If you can break a fraction into parts, you can put parts together. Adding fractions with the same denominator is counting pieces of the same size.
Key idea
To add or subtract fractions with the same denominator, add or subtract the numerators and keep the denominator: $\dfrac{1}{5} + \dfrac{2}{5} = \dfrac{3}{5}$ and $\dfrac{7}{8} - \dfrac{3}{8} = \dfrac{4}{8}$.
Why it works
The denominator names the size of the pieces (fifths); the numerator counts them. 1 fifth and 2 fifths make 3 fifths, the same way 1 apple and 2 apples make 3 apples. Putting pieces together does not change their size, so the denominator stays 5.
Worked example: Add
Find $\dfrac{2}{8} + \dfrac{5}{8}$.
- 2 eighths and 5 eighths make 7 eighths.
- The pieces are still eighths.
Answer: $\dfrac{7}{8}$
Worked example: Subtract from a whole
Find $1 - \dfrac{3}{8}$.
- Write 1 as eighths: $1 = \dfrac{8}{8}$.
- $\dfrac{8}{8} - \dfrac{3}{8} = \dfrac{5}{8}$.
Answer: $\dfrac{5}{8}$
Common mistake
Adding the denominators too, so $\dfrac{1}{5} + \dfrac{2}{5}$ becomes $\dfrac{3}{10}$. But $\dfrac{3}{10}$ is less than $\dfrac{2}{5}$ alone, so adding cannot give it. The pieces do not shrink when you put them together.