A fraction multiplied by a whole number

When we say 'one half of 10', we mean that we divide 10 into two equal parts and take one part. Thus

one half of 10 is 5 or \(\dfrac{1}{2}\) of 10 = 5

In the arithmetic of whole numbers, we know that 3 × 5 = 5 × 3. This is called the commutative law of multiplication.

Note here that \(\dfrac{1}{2}\) of 10 produces the same result as 10 lots of \(\dfrac{1}{2}\).

Similarly, \(\dfrac{1}{4}\) of 12 is 3, and this is the same answer as 12 lots of \(\dfrac{1}{4}\), which is the same as 12 × \(\dfrac{1}{4}\).

Thus \(\dfrac{1}{4}\) of 12, 12 lots of \(\dfrac{1}{4}\), 12 × \(\dfrac{1}{4}\) and \(\dfrac{1}{4}\) × 12 are all equal to 3.

Cancelling

Consider the following example.

Example 2

Find \(\dfrac{2}{3}\) of 18.

Solution

\begin{align}\dfrac{2}{3}\ \text{of}\ 18&=\dfrac{2}{3}×18\\\\ &=\dfrac{36}{3}\\\\ &=12\end{align} or \begin{align}\dfrac{1}{3}\ \text{of}\ 18&=\dfrac{1}{3}×18\\\\ &=6\end{align} so \(\dfrac{2}{3}\ \text{of}\ 18=12\)

This example shows that we could have divided 18 by 3 to find one-third and then doubled the result. This is called cancelling and it is a very convenient approach. We can set it out as

\(\dfrac{2}{\color{darkred}{\setminus}\hspace{-3mm}{3}}×\color{darkred}{\setminus}\hspace{-4mm}{18^6} = 12\)