124. RULE III.

To divide a fraction by an integer: divide the numerator, if it be divisible, by the integer; or, if the numerator be not divisible, multiply the denominator by that integer.

The rule may be proved as follows:

ac

(1) represents ac equal parts, b of which make up the unit;

b

α

represents a equal parts, b of which make up the unit.

The number of parts taken in the first fraction is c times the number taken in the second. Therefore the second fraction is the quotient of the first fraction divided by c;

(2) But if the numerator be not divisible by c, we have

α

Ъ

C

Б

C

of represents a equal parts, b of which make up.

126. RULE IV. To multiply together two or more fractions: multiply the numerators for a new numerator, and the denominator for a new denominator.

The definition of multiplication, in the strict sense of the word, supposes that a quantity is to be added to itself a certain number of times. But when the multiplier is a fraction this definition ceases to be intelligible; the operation can therefore be only understood in some extended sense. To find a meaning

α с

for %

X we may consider that to obtain

a

с

times

α

we must perform on an operation similar to that we perform on the unit to obtain. In this case we take of 1; hence in the

с

127. RULE V. To divide one fraction by another: invert the divisor, and proceed as in multiplication.

Since division is the inverse of multiplication, we may define

α

the quotient x, when is divided by, to be such that

Example 2. Simplify

2a-3
"
12a

by cancelling those factors which are common to both numerator and

denominator.

Thus in each case we divide the unit into bd equal parts, and we take first ad of these parts, and then be of them; that is, we take ad+bc of the bd parts of the unit; and this is expressed ad+bc by the fraction bd

Similarly,

ad+be

bd

129. Here the fractions have been both expressed with a common denominator bd. But if b and d have a common factor, the product bd is not the lowest common denominator, and the fraction will not be in its lowest terms. To avoid working with fractions which are not in their lowest terms, some modification of the above will be necessary. In practice it will be found advisable to take the lowest common denominator, which is the lowest common multiple of the denominators of the given fractions.

RULE I. To reduce fractions to their lowest common denominator: find the L.C.M. of the given denominators, and take it for the common denominator; divide it by the denominator of the first fraction, and multiply the numerator of this fraction by the quotient so obtained; and do the same with all the other given fractions.

Example. Express with lowest common denominator

The lowest common denominator is 6ax (x − a) (x+a).

We must therefore multiply the numerators by 3x (x+a) and 2a respectively.

130. We may now enunciate the rule for the addition or subtraction of fractions.

RULE II. To add or subtract fractions: reduce them to the lowest common denominator; add or subtract the numerators, and retain the common denominator.

since the terms in the numerator destroy each other.

NOTE. To ensure accuracy the beginner is recommended to use brackets as in the first line of work above.