# Find the Common Denominator by Multiplying Denominators

If you want to find the common denominator, this method will always work. The downside is that you can get unnecessarily large numbers. I say “unnecessary” because you can often find a lower common denominator, and then the calculation becomes much simpler. Regardless, this method is a good place to start.

Rule

### ToMultiplytheDenominatorswithEachOther

You expand the fractions by the other fraction’s denominators, which means that you multiply the numerator and the denominator in each of the fractions by the denominators of the other fractions.

Example 1

Find $\frac{3}{7}-\frac{2}{5}$.

The denominators are different from each other, so you expand each fraction by the denominator of the other fraction. Then you can subtract the numerators from each other. $\begin{array}{llll}\hfill \frac{3}{7}-\frac{2}{5}& =\frac{3\cdot 5}{7\cdot 5}-\frac{2\cdot 7}{5\cdot 7}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{15}{35}-\frac{14}{35}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{1}{35}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$