Adding Fractions with Unlike Denominators

Make the pieces match first

Add $\frac{1}{2}$1/2 and $\frac{1}{3}$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}$1/2 = 3/6 and $\frac{1}{3}=\frac{2}{6}$1/3 = 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}$3/6 and $\frac{2}{6}$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}$1/2 + 1/3 = 3/6 + 2/6 = 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}$1/2 + 1/3 = 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}$1/3 + 1/4, the LCM of 3 and 4 is 12. $\frac{1}{3}=\frac{4}{12}$1/3 = 4/12 and $\frac{1}{4}=\frac{3}{12}$1/4 = 3/12, so $\frac{1}{3}+\frac{1}{4}=\frac{4}{12}+\frac{3}{12}=\frac{7}{12}$1/3 + 1/4 = 4/12 + 3/12 = 7/12.

The bars open as $\frac{1}{3}$1/3 and $\frac{1}{4}$1/4. Cut both into twelfths: $\frac{4}{12}+\frac{3}{12}=\frac{7}{12}$4/12 + 3/12 = 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}$2/5 + 3/10, write $\frac{2}{5}=\frac{4}{10}$2/5 = 4/10. Then $\frac{4}{10}+\frac{3}{10}=\frac{7}{10}$4/10 + 3/10 = 7/10.

Multiplying the two denominators always gives a common denominator, but not always the smallest. For $\frac{1}{6}+\frac{1}{4}$1/6 + 1/4, 6 × 4 = 24 works, but the LCM of 6 and 4 is 12. In twelfths, $\frac{1}{6}=\frac{2}{12}$1/6 = 2/12 and $\frac{1}{4}=\frac{3}{12}$1/4 = 3/12, so the sum is $\frac{5}{12}$5/12. In 24ths it comes out as $\frac{10}{24}$10/24, which simplifies to $\frac{5}{12}$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}$1/2 + 1/3 = 2/5. That cannot be right. $\frac{2}{5}$2/5 is less than $\frac{1}{2}$1/2, and $\frac{1}{2}+\frac{1}{3}$1/2 + 1/3 must be more than the $\frac{1}{2}$1/2 it starts from.

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