2.01 Multiply fractions

Fractions describe parts of a whole, but they can also describe parts of a quantity.

Find \dfrac{1}{12} of 36.

Let's start by drawing a grid of 36.

To find one twelfth, we split this grid into 12 equal parts.

Looking at the pieces, each piece has 3 squares. So \dfrac{1}{12} of 36 is 3.

We can also work this out using arithmetic. We know that \dfrac{1}{12} of 36 can be written using multiplication, \dfrac{1}{12} \times 36.

This is the same as \dfrac{1}{12} \times \dfrac{36}{1} because the whole number 36 can be written as a fraction over 1.

First, if we evaluate the multiplication of the numerators we get 36 . And if we evaluate the multiplication in of the denominators we get 12.

Next we can simplify the fraction by factoring out the greatest common factor, which is 12. This gives us \dfrac{3}{1} which is the same as 3.

We can check this answer by multiplying back. 12 \times 3 = 36, so we know that 3 is \dfrac{1}{12} of 36.

Examples

To multiply two fractions together, we'll start by thinking of the fractions as multiples of unit fractions, and work towards a more efficient strategy.

Let's take an example of \,\dfrac23 \times \dfrac45 . We can rewrite these fractions as

\dfrac23 = 2 \times \dfrac13 \quad\text{and}\quad \dfrac45 = 4 \times \dfrac15

We can then multiply the whole parts together:

\begin{aligned} \dfrac23 \times \dfrac45 &=2 \times \dfrac13 \times 4 \times \dfrac15\\\\ &=8 \times \dfrac13 \times \dfrac15 \end{aligned}

What can we do with the product of the unit fractions \dfrac13 and \dfrac15?

Well, this is like taking one whole, dividing it into 3 pieces to get thirds.

Then dividing each of those thirds into 5 pieces.

The result is that the whole has been divided into 15 pieces where we only want 1 piece.

This image represents the fraction \dfrac{1}{15}.

We can now finish our multiplication:

\begin{aligned} \dfrac23 \times \dfrac45 &=8 \times \dfrac13 \times \dfrac15\\\\ &=8 \times \dfrac{1}{15}\\\\ &=\dfrac{8}{15} \end{aligned}

Do you notice the pattern that has happened here?

In a fraction, the denominator tells us the size of the pieces, and the numerator tells us how many pieces there are. When we multiply two fractions, the denominators multiply together to tell us the new size of the pieces, and the numerators also multiply together to tell us how many of the new pieces there are.

That is:

\begin{aligned} \dfrac23 \times \dfrac45 &=\dfrac{2\times4}{3\times5}\\\\ &=\dfrac{8}{15} \end{aligned}

Examples