150 
thrown cut of both. Or any numbers may be 
divided by their greatest common divisor, and 
the quotients taken instead of them. 
k i.j If lti horses can eat up 9 bushels of oats 
in b days, how many horses will eat up 24 
bushels in 7 days at the same rate? 
-{- 9 bushels * 16 horses ** 24 bushels 
0 days ; ; j 7 days -f- 
6 X 16 x 24 , . 2 x 16 X 24 
9 X 7 ^ contract ’ on — 
2 X 16 X 8 
3X? 
2 56 
, „ — - = 364 horses, the answ. 
1X7 7 7 ’ 
The reason of this rule may be readily shown 
from the nature of direct andinverse proportion: 
for every row' in this case is a particular stating 
in one of those rules; and therefore if all the se- 
parate dividends be collected together into one 
dividend, and all the divisors into one divisor, 
their quotient must be the answer sought. Thus, 
in the example: 
24 V 1 6 
As 9 bush. * 16 horses ** 24 bush. ‘ - 
by rule of three direct. As 6 days 
16 
. 24 x 16X6 
7 days . by rule of 3 
J 9x7 1 
horses 
inverse, which is the same as the rule. 
Practice. 
Practice is a contraction of the rule of three 
direct, when the first term happens to be an 
unit, or one; and has its name from its daily use 
amongst merchants and trademen, being an easy 
and concise method of working most questions 
that occur in trade and business. 
The method of proof is by the rule of three 
direct. 
An aliquot part of any number, is such a part 
of it, as being taken a certain number of times 
doth exactly make the number. 
General Rule. 
( 1 ) Suppose the price of the given quantity 
to be 1 1 . or Is. as is most convenient; then will 
the quantity itself be the answer at the supposed 
price. ( 2 .) Divide the given price into aliquot 
parts, either of the supposed price, or of ano- 
ther, and the sum of the quotients belonging to 
each will be the true answer required. 
£v.] What is the value of 526’ yards of cloth, 
at 3s. lCfjd. per yard. 
526 AnS. at ll. 
35. 4d. is ± 87 13 4 ditto tit 0 3 4 
fi 
4d. is JL_ 8 15 4 ditto at 0 0 4 
2d. is \ 4 7 8 ditto at 0 0 2 
i is 4 0 10 111 ditto at 0 0 
101 -7 3| ditto at 0 3 10^ 
the full price. 
In the above example, it is plain that the 
quantity 526 is the answer at 11. consequently, 
as 3s. 4d. is the *. of a pound, 1. part of that 
quantity, or 871. 13s. 4d. is the price of 3s. 4d. 
in like manner, as 4d. is the part of 3s. 4d. so 
i ° . 
,J_ of 871. 13s. 4d. or 81. 15s. 4d. is the answer at 
1 ° 
4d. And by reasoning in this way 41. 7s. 8 d. 
will be shewn to be the price at 2 d. and 10 s. 1 l^d. 
the price at J. Now as the sum of all these 
parts is equal to the whole price (3s. 10-|d.), so 
the sum of the answers belonging to each price 
will be the answer at the full price required. 
And the same will be true in any example what- 
ever. 
Vulgar Fractions. 
In order to understand the nature of Vulgar 
Fractions, we must suppose unity (or the num- 
ber 1) divided into several equal parts. One or 
more of these parts is called a fraction, and is 
represented by placing one number in a smaller 
character above a line, and another under it : 
AftmiMEtte. 
For example, a two fifth part is written thus, 
The number under the line (5) show's- how many 
parts unity is divided into, and is called the de- 
nominator. The number above the line ( 2 ) 
show's how many of these parts are represented, 
and is Called the numerator. 
It foliow'S from the manner of representing 
fractions, that when the numerator is increased, 
the value of the fraction becomes greater ; but, 
when the denominator is increased, the value 
becomes less. Hence we may infer, that, if the 
numerator and denominator be both increased, 
or both diminished, in the same proportion, the 
value is not altered; and, therefore, if We mul- 
tiply both by any number whatever, or divide 
them by any number which measures both, w r e 
shall obtain other fractions of equal value. Thus, 
every fraction may be expressed in a variety of 
forms, which have all the same signification. 
A fraction annexed to an integer, or whole 
number, makes a mixed number : For example, 
five and tw'o third-parts, or 5|-. A fraction, 
whose numerator is greater than its denominator, 
is called an improper fraction : For example, 
seventeen third-parts, or . Fractions of this 
kind are greater than unity. Mixed numbers 
may be represented in the form of improper 
fractions, and improper fractions may be reduced 
to mixed numbers, and sometimes to integers. 
A whole number may be treated as a fraction by 
making ils denominator unity. 
“ To reduce mixed numbers to hnproper frac- 
tions: Multiply the integer by the denominator 
of the fraction, and to the product add the nu- 
merator. The sum is the numerator of the im- 
proper fraction sought, and is placed above the 
given denominator.” 
