You can add and subtract mixed numbers when the fractions have the same denominator, and you can rename unlike fractions. Put those together for mixed numbers with unlike denominators.
Key idea
Rename the fraction parts with a common denominator. Add or subtract the whole numbers and the fractions. When adding gives a fraction of 1 or more, carry a whole. When subtracting needs more pieces than you have, regroup one whole into pieces first.
Why it works
A mixed number is wholes plus a fraction, so you can work the wholes and the fractions separately, as long as you keep track of the pieces. One whole is the same as $\frac{12}{12}$, so $5\frac{3}{12}$ can be written as $4\frac{15}{12}$: same amount, more pieces.
Worked examples
Worked example: Add and carry
Find $2\frac{3}{4} + 1\frac{1}{2}$.
- Rename: $\frac{1}{2} = \frac{2}{4}$.
- Wholes: $2 + 1 = 3$. Fractions: $\frac{3}{4} + \frac{2}{4} = \frac{5}{4}$.
- $\frac{5}{4} = 1\frac{1}{4}$, so carry 1: $3 + 1\frac{1}{4} = 4\frac{1}{4}$.
Answer: $4\frac{1}{4}$
Worked example: Subtract with regrouping
Find $5\frac{1}{4} - 2\frac{2}{3}$.
- Rename with 12: $5\frac{3}{12} - 2\frac{8}{12}$.
- $\frac{3}{12}$ is less than $\frac{8}{12}$, so regroup a whole: $5\frac{3}{12} = 4\frac{15}{12}$.
- Wholes: $4 - 2 = 2$. Fractions: $\frac{15}{12} - \frac{8}{12} = \frac{7}{12}$.
- Check: $2\frac{7}{12} + 2\frac{8}{12} = 4\frac{15}{12} = 5\frac{3}{12}$.
Answer: $2\frac{7}{12}$
Common mistake
Subtracting the smaller fraction from the larger, whichever number it belongs to: $5\frac{3}{12} - 2\frac{8}{12} = 3\frac{5}{12}$. The $\frac{8}{12}$ is being taken away, so you need to regroup. Check by adding back: $3\frac{5}{12} + 2\frac{8}{12} = 6\frac{1}{12}$, not $5\frac{1}{4}$.