2.05 Multiplying fractions

We've seen how to multiply whole numbers by fractions. Can we use the same techniques to multiply fractions by fractions?

Evaluate \dfrac{2}{3} \times \dfrac{4}{5}.

Finding \dfrac{2}{3} \times \dfrac{4}{5} is the same as finding \dfrac{2}{3} of \dfrac{4}{5}.

We can find this by starting with a diagram of \dfrac{4}{5}.

Then we can split each of these fifths into thirds.

Notice that the circle is now divided into fifteenths (that is, 3 \times 5) and twelve parts have been shaded (3 \times 4).

Now we can shade in two thirds of each of the original pieces.

And we finish with eight fifteenths. So \dfrac{2}{3} \times \dfrac{4}{5} = \dfrac{8}{15}.

Each of these steps we've done before. We can think of \dfrac{2}{3} of \dfrac{4}{5} as \dfrac{2}{3} of \dfrac{12}{15} (since \dfrac{4}{5} and \dfrac{12}{15} are

equivalent fractions). Since \dfrac{12}{15} is 12 fifteenths we then want to find \dfrac{2}{3} of 12, and this is the number of fifteenths we are left with.

This suggests another method for multiplying fractions. By equivalent fractions, \dfrac{2}{3} \times \dfrac{4}{5} = \dfrac{2}{3} \times \dfrac{4 \times 3}{5 \times 3}.

Since this is \dfrac {2}{3} \times 4 \times 3 fifteenths, we are multiplying a fraction by a whole number, so we can write

\dfrac {2}{3} \times 4 \times 3 = \dfrac{2 \times 4 \times 3}{3}.

If we cancel the common factor of 3, we get 2 \times 4 fifteenths which is \dfrac{8}{15}.

So \dfrac{2}{3} \times \dfrac{4}{5} = \dfrac{2 \times 4}{3 \times 5}.

We can generalize this method to any fractions. So whenever we want to multiply two fractions, we can multiply the numerators and the denominators separately. Sometimes we might have to simplify the resulting fraction afterwards.

Let's use this method from now on.

Examples