Examples. : 8 - 5 - : 9^ 
4 5 2 
Ex. Required th& greatest number v/hicll 
measures 475 and 569? 
29 
The answers are 
43 
19 
and 
. 4 3 
gr’ T 
Because one is equal to two halves, or 3 third- 
parts, or four quarters, and every integer is 
equal to twice as many halves, or four times as 
many quarters, and so on ; therefore, every inte- 
ger maybe expressed in the form of an improper 
fraction, having any assigned denominator. The 
numerator is obtained by multiplying the integer 
into the denominator. Hence the reason of the 
foregoing rule is evident. 5, reduced to an im- 
proper fraction, whose denominator is 3, makes 
i_5 , and this added to 4, amounts to \J . 
3 ’ . 3 . 3 
“ To reduce improper fractions to whole or 
mixed numbers: Divide the numerator by the 
denominator.” 
5 7 • 8 4 • in . 
Examples. 
9)57 
11)84 
18)91 (5 T V 
90 
The answers are 6JL, 7 7 , and 5JL. 
9’ 11 18 
This problem is the converse of the former, 
and the reason may be illustrated in the same 
manner. 
“ To reduce fractions to lower terms : Divide 
both numerator and denominator by any num- 
ber which measures both, and place the quotients 
in the form of a fraction.” 
Examples. JL£ : : 5 7 0 . 
1 27 16 360 
The answers are 2 ~3 ; 16; for both the 
humerator and denominator of the first fraction 
is divisible by 3 ; of the second by 16; and of 
the third by 90 : but the answers are of pre- 
cisely the same value as the original fractions. 
To find the greatest common measure of two 
numbers: Divide the greater by the less, and the 
divisor by the remainder continually, tillnothing 
remain ; the last divisor is the greatest common 
measure. 
4751589(1 
475 
114)475(4 
456 
Here divide 589 by 47.7, 
and the remainder is 114; 
then divide 475 by 114, and 
the remainder is 19 ; then 1 14 
by 19, and there is ro re- 
19)11476 mainder : from which we in* 
114^ fer, l ^ at the' last divisor, 
— — is the greatest common mca j 
0 sure. 
To explain the reason of this, we must ob- 
serve, that any number which measures two 
others, will also measure their sum, and their 
difference, and will measure any multiple of ei- 
ther. In the foregoing example, any irumbef 
which measures 589 and 475, will measure their 
difference 114, and will measure 456, which 
is a multiple of 114; and any number which 
measures 475, and 456, will also measure 
their difference 19. Consequently, no number 
greater than 19 can measure 589 and 475. Again 
19 will measure them both, for it measures 114, 
and therefore measures 456, which is a multiple 
of 114 and 475, which is just 19 more than 456 : 
and, because it measures 475, and 114, it will 
measure their sum 589. To reduce to the 
lowest possible terms, we divide both number* 
by 19, and it comes to 
If there be no common measure greater than 
1, the fraction is already in the lowest terms. 
If the greatest common measure of 3 numbers 
be required, we find the greatest measure of the 
two first, and then the greatest measure of that 
number, and the third. If there be more num- 
bers, we proceed in the same manner. 
« To reduce fractions to others of equal value 
that have the same denominator: 1st. Multiply 
the numerator of each fraction by all the deno- 
minators except its own. The products are 
numerators to the respective fractions sought.” 
2d. “ Multiply all the denominators into each 
other; the product is the common denominator.’* 
Ex. 4 and 7 and f = ff « and and 
4 X 9 X 8 =r 288 first numerator. 
7 X 5 X 8 =r 280 second numerator. 
3 x 5 X 9 = 135 third numerator. 
5 X 9 X 8 = 360 common denominator. 
Here we multiply 4, the numerator of the first 
fraction, by 9 and 8, the denominators of the two 
others; and the product 288 is the numerator of 
the fraction sought, equivalent to the first. The 
other numerators are found in like manner, and 
the common denominator 360, i&^obtauied by . 
multiplying the given denominators 5, 9, 8, into 
each other. In the course of the whole operation 
the numerators and denominators of each fraction 
are multiplied by the same numbers, and there- 
fore their value is not altered. 
Addition of Vulgar Fractions. 
Rule. Reduce them, if necessary, to a com- 
mon denominator; add the numerators, and 
place the sum above the denominator. 
Ex. 1. 3. a. — 2_7 4- i° = 37 . 
5 * 9 4 5 1 4 5 4 S 
2 5 _1_ 8 I _2_ — 4.5 o _L 510 5 6 7 
7 ‘ T ~ 10 6 3 0 1 6 30 1 630 
— 1 16 7 7. 
6 3 0 
The numerators of fractions that hax'e the 
same denominator signify like parts; and the 
reason for adding them is equally obvious, as 
that for adding shillings or any other inferior 
denomination. 
Subtraction of Vulgar Fractions. 
Rule. “ Reduce the fractions to a common 
denominator; subtract the numerator of the 
subtrahend from the numerator of the minuend, 
and place the remainder above the denomi- 
nator.” 
