Adding Fractions with Unlike Denominators

Make the pieces match first

Add $\frac{1}{2}$ and $\frac{1}{3}$. A half and a third are pieces of different sizes, so they cannot be counted together yet: 1 piece and 1 piece make 2 pieces, but 2 of what? First rewrite both fractions with a common denominator.

The lowest common multiple (LCM) of 2 and 3 is 6, so use sixths. Cut every half into 3 pieces and every third into 2 pieces: $\frac{1}{2} = \frac{3}{6}$ and $\frac{1}{3} = \frac{2}{6}$. Nothing colored has changed size; it is only cut into smaller pieces.

The bars open as a half and a third, and their pieces do not match. Cut both into sixths, and the colored parts are $\frac{3}{6}$ and $\frac{2}{6}$.

Then count

Now every piece is a sixth, so count the colored pieces: 3 + 2 = 5. That makes $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$. Add the numerators and keep the denominator, as with any fractions whose pieces match.

5 of the 6 sixths are colored: $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$.

Three steps every time

Every sum of fractions with unlike denominators goes the same way: find a common denominator, rewrite both fractions with it, then add the numerators. For $\frac{1}{3} + \frac{1}{4}$, the LCM of 3 and 4 is 12. $\frac{1}{3} = \frac{4}{12}$ and $\frac{1}{4} = \frac{3}{12}$, so $\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$.

The bars open as $\frac{1}{3}$ and $\frac{1}{4}$. Cut both into twelfths: $\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$.

The smallest common denominator

When one denominator is in the other one's times table, only one fraction needs rewriting. 10 is in the 5 times table, so for $\frac{2}{5} + \frac{3}{10}$, write $\frac{2}{5} = \frac{4}{10}$. Then $\frac{4}{10} + \frac{3}{10} = \frac{7}{10}$.

Multiplying the two denominators always gives a common denominator, but not always the smallest. For $\frac{1}{6} + \frac{1}{4}$, 6 × 4 = 24 works, but the LCM of 6 and 4 is 12. In twelfths, $\frac{1}{6} = \frac{2}{12}$ and $\frac{1}{4} = \frac{3}{12}$, so the sum is $\frac{5}{12}$. In 24ths it comes out as $\frac{10}{24}$, which simplifies to $\frac{5}{12}$ anyway: the smaller denominator saves the simplifying.

Never add the denominators

The usual mistake is to add the numerators and add the denominators: $\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$. That cannot be right. $\frac{2}{5}$ is less than $\frac{1}{2}$, and $\frac{1}{2} + \frac{1}{3}$ must be more than the $\frac{1}{2}$ it starts from.

$\frac{2}{5}$ falls short of the half line, so it cannot be $\frac{1}{2} + \frac{1}{3}$. The sum, $\frac{5}{6}$, is well past it.