Algebraic fractions add and subtract by exactly the same rule as numeric fractions: you can only combine them once they sit over a common denominator. Get the bottoms to match, then add (or subtract) the tops and keep the denominator.

Take \frac{x}{3} + \frac{x}{4}. The denominators differ, so rewrite each as an equivalent fraction over 12 (the lowest common denominator):

Once the pieces are the same size, the tops just combine: 4x + 3x = 7x.

The rule

For any two algebraic fractions, the safe common denominator is always the product of the two denominators:

Cross-multiply each numerator by the other denominator, add the results, and put the lot over pq. Subtraction is identical — just carry the minus sign into the top:

The denominator xy is the size of the pieces, so it rides along unchanged; only the numerators are added.

See it built

Watch \frac{x}{3} + \frac{x}{4} get pulled onto a common denominator of 12, so the numerators can finally be added. Step through it.

Simplify the result

After combining, always check whether the answer can be reduced — cancel any common factor of the numerator and denominator, just as in simplifying algebraic fractions. For example \frac{2x}{6} + \frac{x}{6} = \frac{3x}{6} = \frac{x}{2}.

See it explained

Sal Khan adds two rational expressions whose denominators differ, by first rewriting both over a common denominator.