 # Fractions with Different Denominators

You can also add and subtract fractions containing different variables. For fractions with different denominators that are going to be added or subtracted together, you calculate the common denominator in the normal way.

Example 1

Write $\frac{H}{O}+\frac{LL}{A}$ as a single fraction

The variables just represent numbers. Since $O$ is one of the denominators and $A$ is the other, then $AO$ is the common denominator. You find this by expanding the first fraction by $A$, and the other fraction by $O$:

$\begin{array}{llll}\hfill \frac{H}{O}+\frac{LL}{A}& =\frac{AH}{AO}+\frac{OLL}{AO}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{HA+LLO}{AO}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $\frac{H}{O}+\frac{LL}{A}=\frac{AH}{AO}+\frac{OLL}{AO}=\frac{HA+LLO}{AO}.$

Here you can’t do anything else with the fraction, since $HA$ and $LLO$ are not a equal variable combination. Then they can’t be added together, and the fraction must be left as is.