You can make equivalent fractions by multiplying. Going the other way, dividing the top and bottom by a common factor, uses fewer, larger pieces.
Key idea
A fraction is in simplest form when the only number that divides both the numerator and the denominator is 1. To simplify $\dfrac{8}{12}$, divide both by a common factor: $\dfrac{8 \div 4}{12 \div 4} = \dfrac{2}{3}$.
Why it works
Dividing undoes the cutting. In $\dfrac{8}{12}$, put every 4 small pieces back together into one bigger piece. 12 pieces become 3, and the 8 shaded pieces become 2. Same amount, bigger pieces: $\dfrac{2}{3}$.
Worked example: One step
Write $\dfrac{6}{8}$ in simplest form.
- 2 divides both 6 and 8.
- $\dfrac{6 \div 2}{8 \div 2} = \dfrac{3}{4}$.
- Only 1 divides both 3 and 4, so stop.
Answer: $\dfrac{3}{4}$
Worked example: More than one step
Write $\dfrac{12}{18}$ in simplest form.
- Divide by 2: $\dfrac{12}{18} = \dfrac{6}{9}$.
- 3 still divides both: $\dfrac{6}{9} = \dfrac{2}{3}$.
- Using the greatest common factor, 6, gets there in one step: $\dfrac{12 \div 6}{18 \div 6} = \dfrac{2}{3}$.
Answer: $\dfrac{2}{3}$
Common mistake
Dividing the top and bottom by different numbers, like $\dfrac{6}{8} \to \dfrac{3}{2}$ (top ÷ 2, bottom ÷ 4). That changes the amount: $\dfrac{3}{2}$ is more than a whole. Use one number for both, and check that the new fraction is about the same size.