Multiplying Fractions

To multiply fractions, you simply multiply the numerators together and then multiply the denominators together. In other words, you multiply the top numbers with each other and the bottom numbers with each other.

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

No matter what the result is, we need to simplify it to its lowest terms. This can be achieved by dividing both the numerator and the denominator by their Greatest Common Factor (GCF). A fraction is considered to be in its simplest form when the GCF of the numerator and denominator is \(1\).

Example 1: Multiply the fractions below.

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

First, multiply the numerators together, then multiply the denominators. You’ll see that the resulting fraction has a greatest common factor (GCF) of \(2\). To simplify the fraction, divide both the numerator and the denominator by \(2\).

$$\eqalign{
 {2 \over 3} \times {1 \over 4} &= {2 \over {12}}  \cr 
   &= {{2 \div 2} \over {12 \div 2}}  \cr 
  &= {1 \over 6} \cr} $$

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: \(21 \times 2 = 42\)

Observe that both the numerator \(18\) and the denominator \(42\) share a Greatest Common Factor of \(6\). To simplify the fraction, divide both the numerator and the denominator by \(6\).

$$\eqalign{
 {6 \over {21}} \times {3 \over 2} &= {{18} \over {42}}  \cr 
    &= {{18 \div 6} \over {42 \div 6}}  \cr 
  &= {3 \over 7} \cr} $$