square the root, adding in the remainder, (if any,)
and the result will equal the given number.

158
THE SQUARE ROOT.
EXAMPLES.
1. What is the square root of 5499025 ?
5499025(2345 Ans.
4
43)149
129
464)2090
1856
4685)23425
23425
2345
2345
11725
9380
7035
4690
5499025 Proof.
Ans. 327.
2. What is the square root of 106929 ?
3. What is the square root of 451584 ?
4. What is the square root of 36372961?
5. What is the square root of 7596796 ?
Ans. 672.
Ans. 6031.
Ans. 2756.228+
6. What is the square root of 3271.4007?
Ans. 57.19+
7. What is the square root of 4.372594 ?
Ans. 2.091+
8. What is the square root of 10.4976 ? Ans. 3.24
9. What is the square root of .00032754?
10. What is the square root of 10?
Ans. .01809+
Ans. 3.1622+
To extract the Square Root of a Vulgar Fraction.
RULE.
Reduce the fraction to its lowest terms, then extract
tne square root of the numerator for a new numerator,
and the square root of the denominator for a new deno-
minator.
Note.-If the fraction be a surd, that is, one whose
root can never be exactly found, reduce it to a decimal,
and extract the root therefrom.
EXAMPLES
9 2 16
1. What is the square root of 7056
2. What is the square root of 2294
3. What is the square root of
549
เป๊ปเป
Ans, ?.
Ans..
Ans. .93309+

THE SQUARE ROOT.
159
To extract the Square Root of a Mixed Number.
RULE.
Reduce the mixed number to an improper fraction,
and proceed as in the foregoing examples: Or,
Reduce the fractional part to a decimal, annex it to
the whole number, and extract the square root there-
from.
EXAMPLES.
1. What is the square root of 3738
2. What is the square root of 27%
3. What is the square root of 851
4. What is the square root of 85?
APPLICATION.
5
טרי טרי
Ans. 64.
Ans. 54.
Ans. 9.27+
Ans. 2.9519+
1. The square of a certain number is 105625: what
is that number?
Ans. 325.
2. A certain square pavement contains 20736 square
stones, all of the same size: what number is contained
in one of its sides?
Ans. 144.
3. If 484 trees be planted at an equal distance from
each other, so as to form a square orchard, how many
will be in a row each way?
Ans. 22.
4. A certain number of men gave 30s. 1 d. for a chari-
table purpose; each man gave as many pence as there
were men: how many men were there? Ans. 19.
Note. The square of the longest side of a right an-
gled triangle is equal to the sum of the squares of the
other two sides; and consequently the difference of the
square of the longest, and either of the other, is the
square of the remaining one.
5. The wall of a certain fortress is 17 feet high,
which is surrounded by a ditch 20 feet in breadth; how
long must a ladder be to reach from the outside of the
ditch to the top of the wall?
Ans. 26.24+feet.
Wall.
Ladder.
Ditch.

160
THE CUBE ROOT.
6. A certain castle, which is 45 yards nigh, is sur-
rounded by a ditch 60 yards broad; what length must
a ladder be to reach from the outside of the ditch to
the top of the castle?
Ans. 75 yards.
7. A line 27 yards long will exactly reach from the
top of a fort to the opposite bank of a river, which is
known to be 23 yards broad; what is the height of the
fort?
Ans. 14.142+ yards.
8. Suppose a ladder 40 feet long be so planted as to
reach a window 33 feet from the ground, on one side of
the street, and without moving it at the foot, will reach
a window on the other side 21 feet high; what is the
breadth of the street?
Ans. 56.64+ feet.
9. Two ships depart from the same port; one of them
sails due west 50 leagues, the other due south 84
leagues; how far are they asunder?
Ans. 97.75+ Or, 974+ leagues.
THE CUBE ROOT.
The cube of a number is the product of that number
multiplied into its square.
Extraction of the cube root is the finding of such a
number, as, being multiplied into its square, will pro-
duce the number proposed.
RULE.
1. Separate the given number into periods of three
figures each, beginning at the units place.
2. Find the greatest cube contained in the left hand
period, and set its root on the right of the given num-
ber: subtract said cube from the left hand period, and
to the remainder bring down the next period for a di-
vidual.
3. Square the root and multiply the square by 3 for
a defective divisor.
4. Reserve mentally the units and tens of the di-
vidual, and try how often the defective divisor is con-
tained in the rest: place the result of this trial to the
root, and its square to the right of said divisor, supply-
ing the place of tens with a cipher, if the square be
less than ten.

THE CUBE ROOT.
161
5. Complete the divisor by adding thereto the pro-
duct of the last figure of the root by the rest and by 30.
6. Multiply and subtract as in Simple Division, and
bring down the next period for a new dividual; for
which find a divisor as before, and so proceed till all
the periods are brought down.
** See note under the rule for extracting the square
root: it applies equally to this rule.
Note.-Defective divisors, after the first, may be
more concisely found thus: To the last complete divi-
sor add the number which completed it with twice the
square of the last figure in the root, and the sum will
be the next defective divisor.
PROOF.
Involve the root to the third power, adding the re-
mainder, if any, to the result.
EXAMPLES.
1. What is the cube root of 99252.S47 ?
99252.847(46.3
64
Defective divisor & square of 6=4836)35252
+720=complete divisor
5556)33336
Defective divisor & square of 3=634809)1916847
+4140 complete divisor
638949) 1916847
Ans. 253.
Ans. 73.
2. What is the cube root of 16194277?
3. What is the cube root of 389017?
4. What is the cube root of 5735339?
5. What is the cube root of 34328125?
6. What is the cube root of 22069810125?
Ans. 179.
Ans. 325.
Ans. 280.5
7. What is the cube root of 12.977875? Ans. 2.35
8. What is the cube root of 36155.027576 ?
Ans. 33.06+
9. What is the cube root of 15926.972504?
Ans. 25.16+
10. What is the cube root of .001906624?
Ans. .124

162
ROOTS OF ALL POWERS.
Note―The cube root of a vulgar fraction is found
by reducing it to its lowest terms, and extracting the
root of the numerator for a numerator, and of the de-
nominator for a denominator. If it be a surd, extract
the root of its equivalent decimal.
2. A mixed number may be reduced to an improper
fraction, or a decimal, and the root thereof extracted.
3000
686
1. What is the cube root of 648?
2. What is the cube root of 25
3. What is the cube root of 152
4. What is the cube root of 1219
5. What is the cube root of 31-15
SURDS
5130
343
6. What is the cube root of 71 ?
7. What is the cube root of 94 ?
APPLICATION.
เบบ
Ans..
Ans.
Ans. 2
Ans. 21.
Ans. 34.
Ans. 1.93+
Ans. 2.092+
1. The cube of a certain number is 103823; what is
that number?
Ans. 47.
2. The cube of a certain number is 1728; what num-
ber is it?
Ans. 12.
3. There is a cistern or vat of a cubical form which
contains 1331 cubical feet: what are the length, breadth,
and depth of it?
Ans. each 11 feet.
4 A certain stone of a cubical form contains 474552
ɛolid inches; what is the superficial content of one of
its sides?
Ans. 6084 inches.
A GENERAL RULE FOR EXTRACTING THE
ROOTS OF ALL POWERS.
1. Point the given number into periods agreeably to
the required root.
2. Find the first figure of the root by the table of
powers, or by trial; subtract its power from the left
hand period, and to the remainder bring down the first
figure in the next period for a dividend.
3. Involve the root to the next inferior power to that
which is given, and multiply it by the number denoting
the given power, for a divisor; by which find a second
figure of the root.

