Mixed numbers have a whole part and a fraction part. To add or subtract them, use like fractions for the fraction parts and whole-number math for the wholes. Sometimes you need to trade, just as you turned wholes into fractions.
Key idea
Add or subtract wholes with wholes and fractions with fractions. When adding, if the fractions make 1 or more, regroup a whole. When subtracting, if the first fraction is too small, trade 1 whole for $\dfrac{d}{d}$ (like $\dfrac{5}{5}$) first.
Why it works
$3\dfrac{1}{5}$ is 3 wholes and 1 fifth. To take away $\dfrac{3}{5}$ you need more fifths, so cut one whole into 5 fifths: now you have 2 wholes and 6 fifths, still the same amount. It is the same idea as trading 1 ten for 10 ones, except a whole is 5 fifths, not 10.
Worked example: Adding with regrouping
Find $2\dfrac{3}{4} + 1\dfrac{3}{4}$.
- Wholes: $2 + 1 = 3$. Fractions: $\dfrac{3}{4} + \dfrac{3}{4} = \dfrac{6}{4}$.
- $\dfrac{6}{4}$ is $1\dfrac{2}{4}$, so regroup: $3 + 1\dfrac{2}{4} = 4\dfrac{2}{4}$.
Answer: $4\dfrac{2}{4}$ (or $4\dfrac{1}{2}$)
Worked example: Subtracting with a trade
Find $3\dfrac{1}{5} - 1\dfrac{3}{5}$.
- $\dfrac{1}{5}$ is less than $\dfrac{3}{5}$, so trade 1 whole for $\dfrac{5}{5}$: $3\dfrac{1}{5} = 2\dfrac{6}{5}$.
- Fractions: $\dfrac{6}{5} - \dfrac{3}{5} = \dfrac{3}{5}$. Wholes: $2 - 1 = 1$.
- Check: $1\dfrac{3}{5} + 1\dfrac{3}{5} = 2\dfrac{6}{5} = 3\dfrac{1}{5}$.
Answer: $1\dfrac{3}{5}$
Common mistake
Trading a whole for $\dfrac{10}{5}$ (thinking of tens), which gives $2\dfrac{11}{5}$. One whole is $\dfrac{5}{5}$ when the pieces are fifths, $\dfrac{8}{8}$ when they are eighths. Check by adding your answer back.