Adding and Multiplying Algebraic Fractions

The rules you already know

Algebraic fractions add and multiply by the same rules as number fractions. To add, rewrite both fractions over a common denominator, then add the numerators. To multiply, multiply the numerators together and the denominators together.

Take $\frac{x}{2} + \frac{x}{3}$. Halves and thirds are pieces of different sizes, so their numerators cannot be added as they stand. Both fit into sixths, because 6 is a multiple of 2 and of 3.

Let each whole bar stand for x. Cut the half and the third into sixths: half of x is 3 sixths and a third of x is 2 sixths, so together they are 5 sixths of x.

Over a common denominator

Rewrite each fraction over 6 by multiplying its numerator and its denominator by the same number. $\frac{x}{2} = \frac{3x}{6}$, multiplying both by 3, and $\frac{x}{3} = \frac{2x}{6}$, multiplying both by 2. Now the pieces are the same size, and the numerators add: $\frac{3x}{6} + \frac{2x}{6} = \frac{5x}{6}$.

Subtraction works the same way: $\frac{x}{2} - \frac{x}{3} = \frac{3x}{6} - \frac{2x}{6} = \frac{x}{6}$.

Adding the denominators is the usual slip

Adding the numerators and the denominators, $\frac{x}{2} + \frac{x}{3} = \frac{2x}{5}$, is the most common mistake. A denominator is the size of the pieces, and sizes do not add. No fifths appear anywhere: the pieces are sixths.

Substitution catches it. Put x = 6: $\frac{x}{2} + \frac{x}{3} = 3 + 2 = 5$, and $\frac{5x}{6} = 5$ as well. But $\frac{2x}{5} = \frac{12}{5} = 2.4$, which is less than $\frac{x}{2} = 3$ on its own.

Multiplying goes straight across

Multiplying needs no common denominator. Multiply the numerators, then the denominators: $\left(\frac{x}{2}\right) \times \left(\frac{x}{3}\right) = \frac{{x \times x}}{{2 \times 3}} = \frac{x^{2}}{6}$.

A square shows why. Take a square of side x, whose area is $x^{2}$. Cut its top side into halves and its left side into thirds. The square is now cut into 2 × 3 = 6 equal pieces, and one piece is $\frac{x}{2}$ wide and $\frac{x}{3}$ tall. Its area is $\left(\frac{x}{2}\right) \times \left(\frac{x}{3}\right)$, and it is one sixth of $x^{2}$, so $\left(\frac{x}{2}\right) \times \left(\frac{x}{3}\right) = \frac{x^{2}}{6}$.

A square of side x cut into halves across and thirds down. One piece is $\frac{x}{2}$ by $\frac{x}{3}$, and it is one of 6 equal pieces of $x^{2}$, so its area is $\frac{x^{2}}{6}$.

Letters in the denominators

A denominator can contain a letter too. The common denominator is still a multiple of both denominators, and when they share no factor, their product will do.

For $\frac{1}{x} + \frac{1}{2}$ the common denominator is 2 × x = 2x. Multiply the numerator and the denominator of each fraction by the factor its denominator is missing: $\frac{1}{x} = \frac{2}{{2x}}$ and $\frac{1}{2} = \frac{x}{{2x}}$. Then $\frac{1}{x} + \frac{1}{2} = \frac{{2 + x}}{{2x}}$.

Check with x = 4: $\frac{1}{4} + \frac{1}{2} = \frac{3}{4}$, and $\frac{{2 + 4}}{{2 \times 4}} = \frac{6}{8} = \frac{3}{4}$.