ALLIGATION.
163
4. Involve the whole ascertained root to the given
power, and subtract it from the first and second pe-
riods. Bring down the first figure of the next period
to the remainder, for a new dividend; to which, find
a new divisor, as before; and so proceed.
Note. The roots of the 4th, 6th, 8th, 9th, and 12th
powers, may be obtained more readily thus:
For the 4th root take the square root of the square
roct.
For the 6th, take the square root of the cube root.
For the 8th, take the square root of the 4th root.
For the 9th, take the cube root of the cube root.
For the 12th, take the cube root of the 4th root.
EXAMPLES.
1. What is the 5th root of 916132832 ?
916132832(62 Ans.
7776
6×6×6×6×6=7776
6×6×6×6×5=6480 divisor.
6480)13853
916132832
62×62×62×62×62=916132832
916132832
3. What is the sixth root of 782757789696 ?
2. What is the fourth root of 140283207936 ?
Ans. 612.
Ans. 96.
4. What is the seventh root of 194754273881?
Ans. 41.
5. What is the ninth root of 1352605460594688 ?
Ans. 48.
ALLIGATION.
Alligation is a rule for adjusting the prices and sim-
ples of compound quantities.
CASE 1.
To find the mean price of any part of the composition,
when the several quantities and their prices are given.
P

164
ALLIGATION.
RULE.
As the sum of the several quantities,
Is to any part of the composition;
So is their total value,
To the value of that part.
PROOF.
The value of the whole mixture at the mean price
must agree with the total value of the several quantities
at their respective prices.
EXAMPLES.
1. If 6 gallons of wine at 67 cents per gallon; 7 at 80
cents, and 5 at 120 cents per gallon, be mixed together,
what will 1 gallon of the mixture be worth?
G.
6 at
cts. cts.
67=402
7 at 80=560
5 at 120=600
18
1562
G. G.
cts.
cts.
As 18: 1 :: 1562 : 86.77+ Answer.
2. If 19 bushels of wheat at 6 s. per bushel; 40 bushels
of rye at 4 s. per bushel, and 12 bushels of barley at
3 s. per bushel, be mixed together, what will a bushel
of the mixture be worth?
Ans. 4 s. 44 d.
3. If a grocer mix 2 cwt. of sugar at 56 s. per cwt.;
1 cwt. at 43 s. per cwt.; and 2 cwt. at 50 s. per cwt.,
what will be the value of 1 cwt. of the mixture?
Ans. 2 L. 11 s.
4. A farmer mingled 20 bushels of wheat at 5 s. per
bushel, and 36 bushels of rye at 3 s. per bushel, with
40 bushels of barley at 2 s. per bushel; I desire to know
the worth of a bushel of this mixture?
Ans. 3 s.
5. If 4 ounces of silver worth 75 cents per ounce, be
melted with 8 ounces worth 60 cents per ounce, what
will 1 ounce of the mixture be worth? Ans. 65 cts.
6. A wine merchant mixes 12 gallons of wine at
4 s 10 d. per gallon, with 24 gallons at 5 s. 6 d., and 16
gons at 6 s. 34 d.; what is a gallon of the mixture
Worth?
Ans. 5 s. 7 d.

ALLIGATION.
165
CASE 2.
When the prices of several simples are given, to find
how much of each, at their respective rates, must be
taken to make a compound or mixture at any proposed
price.
RULE.
Set the prices of the simples one under another, and
link every price which is not greater than the mean
rate, to one or more that are greater than that rate
place the difference between each price and the mean
rate opposite to the price or prices with which it is
linked then, if only one difference stand opposite to
either particular price, it will be the quantity required
at that price; but if there be several differences, their
sum will be the quantity.
Note. Different modes of linking will produce dif-
ferent answers.
EXAMPLES.
1. How much rye at 4 s. per bushel, barley at 3 s. per
bushel, and oats at 2 s. per bushel, will make a mixture
d. per bushel?
worth 2 s. 6 d.
d. (48
Mean rate 30 36.
Ans.
24
{
bu.
S.
- 6 at 4
6 at 3 Answer.
18+6=24 at 2
2. A vintner has three kinds of wine, viz. one kind
at 160 cents per gallon, another at 180 cents, and an-
other at 240 cents; how much of each kind must he take
to make a mixture worth 190 cents per gallon?
§ 50 gals. at 160 cts., 50 gals. at 180 cts.,
and 40 gals. at 240 cts.
3. How much sugar at 4 d. at 6 d. and at 11 d.
per lb.
must be mixed together to make a composition worth
7 d.
per lb.? Ans. an equal quantity of each kind.
4. It is required to mix several sorts of wine, viz. at
9 s. 15 s. and 21 s. per gallon, with water, that the mix-
ture may be worth 12 s. per gallon; how much of each
sort must be taken?
Ans. (3 gals. 9 s., 3 gals. 15 s., and 12 gals.
at 21 s. with 9 gals. of water.
5. A grocer has several sorts of sugar, viz. one sort

166
ALLIGATION.
at 12 cents per lb., another at 11 cents, a third at 9
cents, and a fourth at 8 cents per lb.; how much of
each sort must he take to make a mixture worth 10
cents per lb.
lb.
cts.
2 at 12
lb.
(3 at 12
cts.
lb. cts.
1 at 12
1 at 11
2 at 11
2 at 11
1 Ans.
2 Ans.
3 Ans.
1 at 9
2 at 9
2 at 9
2 at 8
3 at
S
1 at 8
lb. cts.
lb.
cts.
lb. cts.
(1 at 12
3 at 12
2 at 12
3 at 11
1 at 11
3 at 11
4 Ans.
5 Ans.
6 Ans.
3 at 9
3 at 9
1 at 9
1 at 8
2 at 8
3 at 8
7 Ans. 3 lb. of each sort.
CASE 3.
When the price of all the simples, the quantity of
one of them, and the mean price of the whole mixture
are given, to find the several quantities of the rest.
RULE.
Link the several prices, and place their differences
as in case 2; then
As the difference opposite to the price of the given
quantity,
Is to the differences respectively;
So is the given quantity,
To the several quantities required.
EXAMPLES.
1. A grocer would mix 30 lb. of sugar at 14 cents per
Ib. with some at 9 cents, 10 cents, and 13 cents per Îb.;
how much of each sort must he mix with the thirty lb.
that the mixture may sell at 12 cents per lb.?
9
10
13
Mean price 12<
(14
lb. cts.
1
QA ∞ ∞
2
1
2
3
As 3 : 2 :: 30: 20 at 9 per lb.
3 : 1 :: 30: 10 at 10
3.2 2 :: 30: 20 at 20
Answer

ALLIGATION.
167
2. How much barley at 30 cents per bushel, rye at 36
cents, and wheat at 48 cents, must be mixed with 12
bushels of oats, at 18 cents, to make a mixture worth
22 cents per bushel
Ans. 1 bushel of each sort.
3. How much wine at 5 s., at 5 s. 6 d., and at 6 s. per
gallon, must be mixed with 3 gallons at 4 s. per gallon,
so that the mixture may be worth 5 s. 4 d. per gallon?
Ans. 3 gals. at 5 s., 6 at 5 s. 6 d., and 6 at 6 s.
4. How much tea at 12 s., 10 s., and at 6 s. per lb.
must be mixed with 20 pounds at 4 s. per lb. to make
a mixture worth 8 s. per lb.?
Ans. 10 lb. at 6 s., 10 lb. at 10 s., and 20 lb. at 12 s.
CASE 4.
When the prices of the several simples, the quantity
to be compounded, and the mean price are given, to
find the quantity of each simple.
RULE.
Link the several prices, and place their differences
as before; then,
As the sum of the differences,
Is to the difference opposite to each price;
So is the quantity to be compounded,
To the quantity required.
EXAMPLES.
1. How much sugar at 10 cents, 12 cents, and 15
cents per lb. will be required to make a mixture of 20
lb. worth 13 cents per lb.?
13 12
2 As 8:2::20:
20: 5lb. at 10 cts.
2
S:4:20:10lb. at 15 cts. Ans.
3+1=4 8:2::20: 5lb. at 12 cts.
8 Sum of differences.
2. A brewer has three sorts of beer, viz. at 10 d., 8 d.,
and 6 d. per gallon; how much of each sort must he take
to make a mixture of 30 gallons, worth 7 d. per gallon?
Ans. 5 gals. at 10 d., 5 gals. at 8d., and 20 gals. at 6 d.
3. A goldsmith has gold of 15, 17, 20, and 22 carats
fine, and would melt together of each of these so much
as to make a mass of 40 oz. of 18 carats fine; how much
of each sort is necessary?
Ans.
S16 oz. of 15 carats, 8 oz. of 17 carats, 4 oz.
of 20 carats, and 12 oz. of 22 carats fine.
of 15 1262.0817
P 2

168
POSITION.
4. How many gallons of water must be mixed with
wine, at 4 s. per gallon, so as to fill a vessel of 80 gal-
lons, that may be afforded at 2 s. 9 d. per gallon?
Ans. 25 gallons of water, with 55 of wine.
POSITION.
Position is a rule for nading an unknown number,
by one or more supposed numbers.
two parts, single and double.
It is divided into
SINGLE POSITION.
Single Position teaches to resolve such questions as
require only one supposition.
RULE.
Suppose any number to be the true one and proceed
with it agreeably to the tenor of the question; then,
As the result of the operation,
Is to the number given;
So is the supposed number,
To the number sought.
PROOF.
Work with the answer according to the tenor of 'he
question, and the result must equal the given number.
EXAMPLES.
1. A, B, and C bought a quantity of wine for 340
dollars, of which sum A paid three times more than B,
and B four times more than C; how much did each
pay?
$
Suppose A paid 36
Then B paid
12
And C paid
3
51
S
A paid 240
B paid 80 Ans.
C pa
0
340 Proof.
As 51: 340 :: 36 : 240 sum paid by A.
3
2. A person after spending and 4 of his money, had
60 L. left; how much had he at first? Ans. 144 L.
3. What number of dollars is that, of which the 4,
, and, make 74?
Ans. 120.

