Lesson 8, Topic 2
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# Multiplication & Division of Fractions

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### Multiplication of Fractions:

To multiply fractions, you need to:

1. multiply the denominators and

2. reduce the result to its lowest value, if necessary.

Example 1

$$\frac{3 }{4}\: \times \: \frac{5 }{6} \\ = \frac{3 \: \times \: 5}{4 \: \times \: 6} \\ = \frac{15}{24}$$

= $$\frac{5}{8}$$

Example 2

i. $$\frac{2 }{7}\: \times \: \frac{4}{5}$$

Solution

$$\frac{2 }{7}\: \times \: \frac{4 }{5}\\ = \frac{2 \: \times \: 4}{7 \: \times \: 5}$$

= $$\frac{8}{35}$$

ii. $$\scriptsize 3 \normalsize \frac{5 }{17}\: \times \: \scriptsize 2 \normalsize \frac{5 }{6} \: \times \: \scriptsize 1 \normalsize \frac{4}{8}$$

Solution

= $$\frac{56 }{17}\: \times \: \frac{17}{6} \: \times \: \frac{12}{8} \\ = \scriptsize 7 \: \times \: 2 \\ = \scriptsize 14$$

### Division of Fractions:

In whole numbers, if we say 9 divided by 4.

It means or we write  9  ÷   4  or   9/4

Similarly 7 divided by  2/3 means   7   ÷  2/3

Or  $$\frac{7}{\frac{2}{3}}$$

To make the denominator equal to 1, multiply both the numerator and denominator by  3/2 .

i.e $$\frac{7 \: \times \: \frac{3}{2}}{\frac{2}{3} \: \times \: \frac{3}{2}} \\ = \frac{7 \: \times \: \frac{3}{2}}{1}$$

= $$\scriptsize 7 \: \times \: \normalsize \frac{3}{2}$$

Therfore, $$\scriptsize = 7 \: \div \: \normalsize \frac{2}{3} \\ \scriptsize = 7 \: \times \: \normalsize \frac{3}{2}$$

Notice that the sign  $$\scriptsize \div$$ (division) changes to $$\scriptsize \times$$  (multiplication) and 2/3 is inverted (i.e. turned upside down)  to  3/2.

Thus, 3/2  is called the inverse or the reciprocal of 2/3.

Hence, to divide by a fraction, simply multiply by its inverse (or reciprocal).

Example 3

i. $$\frac{3}{7}\: \div \: \frac{2}{5}$$

Solution

$$\frac{3}{7}\: \div \: \frac{2}{5} \\ = \frac{3}{7}\: \times \: \frac{5}{2}$$

Did you notice that the sign changed to $$\scriptsize \times$$ and $$\frac{2}{5}$$ changed to $$\frac{5}{2}$$

We then Multiply

$$\frac{3 \: \times \: 5}{7\: \times \: 2}\\ = \frac{15}{14}$$

=$$\scriptsize 1 \normalsize \frac{1 }{14}$$

ii. $$\scriptsize 3 \normalsize \frac{2 }{3}\: \div \: \scriptsize 4 \normalsize \frac{3}{9}$$

Solution:

First, change these mixed fractions to importer fractions.

$$\frac{11}{3}\: \div \: \frac{39}{3}$$

$$\frac{11}{3}\: \times \: \frac{3}{39} = \frac{11}{39}$$

iii. $$\frac{15}{20} \scriptsize \: \div \: 5$$

Solution:

This example illustrates how to divide a fraction by a whole number

$$\frac{15}{20} \scriptsize \: \div \: 5 \: means \: \normalsize \frac{15}{20}\scriptsize \: \div \: \normalsize \frac{5}{1}$$

This, $$\frac{15}{20} \scriptsize \: \div \: \normalsize \frac{5}{1} \\ = \frac{15}{20} \scriptsize \: \times \: \normalsize \frac{1}{5} \\ = \frac{3 \: \times \: 1}{20 \: \times \: 1}\\ = \frac{3}{20}$$

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