To multiply fractions, we multiply numerator to numerator and denominator to denominator. In other words, we multiply the top numbers together and the bottom numbers together.

$${a \over b} \times {c \over d} = {{ac} \over {bd}}$$

Whatever the result, we need to reduce it to the lowest terms. It can be done by dividing both the numerator and denominator by the Greatest Common Factor (GCF). We say that a fraction is in its simplest form if the GCF of the numerator and denominator is \(1\).

Example 1: Multiply the fractions below.

$${2 \over 3} \times {1 \over 4}$$

Multiply the top numbers. Then multiply the bottom numbers. Notice that the resulting fraction has a GCF of \(2\). Let's divide the numerator and denominator by \(2\) to reduce it to the simplest form.

$$\eqalign{

{2 \over 3} \times {1 \over 4} &= {2 \over {12}} \cr

&= {{2 \div 2} \over {12 \div 2}} \cr

&= {1 \over 6} \cr} $$

Example 2: Multiply the fractions below.

$${6 \over {21}} \times {3 \over 2}$$

Multiply the top numbers: \(6 \times 3 = 18\)

Multply the bottom numbers: \(21 \times 2 = 42\)

Notice that the numerator \(18\) and the denominator \(42\) have the Greatest Common Factor of \(6\). We will use this to simplify the fraction. Divide both the numerator and denominator by \(6\).

$$\eqalign{

{6 \over {21}} \times {3 \over 2} &= {{18} \over {42}} \cr

&= {{18 \div 6} \over {42 \div 6}} \cr

&= {3 \over 7} \cr} $$