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