POSITION.
169
4. A person having about him a certain number of
crowns, said, if a third, a fourth, and a sixth of them
were added together, the sum would be 45; how many
crowns had he?
Ans. 60.
5. What is the age of a person who says, that if g of
the years I have lived he multiplied by 7, and of them
be added to the product, the sum will be 292 ?
Ans. 60 years.
6. A schoolmaster being asked how many scholars
he had, answered, if to double the number I add ½, 1,
and 4 of them, I shall have 333; how many had he?
Ans. 108.
1
7. A certain sum of money is to be divided among 4
persons in such a manner that the first shall have of
it, the second 4, the third, and the fourth the remain-
der, which is 28 dollars; what is the sum ?
Ans. 112 dollars.
8. What sum, at 6 per cent. per annum, will amount
to 860 L. in 12 years?
Ans. 500 L.
DOUBLE POSITION.
Double Position teaches to find the true number, by
making use of two supposed numbers.
RULE.
Suppose two numbers, and work with each agreea-
bly to the tenor of the question, noting the errors of
the results multiply the errors of each operation into
the supposed number of the other; then,
If the errors be alike, i. e. both too much, or both
too little, take their difference for a divisor, and the
difference of the products for a dividend: but if the
errors be unlike, take their sum for a divisor, and the
sum of the products for a dividend.
PROOF.
As in Single Position.
EXAMPLES.
1. A, B, and C would divide 80 dollars among them
in such a manner, that B may have 5 dollars more than
A, and C 10 dollars more than B; required the share
of each ?

170
POSITION.
Suppose A's share $10
B's
C's
15
25
50
Suppose A's share $15
B's
C's
20
30
65
80-50-30 error too little. 80-65-15 error too little.
Errors.
Er. Sup.
30
30×15-450
15
15×10=150
A 20
Ans.
B 25
15 diff. of er.
15)300 diff. of prod.
C 35
20 A's share.
80
2. D, E, and F would divide 100 L. among them, so
as that E may have 3 L. more than D, and F 4 L. more
than E; what is the share of each?
Ans. D's share 30 L., E's 33 L., F's 37 L.
3. A, B, and C owe 1000 L., of which B is to pay
100 L. more than A, and C is to pay as much as both
A and B: how much is each man's share of the debt?
Ans. A's share is 200 L., B's 300 L., and C's 500 L.
4. Bought linen at 4 s. per yard, and muslin at 2 s.
per yard; the number of yards of both was 8, and the
whole cost 20s.: how many yards were there of each?
Ans. 2 yards of linen, and 6 yards of muslin.
5. The head of a certain fish is 9 inches long; its tail
is as long as its head and half of its body; and the length
of its body is equal to the length of its head and tail:
what is the whole length?
Ans. 6 feet.
6. A labourer hired for 40 days upon this condition,
that he should receive 20 cents for every day he
wrought, and should forfeit 10 cents for every day he
was idle; at settlement he received 5 dollars. How
many days did he work, and how many days was he
idle?
Ans. Wrought 30 days, idle 10.
7. A father dying, left to his three sons A, B, and
Chis estate in money, dividing it as follows, viz. to
A he gave half the estate, wanting 44 L.; to B he gave
a third of it, and 14 L. over; and to C he gave the re-
mainder, which was 82 L. less than the share of B.
What was the whole sum left, and what was each son's
share?
The sum left was 588 L., of which A
Ans. {
had 250 L., B 210L., and C 128 L.

ARITHMETICAL PROGRESSION.
171
8. Two persons, A and B, have both the same in-
come; A saves one fifth of his every year; but B, by
spending 150 dollars per annum more than A, at the
end of 8 years finds himself 400 dollars in debt: What
is their income, and what does each spend per annum?
Their income is 500 dollars per annum.
Ans. A spends 400 dols., and B‍550.
ARITHMETICAL PROGRESSION.
Any rank or series of numbers, increasing or de-
creasing by a common difference, is said to be in arith-
metical progression; as 2, 4, 6, 8, 10, and 6, 5, 4, 3,
2, 1.
The numbers which form the series are called the
terms. The first and last terms are called the ex-
tremes.
Note. In any series of numbers in Arithmetical Pro-
gression, the sum of the two extremes is equal to the
sum of any two terms equally distant from them; as in
the latter of the above series 6+1=4+3, and ≈5+2.
When the number of terms is odd, the double of the
middle term is equal to the sum of the two extremes,
or any two terms equally distant from the middle term ;
as in the former of the foregoing series 6×2=2+10,
and 4+8.
CASE 1.
The first term, common difference, and number of
terms given, to find the last term, and sum of all the
terms.
RULE.
1. Multiply the number of terms, less 1, by the
common difference, and to the product add the first
term, the sum is the last term.
2. Multiply the sum of the two extremes by the
number of terms, and half the product will be the sum
of all the terms.
EXAMPLES.
1. The first term of a certain series in arithmetical
progression is 2, the common difference is 2, and the

172
ARITHMETICAL PROGRESSION.
number of terms 15; what is the last term, and the sum
of all the terms?
15 number of terms.
30
two extremes.
1
2
28
14 number of terms less 1.
2 common difference.
2 first term.
32
15 number of terms.
160
32
30 last term.
2)480
240 Sum of all the terms.
2. Bought 15 yards of linen, at 2 cents for the first
yard, 4 for the second, 6 for the third, &c., increasing
2 cents every yard; what was the cost of the last yard,
and what was the cost of the whole?
Ans. The last yard cost 30 cts., the whole cost $2.40.
3. Sold 20 yards of silk, at 3 d. for the first yard, 6 d.
for the second, 9 d. for the third, &c., increasing 3d.
every yard; what sum did it amount to?
Ans. 2 L. 12 s. 6 d.
4. Sixteen persons gave charity to a poor man; the
first gave 5 d., the second 9 d., and so on in an arith-
metical progression; how much did the last person give,
and what sum did the man receive?
Ans. The last gave 5s. 5d., sum received, 2 L. 6 s. 8 d.
5. If 100 stones be laid two yards distant from each
other in a right line, and a basket placed two yards from
the first stone; what distance must a person travel to
gather them singly into the basket?
Ans. 11 miles, 3 fur. 180 yds.
6. A merchant sold 1000 yards of linen, at 2 pins for
the first yard, 4 for the second, and 6 for the third, &c.
increasing two pins every yard; how much did the
linen produce, when the pins were afterwards sold at
12 for a farthing?
Ans. 86 L. 17 s. 10 d.
CASE 2.
When the two extremes and number of terms are
given, to find the common difference.

ARITHMETICAL PROGRESSION.
173
RULE.
Divide the difference of the extremes by the number
of terms, less one; the quotient will be the common
difference.
EXAMPLES.
1. Twenty and sixty are the two extremes of a cer-
tain series in arithmetical progression, and 21 is the
number of terms; what is the common difference?
Ans. 2.
60
extremes.
20
21-1=20)40 Difference of extremes.
2 Common difference.
2. There are 21 men whose ages are equally distant
from each other in arithmetical progression; the young-
est is 20 years old, and the eldest 60; what is the com-
mon difference of their ages, and the age of each man?
Ans. Common difference 2 years
60 years is the age of the first man.
age of the second.
age of the third, &c.
60-2=58
58-2=56
3. A debt is to be paid at 16 different payments in
arithmetical progression; the first payment to be 14 L.
and the last 100 L.: what is the common difference,
each payment, and the whole debt?
Common difference 5 L. 14 s. 8 d.
Ans. First payment 14 L. Second, 19 L.
14 s. 8 d. Third, 25 L. 9 s. 4 d. &c.
4. A person is to travel from Philadelphia to a cer-
tain place in 16 days, and to go but 4 miles the first
day, increasing every day by an equal excess, so that
the last day's journey may be 79 miles; what is the
common difference; and what the whole distance?
Ans. (Common difference 5 miles.
Distance 664 miles.
صہ

174
GEOMETRICAL PROGRESSION.
GEOMETRICAL PROGRESSION.
Any series of numbers, the terms of which increase
by a common multiplier, or decrease by a common di-
visor, are said to be in geometrical progression; as 3,
6, 12, 24, 48; and 48, 24, 12, 6, 3.
The multiplier or divisor by which the series is in-
creased or decreased is called the ratio.
The last term and sum of the series is found ry
this
RULE.
1. Raise the ratio to the power whose index is one
less than the number of terms given, which, being
multiplied by the first term, will give the last term,
or greater extreme.
2. Multiply the last term by the ratio, from the pro-
duct subtract the first term, and divide the remainder
by ratio less one for the sum of the series.
EXAMPLES.
1. A thresher wrought 20 days, and received for
the first day's labour 4 grains of wheat; for the second,
12; for the third, 36, &c. How much did his wages
amount to, allowing 7680 grains to make a pint, and
the whole to be disposed of at one dollar per bushel ?
Note.-The first term in this question is 4, the ratio
3, the number of terms 20: therefore raise the ratio
to the 19th power, which is one less than the number
of terms.

