To add two fractions, $\frac{a}{b}$ and $\frac{c}{d}$ we use this process:

We want the denominator to be the same in each case
The easiest common denominator would be $bd$
We multiply the first fraction by $\frac{d}{d}$ giving $\frac{ad}{bd}$
We multiply the second fraction by $\frac{b}{b}$ giving $\frac{bc}{bd}$
Now we are adding "the same types of things", specifically

The first fraction is $ad$ lots of $\frac{1}{bd}$
The second fraction is $bc$ lots of $\frac{1}{bd}$
So in total we have $(ad+bc)$ lots of $\frac{1}{bd}$

The result is therefore $\frac{ad+bc}{bd}$

It may be the case that this result is not in lowest terms, so we might then continue by reducing to lowest terms. We do this by the process of cancelling fractions until the numerator and denominator have no common factors except 1.

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