You know division with remainders and mixed numbers. Here is a link between them: the fraction bar is a division sign. $\frac{3}{4}$ means $3 \div 4$.
Key idea
$\frac{a}{b} = a \div b$. Sharing $a$ wholes equally among $b$ people gives each person $\frac{a}{b}$ of a whole, even when $a$ is smaller than $b$.
Why it works
Share 3 pizzas among 4 people. Cut each pizza into 4 equal slices. Each person takes one slice from each pizza: 3 slices, each $\frac{1}{4}$ of a pizza. So each person gets $\frac{3}{4}$, and $3 \div 4 = \frac{3}{4}$.
Worked examples
Worked example: A smaller number divided by a larger one
Write $3 \div 4$ as a fraction.
- The number being divided goes on top: $\frac{3}{4}$.
- Check: 4 shares of $\frac{3}{4}$ make $\frac{12}{4} = 3$.
Answer: $\frac{3}{4}$
Worked example: More than one whole each
5 people share 7 pounds of clay. How much does each get?
- $7 \div 5 = \frac{7}{5}$.
- As a mixed number: 5 goes into 7 once with 2 left, and the 2 are shared too: $1\frac{2}{5}$.
Answer: $1\frac{2}{5}$ pounds
Worked example: Between which whole numbers?
Between which two whole numbers is $\frac{17}{5}$?
- $\frac{17}{5} = 17 \div 5$.
- $5 \times 3 = 15$ and $5 \times 4 = 20$, so the answer is 3 with 2 left over: $3\frac{2}{5}$.
Answer: between 3 and 4
Common mistake
Believing a smaller number cannot be divided by a larger one, so $3 \div 4$ “can't be done” or is 0 R 3. It can: each share is just less than 1. Check by multiplying: $4 \times \frac{3}{4} = 3$.