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?
Exploration
Evaluate $\frac{2}{3}\times\frac{4}{5}$23×45.
Finding $\frac{2}{3}\times\frac{4}{5}$23×45 is the same as finding $\frac{2}{3}$23 of $\frac{4}{5}$45. We can find this by starting with a diagram of $\frac{4}{5}$45:
Then we can split each of these fifths into thirds:
Notice that the circles is now divided into fifteenths (that is, $3\times5$3×5) and twelve parts have been shaded ($3\times4$3×4).
Now we can shade in two thirds of each of the original pieces:
And we finish with eight fifteenths. So $\frac{2}{3}\times\frac{4}{5}=\frac{8}{15}$23×45=815.
Each of these steps we've done before. We can think of $\frac{2}{3}$23 of $\frac{4}{5}$45 as $\frac{2}{3}$23 of $\frac{12}{15}$1215 (since $\frac{4}{5}$45 and $\frac{12}{15}$1215 are equivalent fractions). Since $\frac{12}{15}$1215 is $12$12 fifteenths we then want to find $\frac{2}{3}$23 of $12$12, and this is the number of fifteenths we are left with.
This suggests another method for multiplying fractions. By equivalent fractions, $\frac{2}{3}\times\frac{4}{5}=\frac{2}{3}\times\frac{4\times3}{5\times3}$23×45=23×4×35×3.
Since this is $\frac{2}{3}\times4\times3$23×4×3 fifteenths, we are multiplying a fraction by a whole number, so we can write $\frac{2}{3}\times4\times3=\frac{2\times4\times3}{3}$23×4×3=2×4×33.
If we cancel the common factor of $3$3, we get $2\times4$2×4 fifteenths which is $\frac{8}{15}$815.
So $\frac{2}{3}\times\frac{4}{5}=\frac{2\times4}{3\times5}$23×45=2×43×5.
We can generalise 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.
Worked examples
Example 1
Evaluate $\frac{6}{11}\times\frac{5}{9}$611×59.
Think: First we can write this as one fraction by multiplying the numerators and the denominators separately.
Do: Writing the multiplication as one fraction gives us $\frac{6\times5}{11\times9}$6×511×9.
Evaluating the numerator gives $6\times5=30$6×5=30 and evaluating the denominator gives $11\times9=99$11×9=99.
So $\frac{6}{11}\times\frac{5}{9}=\frac{30}{99}$611×59=3099. Since $3$3 is a common factor of $30$30 and $99$99 we can also simplify the fraction further. Cancelling $3$3 gives $\frac{11}{30}$1130.
Reflect: We could also cancel the common factors a step earlier. Sometimes this makes the multiplication easier to evaluate.
Example 2
Evaluate $\frac{3}{7}\times\frac{7}{5}$37×75.
Think: First we can write this as one fraction by multiplying the numerators and the denominators separately. Instead of evaluating the multiplications, we can look for common factors to cancel first.
Do: Writing the multiplication as one fraction gives us $\frac{3\times7}{7\times5}$3×77×5.
We can see that $7$7 is a factor in both the numerator and denominator. Cancelling this gives us $\frac{3}{5}$35.
So $\frac{3}{7}\times\frac{7}{5}=\frac{3}{7}$37×75=37.
Reflect: In this case, it was definitely quicker to cancel the common factors first, so it is worth checking for common factors.
To multiply two fractions, multiply the numerators and the denominators separately.
Practice questions
Question 1
Evaluate $\frac{3}{5}\times\frac{4}{7}$35×47.
Question 2
Evaluate $\frac{5}{3}\times\frac{21}{2}$53×212.