GEOMETRICAL PROGRESSION.
175
1st power.
2d power.
3d power.
4th power.
Ratio 3, 9, 27, 81
!
81
81
648
*
6561 8th power of the ratio.
6561
6561
39366
32805
39366
43046721 16th power.
27 3d power.
301327047
86093442
1162261467 19th power.
4 1st term.
4649045868
3 Ratio.
13947137604
4 First term.
Ratio less one 2)13947137600
7680)6973568800 Sum of the series.
908016 pints,
14187 bushels.
14187 bushels, at 1 dol. p. bu. amount to 14187 dols. Ans.
Q

176
GEOMETRICAL PROGRESSION.
2. Sold 24 yards of Holland, at 2 d. for the first yard,
4 d. for the second, 8 d. for the third, &c.; how much
did it amount to?
Ans. 139810 L. 2 s. 6 d.
3. Bought 30 bushels of wheat, at 2 d. for the first
bushel, 4 d. for the second, 8 d. for third, &c.; what
does the whole amount to, and what is the price per
bushel on an average?
Ans.
8947848 L. 10 s. 6 d. Amount.
298261 L. 12 s. 4 d. per bushel.
4. A merchant sold 30 yards of lace, at 2 pins for the
first yard, 6 for the second, 18 for the third, &c., and
disposed of the pins at 1000 for a farthing; how much
did he receive for the lace? and how much did he gain
by the sale, supposing the lace cost him 100 L. per yd.?
Received 214469929 L. 5 s. 3½ d.
Gained 214466929 L. 5 s. 3 d.
5. A goldsmith sold 1 lb. of gold, at a farthing for the
first ounce, a penny for the second, 4 d. for the third,
&c., in quadruple proportion; how much did he receive
for the whole, and how much did he gain by the sale,
supposing he gave 4 L. per ounce for the gold?
Ans.
He received 5825 L. 8 s. 5 d.
Ans. And gained 5777 L. 8 s. 54 d.
6. What sum would purchase a horse with 4 shoes,
and eight nails in each shoe, at one farthing for the first
nail, a halfpenny for the second, a penny for the third,
&c. doubling to the last? Ans. 4473924 L. 5 s. 34 d.
7. A person married his daughter on new year's day,
and gave her one dollar towards her portion, promising
to double it on the first day of every month for one
year; what was her portion?
Ans. 4095 dols.
8. Suppose a man wrought 20 days, and received
for the first day's labour 4 grains of corn, for the
second 12, for the third 36, &c.; what did he receive
for his labour, supposing 7680 grains to make a pint,
and the whole to be sold at 2 s. 6 d. per bushel?
Ans. 1773 L. 7 s. 6 d.

COMPOUND INTEREST BY DECIMALS.
177
COMPOUND INTEREST,
BY DECIMALS.
The ratio in compound interest is the amount of one
pound or dollar for one year; which is thus found :
As 100 : 1 :: 105: 1.05 As 100: :: 105.5 1.055.
For quarterly amounts, take the 4th root of the ratio;
for half yearly, the square root; and, for 3 quarters, the
product of the quarterly and half yearly.
4
2
Thus, 1.03-1.007417; 1.03-1.014889; and
1.007417×1.014889-1.022416, for 3 quarters.
TABLE I.
Amounts of L.1 for a year and for
quarters, at Compound Interest.
Rate
per
cent.
Simp. Int.
of L. 1 for
Ratio.
For 3 qrs.
For 2 qrs.
For 1 qr.
1 month.
3
1.03 1.022416
1.014889
1.007417 .002500
31/ 1.035 1.026137 1.017349 1.008637) .002917
4
5
1.04 1.029852 1.019804 1.009853 .003333
4 1.045 1.033563 1.022252 1.011065 .003750
1.05 1.037270 1.024695 1.012272 .004167
5/1/ 1.055 1.040973 1.027132 1.013475 .004583
1.06 1.044671 1.029536 1.014674 .005000
6/12/2 1.065 1.048364 1.031988 1.015868 .005417
6
7
1.07 1.052053 1.034408 1.017058 .005833

178
COMPOUND INTEREST BY DECIMALS.
The ratio involved to the time is the amount of 1 L.
or dollar for the time given; as a square for 2 years, a
cube for 3, &c. thus, 1.06×1.06×1.06×1.06.
k
or 1.06-1.262477-the 4th power of 1.06, or the
ratio involved to 4 years.
When the ratio is to be involved to years and quar-
ters, the power for the years is to be multiplied by the
proper quarterly amount; as, 1.262477×1.044671—
1.318873 for 4 years, &c.
The power or the amount of 1 L. or dollar may also
be obtained for months and-days (nearly) by adding the
monthly simple interest of 1 L. or dollar, or proper
parts thereof, to the amount of the quarter next pre-
ceding the expiration of the given time, for what that
time exceeds the said quarter; thus,
Amount for yr.
1.029563: For 44 yrs.=1.318873
Int. of 1 L. for 1 mo.= .005000
One sixth for 5 dys.= .000833
.005000
.000833
For 7 mo. 5 dys.=1.035396: For 4 years, 10 mo. 5 dys. 1.324706
The ratio may be thus involved to any time what-
ever; but the operation is facilitated by the following
tables; which may be extended to 100 years, or up-
wards, by multiplying the amount for 46, by that for
the time above 46, &c.

