# How do you add and subtract fractions with unlike denominators?

Oct 22, 2016

Multiply by $1$ to get like denominators.

#### Explanation:

If we are trying to add $\frac{1}{2} + \frac{2}{4}$, we can change $\frac{2}{4}$ into its reduced form:

$\frac{2}{4} = \frac{1 \times 2}{2 \times 2} = \frac{1}{2} \times \frac{2}{2} = \frac{1}{2} \times 1 = \frac{1}{2}$

so $\frac{1}{2} + \frac{2}{4} = \frac{1}{2} + \frac{1}{2} = 1$

Because the numerator and denominator of $\frac{2}{4}$ have a common factor of $2$, we could treat it as multiplying by $1$ and cancel them. We can also do this in reverse to add any fractions we want.

Suppose we want to add $\frac{2}{9} + \frac{1}{6}$. We can't reduce here, but we can multiply by $1$ to give us a common denominator:

$\frac{2}{9} \times 1 = \frac{2}{9} \times \frac{2}{2} = \frac{2 \times 2}{9 \times 2} = \frac{4}{18}$

$\frac{1}{6} \times 1 = \frac{1}{6} \times \frac{3}{3} = \frac{1 \times 3}{6 \times 3} = \frac{3}{18}$

so $\frac{2}{9} + \frac{1}{6} = \frac{4}{18} + \frac{3}{18} = \frac{7}{18}$

While you can always multiply each fraction by a $1$ formed from the other exponent, notice that we didn't multiply $\frac{2}{9}$ by $\frac{6}{6}$ or $\frac{1}{6}$ by $\frac{9}{9}$. This is because we knew that the least common multiple (LCM) of $6$ and $9$ is $18$, and so we only multiplied by what was necessary to get that. In general, it's easier to multiply to the LCM, as we work with the lowest necessary values.

Even in a case where you can't find the LCM (or there isn't one), this technique still works. For example, even when we have variables or irrational numbers in the mix:

$\frac{2}{\pi} + \frac{3}{x} = \left(\frac{2}{\pi} \times \frac{x}{x}\right) + \left(\frac{3}{x} \times \frac{\pi}{\pi}\right)$

$= \frac{2 x}{\pi x} + \frac{3 \pi}{\pi x}$

$= \frac{2 x + 3 \pi}{\pi x}$