Now that you can divide fractions, the harder part of a story problem is noticing that it is division. Look for one of two questions: "how many of these fit in that?" or "how much is one whole group?"
Key idea
Divide when you know a total and either the size of one group (to find how many groups) or how many groups there are, even part of a group (to find the size of one group).
Why it works
Every division undoes a multiplication: $\text{groups} \times \text{size of a group} = \text{total}$. If you know the total and one of the other two, division finds the missing one. That is true for $3 \times 4 = 12$, and it is just as true when the numbers are fractions.
Worked example: How many pieces?
A ribbon is $2\frac{1}{4}$ yards long. It is cut into pieces $\frac{3}{8}$ yard long. How many pieces are there?
- How many $\frac{3}{8}$s fit in $2\frac{1}{4}$? That is $2\frac{1}{4} \div \frac{3}{8}$.
- $\frac{9}{4} \times \frac{8}{3} = \frac{72}{12} = 6$.
- Check: $6 \times \frac{3}{8} = \frac{18}{8} = 2\frac{1}{4}$.
Answer: 6 pieces.
Worked example: How much is one whole?
$\frac{3}{4}$ cup of water fills $\frac{1}{2}$ of a bottle. How much water fills the whole bottle?
- The total for half a bottle is known. One whole bottle is one group, so divide: $\frac{3}{4} \div \frac{1}{2}$.
- $\frac{3}{4} \times 2 = \frac{6}{4} = 1\frac{1}{2}$.
- Makes sense: two halves of $\frac{3}{4}$ cup each make $1\frac{1}{2}$ cups.
Answer: $1\frac{1}{2}$ cups.
Worked example: A missing side
A rectangle has an area of $\frac{3}{5}$ square meter. Its length is $\frac{3}{4}$ meter. How wide is it?
- Area = length × width, so width = area ÷ length.
- $\frac{3}{5} \div \frac{3}{4} = \frac{3}{5} \times \frac{4}{3} = \frac{12}{15} = \frac{4}{5}$.
- Check: $\frac{3}{4} \times \frac{4}{5} = \frac{12}{20} = \frac{3}{5}$.
Answer: $\frac{4}{5}$ meter.
Common mistake
Dividing in the wrong order. In the bottle story, $\frac{1}{2} \div \frac{3}{4} = \frac{2}{3}$ would mean the whole bottle holds less than its half. Quick check: say what each number means, then ask if the answer is sensible. The whole bottle must hold more than the $\frac{3}{4}$ cup in half of it.