That is, to divide one fraction by a second fraction, divide the numerator of the first by the numerator of the second for a new numerator, and the denominator of the first by the denominator of the second for a new denominator. (Compare with § 36.)

This process is useful only in those cases in which the numerator and denominator of the divisor are factors of the numerator and denominator of the dividend as shown above.

43. The reciprocal of a fraction is unity divided by the fraction.

The fraction b/a is commonly referred to as being the reciprocal of the fraction a/b.

It should be observed that the reduced value, b/a, of the reciprocal, 1÷a/b, of the fraction, a/b, may be obtained by interchanging the terms a and b of the given fraction a/b.

The product obtained by multiplying the reciprocal of a fraction by the fraction is unity.

has the same effect as that of d

multiplying a by the reciprocal of the divisor, that is by (see § 41).

Hence, for the operation of division by a fraction may be substituted that of multiplication by the reciprocal of the fraction.

Division of One Fraction by Another

45. Let it be required to divide a/b by c/d; a, b, c, and d representing positive integers.

Applying Principles I. and II., § 34, and employing a course of reasoning similar to that used above, we obtain the result as follows:

We may either separate the fraction a/b into c equal parts, and then take d of these parts as summands, obtaining ad/bc, or we may substitute for the operation of division that of multiplication, using the reciprocal of the divisor as a multiplier, and immediately a d ad

write

a с
= = X =
b d b
с bc

Hence it follows that the quotient obtained by dividing one fraction by another is equal to the product of the dividend and the reciprocal of the divisor.

Since the factors are removed from the numerator by division, not by subtraction, it follows that when we strike out the last factor, whichever that may be, the quotient 1 remains.

46. Mixed numbers appearing in either dividend or divisor should be reduced to fractional form before the division is carried out.

Expressing the dividend as an improper fraction and multiplying by

the reciprocal of the divisor, we have

Express each of the following quotients in simplest form :

EXERCISE XV. 11

Express the following quotients as single fractions in lowest terms, checking all results numerically :

47. Since in algebraic expressions we cannot restrict ourselves to using positive integers only,.we shall find it necessary to admit to our calculations fractions whose numerators and denominators, either or both, contain terms which in themselves may be positive or negative, integral or fractional.

By the Principle of No Exception we shall so extend our ideas concerning fractions that whatever may be the nature of the numbers entering into the terms of any particular fraction, the operands must in all cases be governed by the laws demonstrated for fractions whose terms consist of positive whole numbers only.

48. A complex fraction is a fraction containing one or more fractions among the terms of its numerator and denominator.

E. g. The following expressions are complex fractions:

49. When simplifying a complex fraction it is often convenient to obtain the reduced value of the numerator alone and of the denominator alone before attempting to perform the indicated division.