You can divide a unit fraction by a whole number and a whole number by a unit fraction. In a story, the hard part is deciding which one it is.
Key idea
Ask what the story does. If a fraction is shared into equal parts, divide the fraction by the number of parts: $\frac{1}{2} \div 4$. If you measure how many pieces of a fraction size fit in an amount, divide the amount by the fraction: $4 \div \frac{1}{2}$.
Why it works
Both are division, but they answer different questions. Sharing $\frac{1}{2}$ of a pizza among 4 friends gives each a small piece, $\frac{1}{8}$. Measuring how many $\frac{1}{2}$-cup scoops are in 4 cups gives a count, 8. Swapping the numbers swaps the question, so the order matters.
Worked examples
Worked example: Sharing
Ana has $\frac{1}{3}$ gallon of paint. She pours it equally into 2 cans. How much is in each can?
- A fraction is shared into 2 parts: $\frac{1}{3} \div 2$.
- $\frac{1}{3 \times 2} = \frac{1}{6}$.
- Check: $2 \times \frac{1}{6} = \frac{1}{3}$.
Answer: $\frac{1}{6}$ gallon
Worked example: Measuring
Ben has 3 yards of fabric. Each bandana takes $\frac{1}{4}$ yard. How many bandanas can he make?
- How many $\frac{1}{4}$s fit in 3? $3 \div \frac{1}{4}$.
- 4 quarters in each yard: $3 \times 4 = 12$.
- Check: $12 \times \frac{1}{4} = 3$.
Answer: 12 bandanas
Common mistake
Writing the division in the wrong order, like $4 \div \frac{1}{2}$ for “$\frac{1}{2}$ of a pizza shared by 4.” Check with sense: each friend should get less than $\frac{1}{2}$ of the pizza, not 8 pizzas.