Multiplying pairs of proper fractions

​​In a nutshell

Multiplying proper fractions is simply a case of multiplying the numerators together and then multiplying the denominators together.

Vocabulary reminder

As a reminder, the numerator is the number on the top of a fraction and the denominator is the number on the bottom. For example:

​$$\dfrac58$$​​

The numerator is $$5$$​ and the denominator is $$8$$​.

$$5$$

A proper fraction is one where the numerator is smaller than the denominator.. This means that the fraction's value is less than one. So $$\frac58$$ is an example of a proper fraction.

​​Multiplying proper fractions

Unlike with adding and subtracting, to multiply fractions, the denominators do not need to be the same. 

procedure

Example 1

Calculate

​$$\dfrac58\times\dfrac45$$​​

Start by multiplying the numerators:

$$5\times4=20$$​​

Next multiply the denominators:

$$8\times5=40$$​​

Put these two steps together:

$$\dfrac58\times\dfrac45=\dfrac{20}{40}$$​​

This can be simplified by dividing the numerator and the denominator by $$20$$:

$$\dfrac58\times\dfrac45=\dfrac{20}{40}=\underline{\dfrac12}$$​​

A shortcut?

If you spot any cancellations to be made before you multiply, this can make the calculation easier. Start by making sure both of your fractions are simplified fully. Then check if the numerator of one fraction and the denominator of the other fraction can both be divided by the same number.

Example 2

Calculate 

$$\dfrac58\times\dfrac45$$​​

This time, check for any simplifications first. Notice that the numerator of the first fraction and the denominator of the second fraction can both be divided by $$5$$:

$$\dfrac58\times\dfrac45=\dfrac{\cancel{5}^{\space{1}}}8\times\dfrac4{\cancel{5}^{\space{1}}}=\dfrac18\times\dfrac41$$​​

Also notice that the numerator of the second fraction and the denominator of the first fraction can both be divided by $$4$$:

​$$\dfrac18\times\dfrac41=\dfrac1{\cancel{8}^{\space{2}}}\times\dfrac{\cancel{4}^{\space{1}}}1=\dfrac12\times\dfrac11$$​

Now you have the simple multiplication $$\frac12\times\frac11$$, which gives: 

$$\dfrac12\times\dfrac11=\dfrac{1\times1}{2\times1}=\underline{\dfrac12}$$​​

This is the same answer that was found in the earlier example.

Note: This method of simplifying first will sometimes mean you avoid having to do tricky multiplications.