To divide fractions, we will have to rewrite the division problem into a multiplication problem. To do that, we change the operation from division to multiplication and then take the reciprocal of the second fraction. The reciprocal of a fraction is just flipping the fraction 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 first step is to convert the division problem into a multiplication problem. Change the operation from division to multiplication then invert the second fraction.

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

Proceed with regular multiplication. Multiply the numerators together and the denominators together. Reduce to lower terms by dividing the top and bottom with GCF which in this case is \(6\).

$$\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 transform the division problem to multiplication problem by replacing the division to multiplication operation and then take 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} $$