You can add fractions with like denominators: $\frac{1}{5} + \frac{2}{5} = \frac{3}{5}$, because fifths add like apples. When the pieces are different sizes, first rename them with a common denominator.
Key idea
To add fractions with unlike denominators, rename them with a common denominator, then add the numerators. The denominator stays the same, because it names the size of the pieces.
Why it works
Adding counts pieces, and you can only count pieces of one size. $\frac{1}{3}$ is one big piece and $\frac{1}{4}$ is one smaller piece; together they are not “2 sevenths.” As twelfths they are 4 pieces and 3 pieces of the same size: 7 twelfths.
Worked examples
Worked example: Unit fractions
Find $\frac{1}{2} + \frac{1}{3}$.
- A common denominator of 2 and 3 is 6.
- $\frac{1}{2} = \frac{3}{6}$ and $\frac{1}{3} = \frac{2}{6}$.
- Add the numerators: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$.
Answer: $\frac{5}{6}$
Worked example: A sum bigger than 1
Find $\frac{2}{3} + \frac{3}{4}$.
- Use 12: $\frac{2}{3} = \frac{8}{12}$ and $\frac{3}{4} = \frac{9}{12}$.
- $\frac{8}{12} + \frac{9}{12} = \frac{17}{12}$.
- $\frac{17}{12} = 1\frac{5}{12}$, since 12 twelfths make one whole.
Answer: $1\frac{5}{12}$
Common mistake
Adding the tops and the bottoms: $\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$. A quick check shows it is wrong: $\frac{2}{5}$ is less than $\frac{1}{2}$, but adding $\frac{1}{3}$ to $\frac{1}{2}$ must give more than $\frac{1}{2}$.