COMPOUND INTEREST BY DECIMALS.
179
TABLE II.
Showing the amount of 1 L. or Dollar
from 1 year to 46.
Y.3 per cent. 4 per cent. 4 per cent. 5 per cent. 5 per cent. 6 per cent.
11.0350000 1.0400000 1.0450000 1.0500000 1.0550000 1.0600000
1.0712250 1.0816000 1.0920250 1.1025000 1.1130250 1.1360000
31.1087178 1.1248640 1.1411661 1.1576250 1.1742413 1.1910160
41.1475230 1,1698585 1.1925186 1.2155062 1.2388246 1.2624769
51.1876863 1.2166529 1.2461819 1.2762815 1.3069598 1.3382256
61.2292553 1.2653190 1.3022601 1.3400956 1.3788426 1.4185191
1.2653190|1.3022601|1.3400956|
71.2722792 1.3159317 1.3608618 1.4071004 1.4546789 1.5036302
81.3168090 1.3685690 1.4221006 1.4774554 1.5346862 1.5938480
91.3628973 1.4233118 1.4860951 1.5513282 1.6190939 1.6894789
10 1.4105987 1.4802442 1.5529694 1.6238946 1.7081440 1.7908476
111.4599697 1.5394540 1.6228530 1.7103393 1.8020919 1.8982985
12 1.5110686 1.6010322 1.6958814 1.7958563 1.9012069 2.0121964
131.5630560 1.6650735 1.7721961 1.8856491 2.0057732 2.1329282
141.6186945 1.7316764 1.8519449 1.9799316 2.1160907 2.2609039
15 1.6753488 1.8009435 1.9352824 2.0789281 2.2324756 2.3965581
161.7339860 1.8729812 2.0223701 2.1828745 2.3552617 2.5403517
171.7946755 1.94790052.1133768 2.2920183 2.4848011 2.6927727
18 1.8574892 2.0258161|2.2084787 2.4066192 2.6214652 2.8543391
191.9225013 2.1068491 2.3078603 2.5269502 2.7656458 3.0255995
20 1.9897888 2.1911231 2.4117140 2.6532977 2.9177563 3.2071355
212.0594314 2.2787680 2.5202411 2.7859625 3.0782329 3.3995636
222.1315115 2.36991872.6336520 2.9252607 3.2475357 3.6035374
23 2.2061144 2.4647155 2.7521663 3.0715237 3.4261502 3.8097496
24 2.2833284 2.5633041 2.8760138 3.2250999 3.6145885 4.0489346
252.3632449 2.6658363 3.0054344 3.3863549 3.8133910 4.2918707
26 2.4459985 2.7724697 3.1406790 3.5556726 4.0231279 4.5493829
272.5315671 2.8833685 3.2820095 3.7334563 4.2443999 4.8223459
282.6201719 2.9987033 3.4296999 3.9201291 4.4778419 5.1116867
292.7118779 3.1186514 3.5840364 4.1161356 4.7241232 5.4183870
30 2.8067937 3.2433975 3.7453181 4.3219423 4.9839469 5.7434912
312.9050314 3.37313343.9138574 4.5380394 5.2580671 6.0881007
32 3.0067075 3.5080587 4.0899810 4.7649414 5.5472608 6.4533867
33 3.1119423 3.6481831 4.2740301 5.0031885 5.8523600 6.8405899
34 3.2208603 3.7943163 4.46630155.2533479 6.1742398 7.2510253
35 3.3335904 3.94608894.6673478 5.5160152 6.5138230 7.6860868
36 3.4502661 4.1030325 4.97737845.7918101 6.8720832 S.1472520
37 3.5710254 4.2680898 5.0368604 6.0814069 7.2500478 8.6360871
383.6960113 4.43881345.3202192 6.3854772 7.648S004 9.1542523
393.82537174.6163659 5.5658990 6.7047511 8.0694844 9.7035074
40 3.95925974.80102065.8163645 7.0399887 S.5133060 10.2857178
41 4.09783374.9930614|6.07810097.3919881| 8.9815378|10.9028608
1
424.2412579 5.1927838 6.3514240.7615871
9.4755224 11.5570325
9.9966761 12.2504547
43 4.3897020 5.4004952 6.6375523 8.1496669
444.5433415 5.61651506.9362421|8.5571502|10.5464933|12.9854817
45 4.7023585 5.8411756 7.2483730 8.9850077 11.1265504 13.7646107
46 4.8669411 6.0748236 7.5745497 9.4342581 11.7385217 14.5904873
Q 2
180
COMPOUND INTEREST BY DECIMALS.
CASE 1.
The principal, time, and rate given, to find the
amount, or interest:
RULE.
Multiply the principal by the ratio involved to the
time, (found either by involution, or in table II.) and
the product will be the amount; from which subtract
the principal, for the compound interest.
EXAMPLES.
1 What will 225 L. amount to in 3 years, at 5 per
cent. per annum ?
1.05x1.05x1.05-1.157625 raised to the third
power;
then, 1.157625×225=260 L. 9 s. 3 d. 3 qrs. the Ans.
2. What will 480 L. amount to in 6 years, at 5 per
cent. per annum?
Ans. 643 L. 4 s. 11.0178 d.
3. What is the amount of 500 L. at 44 per cent. per
annum, for 4 years? Ans. 590 L. 11 s. 5 d. 2.95+qrs.
4. What is the compound interest of a bond for 764
dollars, for 4 years and 9 months, at 6 per cent. per
Ans 243 dols. 61 cts +
annum ?
CASE 2.
DISCOUNT,
Or, the amount, rate, and time given, to find the
principal:
1
RULE.
Divide the amount by the ratio involved to the time.
EXAMPLES.
1. What principal must be put to interest, to amount
to 260 L. 9 s. 3 d. 3 qrs. in 3 years, at 5 per cent. per
annum?
260 L. 9 s. 3 d. 3 qrs. 260.465625 1.
1.05x1.05x1.05-1.157625 ratio raised to the 3d
power.
1.157625)260.465625(225 L. Ans.
2. What principal will amount to 547 L. 9 s. 10 d.
2.0528 qrs. in 5 years, at 4 per cent. per annum ?
3. What principal will amount to
3.809 qrs. in 4 years, at 5 per cent.?
Ans. 450 L.
619 L. 8 s. 2 d.
Ans. 500 L.

ANNUITIES AT COMPOUND INTEREST.
181
An annuity is a sum of money payable yearly, half
yearly, or quarterly, for a number of years, during life,
or for ever; and may dray interest if it remain unpaid
after it becomes duc.
Tables to facilitate the calculations of Annuities.
1
TABLE III. Showing the amount of 1 L. annuity.
4 per cent. 4 per cent.
1.
5 per cent.
5 per cent. 6 per cent.
1.
1.
2 3
2.04
2.045
1.
2.05
1.
2.055
2.06
2
3
3.1216
3.137025
3.1525
3.168225
3.1836
3
4
4.246464
4.278191
4.310125
4.342266
4.374602 4
5 5.416322
5.47071
5.525631
5.581091
5.637093 5
6
6.632975
6.716892 6.801913
6.888051
6.975318 6
7.898294
8.019152
8.142008
8.266894
8.393837.7
9.214226
9.380014
9.549109
9.721573
9.897468 8
9 10.582795
10.802114
11.026564
11.256259
11.491316 9
10 12.006107
12.28821 12.577892
12.875354
13.180795 10
11 13.486351
12 15.025805
13 16.626838 17.159913 17.712983
14 18.291911 18.932109 19.598632
15 20.023588 20.784054
16 21.824531 22.719337 23.657492
17 23.697512 24.741707 25.840366
18 25.645413
21.578563
13.841179 14.206787
14.583499
14.971643 11
15.464032 15.917126
16.38559
16.869942 12
18.286798
18.882133 13
20.292572
21.015066|14
22.408663
23.27597115
24.64114
25.672528 16
26.996402
28.212881 17
19 27.671229
20 29.778078
44.56521
47.727099
47.570645 51.113454
50.711324 54.669126
66.438847
70.76079
75.298829
80.063771
21 31.969202
22 34.247970
23 36.617888
24 39.082604
25 41.645908
26 44.311745
27 47.084214
28 49.967582 53.993333 58.402583
29 52.966286 57.423033 62.322712
30 56.084938 61.007069
31 59.328335 64.752388
52] 62.701469 68.666245
33 66.209527 72.756226
34 69.357904 77.030256 85.066959
35 73.6a2225 81.496618 90.320307 100.251363
36 77.598314 86.163966 95.836323 106.765188 119.120867 36
37 81.702246 91.041344 101.623139 113.637274 127.268118 37
38 85.970336 96.138205 107.709546 120.887324| 135.904206 38
39 90.40915 101.464424| 114.095023| 128,536127 145.058458 39
40 95.025516| 107.030329|120.799774| 136.605146 154.761966 40
26.855084 28.132385 29.481205
29.063562 30.539004
31.371423 33.065954 34.868318
33.783137 35.719252 37.786075
36.833378 38.505214 40.864309
39.93703 41.430475 44.111846
41.689196 44.501999 47.537998
30.905653 18
32.102671
33.759993 19
36.785592 20
39.992728 21
43.392291 22
46.995828 23
50.815578 24
51.152588
54.864513|25
54.965979
59.156383 26
58.989109
63.705766 27
63.23351
68.528117 28
67.711353
73.639798 29
72.435478
79.058186 30
77.419429
84.801677 31
82.677498
90.589778 32
88.22476
97.343165 33
94.077122
104.183754 34
111.434780 35

