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}.
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.
The following applet explores multiplication of fractions with area models. The intersection of the two area models represents the product.
We can use area models to show a visual representation of the product of two fractions. In area models, the total number of equally partitioned pieces represents the denominator of the product.
Evaluate \dfrac35\times\dfrac47.
Evaluate \dfrac53\times\dfrac{21}{2}.
To multiply two fractions, multiply the numerators and the denominators separately.