 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

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