182
ANNUITIES AT COMPOUND INTEREST.
TABLE IV.
Showing the present worth of 1 L. annuity for any
number of years. from 1 to 40.
Y. 4 per cent. 4 per cent. 5 per cent. 51 per cent.
6
per cent. Y.
0.96154 0.95694
0.95231
0.94786
0.94339
જે સર્વ
2
1.88609
1.87267
1.85941
1.81632
1.83339 2
3
2.77509 2.74876
2.72325
2.69793
2.67301 3
3.62989 3.58752
3.54595
3.50514
3.4651 4
4.45182 4.38997
4.32988
4.27028
4.21236 5
6
5.24214 5.15787
5.07569
4.99553
4.91732 6
7
6.40205 5.8927
5.78637
5.68297
5.58238 y
8
6.73274
6.59589
6.46321
6.33457
6.20979 8
9
7.43533 7.26879
7.10782
6.95220
6.80169 9
10
8.11089 7.91272
7.72173
7.53762
7.36008 10
11
8.76048
8.52892
8.3064 8.09254
7.88687 11
12
9.38500
9.11858
8.86325
8.61852
8.38384 12
13
9.98565 9.68285
9.39357
9.11708
8.85268 13
14 10.56312 10.22282
9.89864
9.58965
9.29498 14
9.71225/15
9.10589 16
18
19
20
10.47726 17
10.8276 18
11.60765
11.15811 19
11.95034
11.46992 20
12.27524
11.76407 21
13.163
12.58317
12.04158 22.
13.48857
12.87504
12.30338 23
13.79864 13.15170
12.55035 24
13.41391
12.78335 25
13.00316 26
15 11.41839 10.73954 10.37965 10.03759
16 11.65229 11.23401 10.83777 10.46216
17 12.16567 11.70719 11.27407 10.86461
12.65929 12.15999| 11.68958
11.68958 11.24607
13.13394 12.59329 12.08532
13.59032 13.00793| 12.46221
21 14.02916 13.40472 12.82115
22 14.45111 13.78442
23 14.85684 14.14777
24 15.24696 14.49548
25 15.62208 14.82821 14.09394
26 15.98277 15.14661 14.37518 13.66250
27 16.32959 15.45130 14.64303 13.89810] 13.21053 27
28 16.66306) 15.74287 14.89813 14.12142] 13.40616|28
29 16.98371 16.02189 15.14107 14.33310
30 17.29203 16.28889] 15.37245 14.53375
31 17.58849 16.54439 15.59281
13.59072 29
13.76483 30
14.72393
13.92908 31
15.80268
14.90420
14.08404 32
16.00255 15.07507
14.23023 33]
16.1929 15.23703
14.36814 34
16.37419
16.54685
15.39055
14.49825 35
15.53607
14.62098 36
14.73678 37
15.80474
14.84602|38
14.94907/39
32 17.87355 16.78889
33 18.14764 17.02286
34 18.41126 17.24676
35 18.66461 17.46101
36 18.90828 17.66604
37 19.14258 17.86224 16.71129 15.67400
15.67400
38 19.36786 18.04999 16.86789
39 19.58448 18.22965 17.01704 15.92866
40 19.79277 18.40158 17.15909 16.04612 14.92640/40

ANNUITIES AT COMPOUND INTEREST.
183
Rate
per ct.
3
TABLE V.
Half yearly
payinents.
Quarterly
payments.
1.007445 1.011181
The construction of this
table is from an algebraic
theorem, given by the learn-
ed A. De Moivre, in his trea-
31.008675 1.013031 tise of Annuities on Lives,
4
5
1.009902 1.014877
1.0111261.016720
1.012348 1.018559
which may be in words,
thus:
For half yearly payments
51.013567 1.020395 take a unit from the ratio,
6 1.014781 1.022257
1.022257 and from the square root of
61.015993 1.024055 the ratio; half the quotient
7
1.017204 1.025880
of the first remainder divided
by the latter, will be the tabular number.
For quarterly payments use the 4th root as above,
and take one quarter of the quotient.
CASE 1.
The annuity, time, and rate of interest given, to find
the amount.
RULE.
From the ratio involved to the time take a unit, or
one, for the dividend; which divide by the ratio less
one; and multiply the quotient by the annuity, for the
amount or answer. Or, by Table III.
Multiply the number under the rate, and opposite to
the time, by the annuity, and the product will be the
amount for yearly payments.
If the payments be half yearly or quarterly, the
amount for the given time, found as above, multiplied
by the proper number in Table V., will be the true
amount.
EXAMPLES.
1. What will an annuity of 50 L. per annum, payable
yearly, amount to in 4 years at 5 per cent.?
1.05×1.05×1.05×1.05-1.21550625
1.05-1.05).21550625
4.310125
50
Ans. L. 215.506250=215 L. 10 s. 1 d. 2 qrs.

184
ANNUITIES AT COMPOUND INTEREST.
2. What will an annuity of 30L. per annum, paya-
ble yearly, amount to in 4 years, at 5 per cent. per an-
num, and what would be the respective amounts, if the
payments were to be half yearly or quarterly?
Ans.
Amount for yearly payments is L. 129.30375
for half yearly
L. 130.CC04
for quarterly
L. 131.7035
¿
3 If a salary of 35L. per annum to be pa yearly,
be omitted for 6 years at 5½ per cent. what is the
amount?
Ans. 241L. 1s. 7d. 2.5+qrs.
CASE 2.
The annuity, time, and rate given, to find the pre-
sent worth:
RULE.
Divide the annuity by the ratio involved to the time,
and subtract the quotient from the annuity; divide the
remainder by the ratio less one, and the quotient will
be the present worth: Or, by Table IV.
Multiply the number under the rate, and opposite
the time by the annuity, and the product will be the
present worth.
When the payments are half yearly or quarterly,
multiply the present worth so found, by the proper
number in Table V.
EXAMPLES.
1. What is the present worth of a pension of 30L.
per annum for 5 years, at 4 per cent.?
Number from Table IV. 4.45182
Ans. 133L. 11s. 1d.
×30 annuity.
L. 133.55460
Or, 133L. 11s. 1.104d.
2. What is the present worth of 20L. a year for 6
years, payable either yearly, half yearly, or quarterly,
computing at 5 per cent. per annum?
L.
Present worth for yearly payments, 101.5138
Ans.
for half yearly
for quarterly
102.7673
103.3978

ANNUITIES TAKEN IN REVERSION.
185
3. What is the yearly rent of 50L. to continue 5
years, worth in ready money, at 5 per cent.?
Ans. 216L. 9s. 10d. 2.24qrs.
ANNUITIES TAKEN IN REVERSION,
AT COMPOUND INTEREST.
Annuities taken in reversion, are certain sums of
money payable yearly for a limited period, but not to
commence till after the expiration of a certain time.
CASE 1.
The annuity, time of reversion, time of continuance,
and rate given, to find the present worth of the annui
ty in reversion:
RULE.
Divide the annuity by the ratio involved to the time
of continuance, and subtract the quotient from the an-
nuity for a dividend; multiply the ratio involved to the
time of reversion by the ratic, less one, for a divisor;
the quotient of this division will be the present worth.
Or,
Take two numbers under the given rate in Table IV.
viz. that opposite the sum of the two given times, and
that against the time of reversion, and multiply their
difference by the annuity of the present worth.
When the payments are half yearly or quarterly, use
Table V.
EXAMPLES.
1. What is the present worth of a reversion of a lease
of 40L. per annum, to continue for six years, but not
to commence till the end of 2 years, allowing 6 per
cent. to the purchaser?
Ratio in-
volved to
the time.
40 annuity.
1.4185191)40.000000000000(28.19842
11.80158
1.06×1.06×.06=.067416)11.80158(175.056+L. Ans.
Or by Table IV. First, the sum of the two given

186
PERPETUITIES.
times is 8 years, and the time of reversion 2 years,
therefore,
Take for 8 years 6.20979
for 2 do. 1.83339
Difference 4.37640
×40 annuity.
L. 175.05600 Ans. as before.
2. A person owns a farm which he proposes to let
for 8 years, at 100 dollars per annum ; but cannot give
possession till after the expiration of two years; what is
the present worth of such a lease, allowing 4 per cent.
for present payment?
Ans. 622.48 dols.
3. What is the present worth of a reversion of a lease
of 60 L. per annum, to continue 7 years, but not to com-
mence till the end of 3 years, allowing 5 per cent. to
the purchaser ?
Ans. 299 L. 18 s. 2.112 d.
PERPETUITIES,
AT COMPOUND INTEREST.
Perpetuities are such annuities as continue for ever.
CASE 1.
The annuity, and rate given, to find the present
worth.
RULE.
Divide the annuity by the ratio less one, for the pre-
sent worth.
Note. For perpetual half yearly, or quarterly pay-
ments, Table V. is to be applied as in similar cases of
temporary annuities.
EXAMPLES.
1. What is an estate of 140 L. per annum, to con-
tinue for ever, worth in present money, allowing 4 per
cent. to the purchaser ?
L.
1.04-1.04)140.00
L. 3500

