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Dividing by a fraction

If we want to divide 8 by \(\dfrac{1}{2}\), we can say 'How many halves in 8 wholes?'

Since there are 2 halves in every whole, there will be 16 halves in 8.

Thus \(8÷\dfrac{1}{2}=16\). If we multiply by the reciprocal, we obtain the same answer since

\(8÷\dfrac{1}{2}=8×\dfrac{2}{1}=16\)

Similarly, \(6÷\dfrac{2}{3}=6×\dfrac{3}{2}=\dfrac{18}{2}=9\).

In these examples, we turn the fraction upside down and multiply.

The reciprocal of a fraction is the fraction obtained by turning it upside down, which entails swapping the numerator and the denominator.

So to divide a number by a fraction, we multiply by the reciprocal.

Dividing a fraction by a fraction

This same method works if we divide a fraction by a fraction.

For example, \(\dfrac{3}{5}÷\dfrac{2}{3}=\dfrac{3}{5}×\dfrac{3}{2}=\dfrac{9}{10}\).

Hence the following rule holds in all situations.

To divide by a fraction, multiply by the reciprocal.