Fractions with Different Denominators

It is possible to 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 same way as you do with fractions that contain only numbers.

Example 1

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

Variables just represent numbers. Since $O$ is one denominator and $A$ is the other one, then $AO$ is the common denominator between the two fractions. You find this by expanding the first fraction by $A$, and the second 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 an equal variable combination. Then they can’t be added together, so the fraction must be left as-is.