A complex fraction, also called a complex rational expression, is one in which the numerator, denominator, or both are fractions. For example, $\frac {\left( \frac {8}{15}\right) }{\left( \frac {2}{3}\right)}$ and $\frac {3}{1-\frac{2}{5}}$ are complex fractions. When dealing with equations that involve complex fractions, it is useful to simplify the complex fraction before solving the equation.

The process of simplifying complex fractions, known as the "combine-divide method," is as follows:

1. Combine the terms in the numerator.
2. Combine the terms in the denominator.
3. Divide the numerator by the denominator.

Example 1

Let's apply this method to the first complex fraction presented above:

$\displaystyle{\frac {\left( \frac {8}{15}\right) }{\left( \frac {2}{3}\right)}}$

Since there are no terms that can be combined or simplified in either the numerator or denominator, we'll skip to Step 3, dividing the numerator by the denominator:

$\displaystyle{ \frac {\left( \frac {8}{15}\right) }{\left( \frac {2}{3}\right)} = \frac {8}{15} \div \frac {2}{3}}$

From previous sections, we know that dividing by a fraction is the same as multiplying by the reciprocal of that fraction. Therefore, we use the cancellation method to simplify the numbers as much as possible, and then we multiply by the simplified reciprocal of the divisor, or denominator, fraction:

$\displaystyle {{\frac 8{15}}\cdot {\frac 32} = {\frac 4{5}}\cdot{\frac 11} ={\frac 4{5}}}$

Therefore, the complex fraction $\frac {\left( \frac {8}{15}\right) }{\left( \frac {2}{3}\right)}$ simplifies to $\frac {4}{5}$.

Example 2

Let's try another example:

$\displaystyle {\frac {\left(\dfrac{1}{2}+\dfrac{2}{3}\right)}{\left(\dfrac{2}{3}\cdot\dfrac{3}{4}\right)}}$

Start with Step 1 of the combine-divide method above: combine the terms in the numerator. You'll find that the common denominator of the two fractions in the numerator is 6, and then you can add those two terms together to get a single fraction term in the larger fraction's numerator:

 $\displaystyle \frac {\left(\dfrac{1}{2}+\dfrac{2}{3}\right)}{\left(\dfrac{2}{3}\cdot\dfrac{3}{4}\right)} = \dfrac {\left(\dfrac{3}{6}+\dfrac{4}{6}\right)}{\left(\dfrac{2}{3}\cdot\dfrac{3}{4}\right)} = \dfrac {\left(\dfrac{7}{6}\right)}{\left(\dfrac{2}{3}\cdot\dfrac{3}{4}\right)}$

Let's move on to Step 2: combine the terms in the denominator. To do so, we multiply the fractions in the denominator together and simplify the result by reducing it to lowest terms:

$\dfrac {\left(\dfrac{7}{6}\right)}{\left(\dfrac{2}{3}\cdot\dfrac{3}{4}\right)} = \dfrac{\left(\dfrac{7}{6}\right)}{\left(\dfrac{6}{12}\right)} = \dfrac{\left(\dfrac{7}{6}\right)}{\left(\dfrac{1}{2}\right)}$

Let's turn to Step 3: divide the numerator by the denominator. Recall, again, that dividing by a fraction is the same as multiplying by the reciprocal of that fraction:

$\frac{\left(\dfrac{7}{6}\right)}{\left(\dfrac{1}{2}\right)}= {\dfrac{7}{6}} \cdot {\dfrac{2}{1}} = \dfrac{14}{6}$

Finally, simplify the resultant fraction:

$\displaystyle \frac{14}{6}=2\frac{2}{6}$

Therefore, ultimately:

$\frac {\left(\dfrac{1}{2}+\dfrac{2}{3}\right)}{ \left(\dfrac{2}{3}\cdot \dfrac{3}{4}\right)} =2\dfrac{2}{6}$