8.06 Multiply fractions by fractions

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Do you remember how to multiply a fraction by a whole number?

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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 $\frac{2}{3}\times\frac{4}{5}$23​×45​. We can rewrite these fractions as

$\frac{2}{3}=2\times\frac{1}{3}$23​=2×13​ and $\frac{4}{5}=4\times\frac{1}{5}$45​=4×15​

We can then multiply the whole parts together:

$\frac{2}{3}\times\frac{4}{5}$23​×45​	$=$=	$2\times\frac{1}{3}\times4\times\frac{1}{5}$2×13​×4×15​
	$=$=	$8\times\frac{1}{3}\times\frac{1}{5}$8×13​×15​

 

What can we do with the product of the unit fractions $\frac{1}{3}$13​ and $\frac{1}{5}$15​? Well, this is like taking one whole, dividing it into $3$3 pieces to get thirds, and then dividing each of those thirds into $5$5 pieces. The result is that the whole has been divided into $15$15 pieces.

We can now finish our multiplication:

$\frac{2}{3}\times\frac{4}{5}$23​×45​	$=$=	$8\times\frac{1}{3}\times\frac{1}{5}$8×13​×15​
	$=$=	$8\times\frac{1}{15}$8×115​
	$=$=	$\frac{8}{15}$815​

 

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:

$\frac{2}{3}\times\frac{4}{5}$23​×45​	$=$=	$\frac{2\times4}{3\times5}$2×43×5​
	$=$=	$\frac{8}{15}$815​

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