Multiplication & Division of Fractions

Multiplication of Fractions:

To multiply fractions, you need to:

1. multiply the denominators and
2. reduce the result to its lowest value, 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. \(\scriptsize 3 \normalsize \frac{5 }{17}\: \times \: \scriptsize 2 \normalsize \frac{5 }{6} \: \times \: \scriptsize 1 \normalsize \frac{4}{8}\)

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

Solution

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

Division of Fractions:

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

It means we write  9  ÷   4  or   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 \( \frac{7 \: \times \: \frac{3}{2}}{\frac{2}{3} \: \times \: \frac{3}{2}} \\ = \frac{7 \: \times \: \frac{3}{2}}{1} \)

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

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

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).

Worked Example 8.2.3:

Solve:

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

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

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

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

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

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