Multiplication and Division of Algebraic Fractions

(1) To multiply algebraic fractions, cancel common terms before multiplying out 

(2) To divide an algebraic fraction, multiply by the inverse of the term after the division (÷) sign.

Example 4.3.1:

Simplify the following:

(a) \( \frac{8}{x} \: \times \: \frac{y}{3}\)

(b) \( \frac{x}{6} \: \times \: \frac{4}{x}\)

(c) \( \frac{5a}{7b} \: \times \: \frac{5b}{3a}\)

(d) \( \frac{c}{d} \: \div \: \frac{c}{a}\)

(e) \( \frac{9x}{5} \: \div \: \frac{3x}{10}\)

(f) \( \frac{z^3}{6xy} \: \div \: \frac{9}{2xy}\)

Solution

(a) \(\normalsize \frac{8}{x} \: \scriptsize \times \: \normalsize \frac{y}{3} \\ = \normalsize \frac{8 \: \times \: y}{x\: \times \: 3} \\ =\normalsize \frac{8y}{3x}\) 

(b) \(\normalsize \frac{x}{6} \scriptsize \: \times \: \normalsize \frac{4}{x} \\ =\normalsize \frac{x\: \times \:4}{6\: \times \: x}\\ = \normalsize\frac{\not{x}\: \times \:4}{6\: \times \: \not{x}} \\ = \normalsize \frac{4}{6} \\ =\normalsize \frac{2}{3} \)

(c) \( \frac{5a}{7b} \: \times \: \frac{5b}{3a} \\ = \frac{5a \: \times \: 5b}{7b \: \times \: 3a} \\ = \frac{5 \: \times \: 5 \: \times \: \not{a} \: \times \: \not{b}}{7 \: \times \: 3 \: \times \: \not{a} \: \times \: \not{b}} \\ = \frac{5 \: \times \: 5}{7 \: \times \: 3} \\ = \frac{25}{71}\)

(d) \( \frac{c}{d} \: \div \: \frac{c}{a}\\ = \frac{c}{d} \: \times\: \frac{a}{c} \\ = \frac{c\: \times\: a}{d \: \times\: c} \\ = \frac{\not{c}\: \times\: a}{d \: \times\: \not{c}} \\ = \frac{a}{d}\)

(e) \( \frac{9x}{5} \: \div \: \frac{3x}{10} \\ =\frac{9x}{5} \: \times \: \frac{10}{3x} \\ = \frac{9x\: \times\:10}{5\: \times\: 3x} \\ = \frac{9\: \times\:10\: \times\: x}{5\: \times\: 3 \: \times\: x} \\ = \frac{\not{9}^{3}\: \times\:10\: \times\: \not{x}}{5\: \times\: \not{3} \: \times\: \not{x}} \\ = \frac{3 \: \times\: 10}{5} \\ = \scriptsize 3 \: \times\: 2 \\ = \scriptsize 6 \)

(f) \( \frac{z^3}{6xy} \: \div \: \frac{9}{2xy} \\ = \frac{z^3}{6xy} \: \times\: \frac{2xy}{9} \\ =\frac{z^3 \: \times \: 2xy}{6xy\: \times \: 9} \\ = \frac{z^3 \: \times \: 2 \: \times \: x \: \times \: y}{ 6\: \times \: x \: \times \: y \: \times \: 9} \\ = \frac{z^3 \: \times \: \not{2} \: \times \: \not{x} \: \times \: \not{y}}{ \not{6}^{3} \: \times \: \not{x} \: \times \: \not{y} \: \times \: 9} \\ = \frac{z^3}{3 \: \times \: 9} \\ = \frac{z^3}{27} \)