PERMUTATION.
187
2. What is the present worth of a freehold estate of
290 dollars per annum, to continue for ever, allowing
4 per cent. to the purchaser?
Ans. 7250 dols.
PERPETUITIES IN REVERSION.
CASE 1.
The rent of a freehold estate, time of reversion, and
rate per cent. given, to find the present worth.
RULE.
Multiply the ratio involved to the time of reversion,
by the ratio, less one, for a divisor; by which divide
the yearly payment, the quotient will be the answer.
EXAMPLES.
1. If a freehold estate of 50 L. per annum, to com-
mence 4 years hence, be put up at sale, what is the
present worth, allowing the purchaser 5 per cent.?
Ans. 822 L. 14 s. 1 d. 2 qrs. +
Ratio involved to the time?
of reversion, viz. 4 years S
1.2155062
.05 ratio less one.
.060775310)50(822 l. 14s. 1 d. 2q.+
2. What is an estate of 696 dols. per annum, to con-
tinue for ever, but not to commence till the expiration
of 4 years, worth in present money, allowance being
made at 4 per cent.?
Ans. 14873.594 dols.
PERMUTATION.
Permutation is a rule for finding how many different
ways any given number of things may be varied in
position, place, or succession; thus, a b c, a c b, ba c,
bca, ca b, c ba, are six different positions of three
letters.
RULE.
Multiply all the terms of the natural series continu-
ally from one to the given number inclusive; the last
product will be the answer required.
R

190
DUODECIMALS.
5. Bought a raft of boards containing 59621 ft. 8 in.,
of which are since sold three parcels, each 14905 ft. 5in.,
how many feet remain ?
Ans. 14905 ft. 5 in.
MULTIPLICATION OF DUODECIMALS.
CASE 1.
When the feet of the multiplier do not exceed 12.
RULE.
Set the feet of the multiplier under the lowest deno-
mination of the n ultiplicand, as in the following exam-
ple; then multiply as in Compound Multiplication by
each denomination of the multiplier separately, observ-
ing to place the right hand figure, or number, of each
product, under that denomination of the multiplier by
which it is produced.
Note 1.-If there are no feet in the multiplier, sup-
ply their place with a cypher; and in like manner sup-
ply the place of any other denomination between the
highest and lowest.
2. Feet multiplied by feet, give feet.
Feet multiplied by inches, give inches.
Feet multiplied by seconds, give seconds.
Inches multiplied by inches, give seconds.
Inches multiplied by seconds, give thirds.
Seconds multiplied by seconds, give fourths.
* It may be remarked that though the feet obtain-
ed by this rule are square feet, the inches are not square
inches, but twelfth parts of a square foot.
EXAMPLES.
1. Multiply 10ft. 6in. by 4ft. 6in.
Ft. I.
I.
10 6
4 6
| 6 | 3 | 10
Or thus;
6
Or Decimally.
Ft. I. Ft.
10
6=10.5
4 6
4
6= 4.5
5 3 0
42 0
525
42 0
5 3
420
47 3 0
47 3
47.25
Ft. I.
Ft. I.
Ft. I.
2. Multiply 9 7
by 3
6
Result 33
6 6
3.
3 11
by 9
5
36 10 7
4.
5.
8 6 9 by 7 3
28 10 6 by 3
8
62 6 7 9
2 4
92 2 10 6
: …

C
DUODECIMALS.
CASE 2.
When the feet of the multiplier exceed 12.
RULE.
191
Multiply by the feet of the multiplier as in Compound
Multiplication, and take parts for the inches, &c.
EXAMPLES.
1 Multiply 112 ft. 3 in. 5" by 42 ft. 4 in. 6".
Ft. I.
112 3 5
6x7=42
673 8 6
7
I.
113
16
4715 11
6
37 5
1
S
00find
4 8
18 6
4758
0 9 4 6
Ft. I. "
Ft. I."
Ft. I.
1. "
///
by 19 10
2 Multiply 76 7
3.
4.
127 6 by 184 8
71 2 6 by 81 18
APPLICATION.
Result 1518 10 10
23545 0 0
5777 9 2 2
1. A certain board is 28 ft. 10 in. 6" long, and 3 ft.
2 in. 4″ wide; how many square feet does it contain?
Ans. 92 ft. 2 in. 10″ 6"".
2. If a board be 23 ft. 3 in. long, and 3 ft. 6 in. wide,
how many square feet does it contain?
3. A certain partition is 82 ft.
many square feet does it contain?
4. If a floor be 79 ft. 8 in. by
square feet are therein?
Note.-Divide the square feet
will be square yards.
6
Ans. 81 ft. 4 in. 6".
in. by 13 ft. 3 in., how
Ans. 1093 ft. 1 in. 6".
38 ft. 11 in., how many
Ans. 3100 ft. 4 in. 4".
by 9, and the quotient
5. If a ceiling be 59 ft. 9 in. long, and 24 ft. 6 in.
broad, how many square yards does it contain?
Ans. 162 yd. 5 ft.+
6. What will the plastering of a ceiling come to, at
10d. per yard, supposing the length 21 ft. 8 in., and the
breadth 14 ft. 10 in.?
Ans. 1 L. 9 s. 9 d.
R 2

192
DUODECIMALS.
7. What will the paving of a court-yard come to, at
4åd. per yard, the length being 58 feet 6 inches, and
the breadth 54 feet 9 inches?
Ans. 7L. Os. 10d.
8. Suppose the dimensions of a bale to be 7 feet 6
inches, 3 feet 3 inches, and 1 foot 10 inches; what is the
solid content?
Ans. 44ft. 8in. 3'
Ft. I.
7 6
3 3
Ft. I.
7
6×3ın.
1
10 6
7
6×3 ft. =22
6.
24 4 6
1 10
Ft. I.
24 4
6x10in.=20
39 0
24 4 6x 1 ft. =24
4 6
Ans. 44 8 3
9. What is the freight of a bale containing 65 feet 9
inches, at 15 dollars per ton of 40 feet?
Ans. 24 dols. 653 cts.
Decimally
20 ft.
5 ft.
AICHT
$15.00 for 40 feet.
7.50
6 in.
1.87.5
ΤΟ
2 in.
18.7
9.3
24.65.5
65.75
15
32875
6575
40)986.25
24.65.6
10. A merchant imports from London 6 bales of the
following dimensions, viz.
Length.
Ft. I.
No. 1 2
10
6
02 0 + 0
3
4
5
2
3
2
10
6
10
2
10
2
11
Height.
Depth.
Ft. 1.
Ft. I.
2
OI OI OI OI O Q
2
4
1 9
6
1
3
2
1
1
У
2
6
1
9
8
1
8

PROMISCUOUS QUESTIONS.
193
What are the solid contents, and how much will the
freight amount to at 20 dollars per ton?
The contents are, viz.
No. 1 11 7
Feet.
2 S 10
71.58
3
12 7
20
4
13 2
5
12
5
40,1431.60
6
13 0
71 7
35.79
Amount $35.79
PROMISCUOUS QUESTIONS.
1. A merchant had 1000 dollars in bank; he drew
out at one time $237.50, at another time $116.09, and
at another $241.06; after which he deposited at one
time 1500 dollars, and at another time $750.50; how
much had he in bank after making the last deposite?
Ans. $2655.841.
2. Sold 8 bales of linen, 4 of which contained 9
pieces each, and in each piece was 35 yards; the other
4 bales contained 12 pieces each, and in each piece was
27 yards: how many pieces and how many yards were
in all ?
Ans. 84 pieces, 2556 yards.
3. A was born when B was 21 years of age, how old
will A be when B is 47; and what will be the age of B
when A is 60?
SA will be 26 when B is 47.
B will be 81 when A is 60.
Ans.
4. If 79 L. 4 s. 10 d. be divided among 4 men, 6
women, and 9 boys, so as that cach man shall receive
twice as much as a woman, and each woman twice as
much as a boy, what will be the share of each.
Each boy must have 2 L. 2 s. 10 d.
Ans. Each woman
Each man
4 L. 5 s. 8 d.
S L. 11 s. 4 d
5. How many bushels of wheat, at $1.12 per bushel,
can I have for $81.76?
Ans. 73.
6. What will 27 cwt. of iron come to, at $4.56 per
Ans. $123 12.
cwt.?

