Multiplication & Division of Fractions

Topic Content:

Multiplication of Fractions:

To multiply fractions, you need to:

1. multiply the numerators,
2. multiply the denominators, and
3. reduce the result to its lowest terms, if necessary.

Worked Example 8.2.1:

Solve:

\(\frac{3 }{4}\: \times \: \frac{5 }{6} \)

Solution

\(\frac{3 }{4}\: \times \: \frac{5 }{6} \\ = \frac{3 \: \times \: 5}{4 \: \times \: 6} \\ = \frac{15}{24} \\ = \frac{5}{8}\)

Worked Example 8.2.2:

Solve:

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

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

Solution

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. \(\scriptsize 3 \frac{5 }{17}\: \times \: 2 \frac{5 }{6} \: \times \: 1 \frac{4}{8}\)

Solution

= \(\frac{51 \: + \: 5 }{17}\: \times \: \frac{12\:+\:5}{6} \: \times \: \frac{8\:+\:4}{8}\\ = \frac{56 }{17}\: \times \: \frac{17}{6} \: \times \: \frac{12}{8} \\ = \frac{56 \times 17 \times 12}{17 \times 6 \times 8}\\= \scriptsize 7 \: \times \: 2 \\ = \scriptsize 14 \)

Division of Fractions:

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

It means we write  9  ÷   4  or   \( \frac{9}{4} \)

Similarly, 7 divided by \( \frac{2}{3} \) means  7   ÷  \( \frac{2}{3} \)

Or  \( \frac{7}{\frac{2}{3}} \)

To make the denominator equal to 1, multiply both the numerator and denominator by \( \frac{3}{2} \)

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

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

Therefore, \(\scriptsize 7 \: \div \: \normalsize \frac{2}{3} \\ \scriptsize = 7 \: \times \: \normalsize \frac{3}{2} \)

Notice that the division sign (÷) changes to multiplication (×) and \( \frac{2}{3} \) is inverted (i.e. turned upside down) to \( \frac{3}{2} \)

Thus, \(\frac{3}{2}\) is the inverse (or reciprocal) of \(\frac{2}{3}\).

Likewise, \(\frac{2}{3}\) is the inverse (or reciprocal) of \(\frac{3}{2}\).

In other words, reciprocal fractions come in pairs. When one fraction is turned upside down, it becomes the reciprocal of the other.

Worked Example 8.2.3:

Solve:

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

ii. \(\scriptsize 3 \frac{2 }{3}\: \div \: 4 \frac{3}{9} \)

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

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. \(\scriptsize 3 \frac{2 }{3}\: \div \: 4\frac{3}{9} \)

Solution

First, change these mixed fractions to improper fractions

\( \scriptsize 3\frac{2}{3} = \normalsize \frac{11}{3}\)

\(\scriptsize 4\frac{3}{9} = 4\frac{1}{3} = \normalsize \frac{13}{3}\)

\(\frac{11}{3} \div \frac{13}{3} = \frac{11}{3} \times \frac{3}{13} \)

\( \frac{11}{\not{3}} \times \frac{\not{3}}{13} \\ = \frac{11}{13}\)

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

Solution

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

\(\frac{15}{20} \scriptsize \: \div \: 5 \: \text{means} \: \normalsize \frac{15}{20}\scriptsize \: \div \: \normalsize \frac{5}{1}\)

Thus, \(\frac{15}{20} \scriptsize \: \div \: \normalsize \frac{5}{1} \\ = \frac{15}{20} \scriptsize \: \times \: \normalsize \frac{1}{5} \\ = \frac{3 \: \times \: 1}{20 \: \times \: 1}\\ = \frac{3}{20}\)

Final Evaluation: