Lesson 4, Topic 3
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# Multiplication and Division of Algebraic Fractions

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(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

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) $$\frac{8}{x} \: \times \: \frac{y}{3} \\ = \frac{8 \: \times \: y}{x\: \times \: 3} \\ = \frac{8y}{3x}$$

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

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

Evaluation

Simplify the following

(a) $$\frac{g}{7} \: \times \: \frac{14}{g^2}$$

(b) $$\frac{3}{x} \: \div \: \frac{6}{2x}$$

(c) $$\frac{x}{y} \: \div \: \frac{x}{y}$$

(d) $$\frac{t^2}{5a} \: \times \: \frac{3a}{t}$$

(e) $$\frac{4x}{5y} \: \times \: \frac{4y}{7}$$

(f) $$\frac{6x^2}{11y} \: \div \: \frac{18x}{33y^2}$$

(a) $$\frac{2}{g}$$

(b) 1

(c) 1

(d) $$\frac{3t}{5}$$

(e) $$\frac{16x}{35}$$

(f) $$\scriptsize xy$$

error: