Quick Tip: Complex Fractions

A lot of times math problems look a lot more difficult than they are. Take complex fractions:

\(\frac{\frac{2}{3}}{\frac{4}{5}}\)

At first glance, this problem looks very intimidating, and there isn’t a student who doesn’t panic a bit seeing those for the first time. At a second glance, the complex fraction can be simplified with methods that 5th graders learn: You find an equivalent fraction that is simpler.

To find equivalent fractions we need to multiply both the denominator (i.e. the bottom) and the numerator (i.e. the top) by the same number. E.g. \(\frac{1}{2}\) is equivalent to \(\frac{2}{4}\) and \(\frac{3}{6}\). The trick is to multiply by the right number: To simplify complex fractions, we multiply the numerator and denominator by the denominator's reciprocal (i.e. the flipped fraction).

\(\frac{\frac{2}{3}\cdot \frac{5}{4}}{\frac{4}{5}\cdot \frac{5}{4}}\)

When a fraction is multiplied by its reciprocal, the product is 1 (try it out). And there we go: the denominator is now 1 and can be ignored.

\(\frac{\frac{2}{3}\cdot \frac{5}{4}}{\frac{4}{5}\cdot \frac{5}{4}}=\frac{\frac{2}{3}\cdot \frac{5}{4}}{1}=\frac{2}{3}\cdot \frac{5}{4}\)

All that is left is the numerator which is a product of two fractions. In the last step, we multiply across and find

\(\frac{2}{3}\cdot \frac{5}{4}=\frac{10}{12}=\frac{5}{6}\)

In short: To simplify complex fractions, find an equivalent fraction with a denominator of 1. And don't let the intimidating looks of a problem deter you.