Dividing Fractions

To divide fractions, we need to convert the division problem into a multiplication problem. To do this, we change the division operation to multiplication and then take the reciprocal of the second fraction. The reciprocal of a fraction is simply flipping it upside down.

$${a \over b} \div {c \over d} \Rightarrow {a \over b} \times {d \over c} = {{ad} \over {bc}}$$

Example 1: \(\large{{4 \over 9} \div {2 \over 3}}\)

The initial step is to transform the division problem into a multiplication problem. To do this, switch the division operation to multiplication and then invert the second fraction.

$${4 \over 9} \div {2 \over 3} \Rightarrow {4 \over 9} \times {3 \over 2}$$

Perform regular multiplication by multiplying the numerators and denominators together. Simplify the result by dividing both the numerator and the denominator by their greatest common factor (GCF), which is \(6\) in this case.

$$\eqalign{
{4 \over 9} \times {3 \over 2} &= {{12} \over {18}}  \cr 
  &= {{12 \div 6} \over {18 \div 6}}  \cr 
  &= {2 \over 3} \cr} $$

Example 2: \(\large{{3 \over 4} \div {9 \over 8}}\)

Let’s convert the division problem into a multiplication problem by changing the division operation to multiplication and then taking the reciprocal of the second fraction.

$${3 \over 4} \div {9 \over 8} \Rightarrow {3 \over 4} \times {8 \over 9}$$

Proceed with regular multiplication. The GCF here is \(12\).

$$\eqalign{
   {3 \over 4} \times {8 \over 9} &= {{24} \over {36}}  \cr 
  &= {{24 \div 12} \over {36 \div 12}}  \cr 
  &= {2 \over 3} \cr} $$