Multiplying Two Fractions

We cut a pie into four equal parts, then we split each of these parts in two.
What fraction of the pie does each of these small parts represent, and how does this example allow us to illustrate the multiplication of two numbers that are written as fractions?

I. Starting example

We cut a pie into four equal parts.

Colored fraction of the pie: $\frac{1}{4}$.


Next we split each of these parts in two.

Colored fraction of the pie: half of  $\frac{1}{4}$; therefore $\frac{1}{2}\times \frac{1}{4}$.


What fraction of the pie does each of these small parts represent?

Answer: $\frac{1}{8}$. From this we can deduce that $\frac{1}{2}\times \frac{1}{4}=\frac{1}{8}$.

II. Rules of calculation

A. General rule

To multiply two numbers that are written as fractions, we multiply the numerators together and the denominators together:


$\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}$, with b ≠  0 and d ≠  0.


Examples:

$\frac{2}{3}\times \frac{4}{5}=\frac{2\times 4}{3\times 5}=\frac{8}{15}$

$\frac{1.6}{3}\times \frac{2}{0.41}=\frac{1.6\times 2}{3\times 0.41}=\frac{3.2}{1.23}$

B. Particular case

If one of the factors is not written as a fraction, we can apply the following rule: $a\times \frac{c}{d}=\frac{a\times c}{d}$, with d ≠ 0. (We can find this rule by thinking that $a=\frac{a}{1}$ and then applying the rule above.)


Example: $3\times \frac{5}{7}=\frac{3\times 5}{7}$

C. Generalization

The rule stated above can be generalized for more than two factors.


Example: $\frac{1}{2}\times \frac{{\displaystyle 3}}{5}\times \frac{7}{11}=\frac{1\times 3\times 5}{2\times 5\times 11}=\frac{21}{110}$

D. Simplification

Before working out the products of the numerators and denominators it can help to simplify.


Examples: $\frac{44}{27}\times \frac{27}{5}=\frac{44\times 27}{27\times 5}=\frac{44}{5}$ (we have simplified by canceling the common factor 27)

III. Example of Multiplying Two Fractions

In a class of 30 students, three-fifths of the students are girls, and five-sixths of the girls study French. What fraction of the students in the class is represented by girls who study French?


1st method:


$\frac{3}{5}\times 30=\frac{3\times 30}{5}=18$; there are 18 girls in the class.


$\frac{5}{6}\times 18=\frac{5\times 18}{6}=15$; there are 15 girls who study French.


$\frac{15}{30}=\frac{1}{2}$; the girls who study French therefore represent half of the class.


2nd method:


$\frac{3}{5}\times \frac{5}{6}=\frac{3\times 5}{5\times 6}=\frac{6}{3}=\frac{1}{2}$; the girls who study French represent half of the class.


This second method is quicker than the first; also, it can be used even when we do not know how many students there are in the class.