### Fraction of a quantity

You might have noticed in the example on a page one that:

\begin{align}\frac{2}{3} × \frac{18}{1} &= \frac{2 × 18}{3 × 1} \\&= \frac{36}{3} \\&= 12\end{align}You can see that multiplying the numerators and multiplying the denominators gives the same result. This is a shortcut to drawing and circling and is used as the written calculation for multiplication of fractions.

Once students are proficient at drawing a collection and circling the fraction 'of ' it, they can take a shortcut and write the multiplication.

To find five-sixths of twelve we write:

\begin{align}\frac{5}{6} × \frac{12}{1} &= \frac{5 × 12}{6 × 1} \text{(we write} \frac{12}{1} \text{to show that we are multiplying by 12 ones)} \\&= \frac{60}{6} \\&= 10\end{align}