194
PROMISCUOUS QUESTIONS.
7. When a man's yearly income is 949 dollars, how
much is it per day?
Ans. $2.60.
8. If a man leave 6509 dollars to his wife and two
sons thus, to his wife, to his elder son of the re-
mainder, and to his other son the rest; what is the
share of cach?
Wife's share $2440.87).
Elder son's $2440.87½.
Other son's $1627.25.
Ans.
9. How many yards of cloth, at 17 s. 6 d. per yard,
can I have for 13 cwt. 2 qrs. of wool, at 14d. per lb.?
Ans. 100 yards, 3 qrs.
10. How many dollars are equal to 980 French
crowns?
Ans. 1078.
11. If goods which cost 10 s. be sold for 11s. 9 d.
what is the gain per cent.?
Ans. 17.
12. Bought 27 bags of ginger, each weighing gross
843 lb. tare 13 lb. per bag, tret 4 lb. per 104 lb.; what
does the whole (neat weight) come to, at 84 d. per lb.?
Ans. 76 L. 13 s. 24 d.
13. My factor sends me word he has bought goods
to the value of 500 L. 13 s. 6 d. upon my account; what
will his commission come to at 3
14. If of an ounce cost of
of a lb. cost?
per cent.?
Ans. 17 L. 10s. 5 d.
a shilling, what will g
Ans. 17 s. 6 d.
15. If of a gallon cost of a L., what will & of a tun
cost?
9
Ans. 105 L.
16. If of a ship be worth 3740 L., what is the worth
of the whole?
Ans. 9973 L. 6 s. 8 d.
17. A person who was possessed of of a vessel, sold
of his share for 375 L., what was the whole vessel
worth at that rate?
Ans. 1500 L.
18. If 4 cwt. be carried 36 miles for 35 s., how
pounds can I have carried 20 miles for the same mo-
ney?
Ans. 907lb. 3 oz. 3 dr. 4
19. What is the interest of 47 L.
52 days, at 42 per cent.?
many
20
10 s. for 4 years and
Ans. 8 L. 17 s. 1d
20. If 100 L. in 5 years gain 20 L. 10s., in what time
will any sum of money double itself at the same rate
of interest?
Ans. 2414 years.
21. What sum will produce as much interest in 34
years, as 210L. 3s. would in 5 years and 5 months?
Ans. 350 L. 5s.

PROMISCUOUS QUESTIONS.
195
22. What is the commission on $2176.50, at 2 per
cent.?
Ans. $54.41.
23. What is the premium of insuring 1650 dollars,
at 15 pcr cent.?
Ans. $255.75.
24. Bought a quantity of goods for 250L. and 3
months afterwards sold them for 275L. how much per
cent. per annum was gained by the transaction?
Ans. 40L.
25. A vintner mixed 20 gallons of Port wine at 5s.
4d. per gallon, with 12 gallons of White wine, at 5s.
per gallon, 30 gallons of Lisbon, at 6s. per gallon, and
20 gallons of Mountain, at 4s. 6d. per gallon; what
was a gallon of the mixture worth? Ans. 5s. 33d.+
26. A person has two silver cups of unequal weight,
having one cover to both, which weighs 50z.; now if
the cover be put on the less cup it will be double the
weight of the greater cup, and put on the greater cup it
will be three times as heavy as the less cup: what is the
weight of each cup?
Ans. The less 3oz. the greater 4oz.
27. A person said he had 20 children, and that it
happened there was a year and a half between each of
their ages; his eldest son was born when he was 24
years old, and the age of his youngest is 21; what was
the father's age?
Ans. 73 years.
1
6
28. In a certain orchard of the trees bear apples,
pears, plums, 6u of them peaches, and 40 cherries;
how many trees are in the orchard? Ans. 1200.
29. If by selling goods at 50s. per cwt. I gain 20 per
cent. what do I gain or lose per cent. by selling at 45s.
Ans. 8L. gain.
per cwt.?
30. Sold goods for 63L. and by so doing lost 17 per
cent. whereas I ought, in dealing, to have cleared 20
per cent. how much under their just value were they
sold?
Ans. 28L. 1s. 8d.
31. A person willing to distribute some money
among a number of beggars, wanted 8d. to give them
3d. a piece; he therefore gave each 2d. and had 3d. left;
how many beggars were there?
Ans. 11.
32. A person being asked the hour of the day, said;
the time past noon is equal to of the time till mid-
night; what was the time? Ans. 20 minutes past 5.
33. A person looking on his watch, was asked what

196
PROMISCUOUS QUESTIONS.
was the time of day, who answered, it is between 4
and 5; but a more particular answer being required, he
said that the hour and minute hands were then exactly
together what was the time?
Ans. 21
1 1
minutes past 4.
34. Two men depart from the same place at the same
time, one travels 30, and the other 35 miles a day;
how far are they distant from each other after 7 days,
supposing them both to travel the same road; and how
far, if they travel in contrary directions?
Ans. {
S35 miles, when going the same way,
455 miles,
contrary ways.
35. A guardian paid his ward 3500 L. for 2500 L.
which he had held in possession 8 years: what rate of
interest did he allow him?
Ans. 5 per cent.
36. In what time will any sum of money double it-
self, at 6 per cent. interest?
Ans. 16 years and 8 months.
37. A owes B 100 L., payable in 34 months, 150 L.
in 4½ months, and 204 L. in 54 months, and is willing
to make one payment of the whole; in what time should
the payment be made? Ans. 4 months 23 days+
38. If the earth be 360 degrees in circumference,
and each degree 69½ miles, how long would a man be
in travelling round it, who agances 20 miles a day;
reckoning 365 days in a year?
Ans. 3 years 1554 days.
39. A minor of 14 had an annuity left him of 400
dollars a year, the proceeds of which, by will, was to
be put out both principal and interest yearly as it fell
due, at 5 per cent., until he should arrive at 21 years
of age: what had he then to receive?
Ans. $3256.80+
40. If a piece of marble be 47 inches long, 47 inches
broal, and 47 inches deep, how many cubical inches
does it contain?
Ans. 103823.
41. There is a cellar dug that is 12 feet every way,
in length, breadth, and depth; how many solid feet of
earth were taken out of it?
Ans. 1728.
42. How many bricks 9 inches long and 4 inches
wide, will pave a yard that is 20 feet square?
Ans. 1600.
43. If A can do a piece of work alone in 7 days, and

PROMISCUOUS QUESTIONS.
197
1
B in 12 days, in what time can they finish it, both.
working together?
Ans. 4,8 days.
44. A and B traded together; A put in 320L. for 5
months, B 460L. for three months, and they gained
1001. what is each man's share of the gain?
Ans. A's share is 53L. 13s. 9d. +
B's share is 46L. 6s. 2d. +
45. What is the value of a slab of marble, the length
of which is 5 feet 7 inches, and the breadth 1 foot 10
inches, at 1 dollar per foot?
Ans. $10.23
46. A certain stone measures 4 feet 6 inches in
length, 2 feet 9 inches in breadth, and 3 feet 4 inches
in depth; how many solid feet does it contain?
Ans. 41 feet 3 inches.
47. Shipped to Jamaica 550 pair of stockings, at 11s.
6d. per pair, and 460 yards of stuff, at 14d. per yard;
in return for which I have received 46cwt. 3qrs. of
sugar, at 24s. 6d. per cwt. and 1570lb. of indigo, at 2s.
4d. per lb.; what remains due to me?
Ans. 102L. 12s. 11 d.
48. If the flash of an ordnance was observed just one
minute and three seconds before the report, what was
the distance; supposing the flash to be seen the instant
of its going off, and admitting the sound to fly at the
rate of 1142 feet in a second? Ans. 13 miles 5 fur. +
49. Which would be preferable, an annual rent of
876 dollars clear, for 12 years, to be received in quar-
terly payments, or 7200 dollars in hand, reckoning in-
terest at 5 per cent.?
Ans. The annuity by 1272dols. +
50. A line 35 yards long will exactly reach from the
top of a fort, standing on the brink of a river, to the
opposite bank, known to be 27 yards from the foot of
the wall; what is the height of the wall?
Ans. 22 yards 9 inches. +
51. Bought 120 apples at 2 for a penny, and 120
more at 3 for a penny, and sold them altogether at 5
for 2d.; what did I gain or lose by the bargain?
Ans. Lost 4d.
52. A cistern for water has two cocks to supply it,
by the first it may be filled in 45 minutes, and by the
second in 55 minutes; it has likewise a discharging
cock, by which it may, when full, be emptied in 30

198
PROMISCUOUS QUESTIONS.
minutes: now if these three cocks be all left open when
the water comes in, in what time will the cistern be
filled?
Ans. 2 hours, 21min. 25 sec.
53. The account of a certain school is as follows, viz.
of the boys learn geometry, learn grammar,
learn arithmetic, learn to write, and 9 learn to read:
what number is there of each?
Ans.
3
20
3
3
5 who learn geometry, 30 grammar, 24
{ arithmetic, 12 writing, and 9 reading.
54. The sales of certain goods amount to $1873.40;
what sum is to be received for thein, allowing 24 per
cent. for commission, and 4 per cent. for prompt pay-
ment of the neat proceeds?
Ans. $1821.998+
FINIS.
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Gardeth Sher



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UNIVERSITY OF MICHIGAN
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Hannah Shearer & Foot





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