You can multiply fractions, and division is fair sharing. What happens when the thing being shared is itself a fraction, like $\frac{1}{3}$ of a pan of cornbread?
Key idea
$\frac{1}{a} \div b = \frac{1}{a \times b}$. Sharing $\frac{1}{3}$ among 4 people gives each $\frac{1}{12}$. Check with multiplication: $\frac{1}{12} \times 4 = \frac{1}{3}$.
Why it works
Cut the $\frac{1}{3}$ into 4 equal pieces. If you did that to every third of the pan, the whole pan would be cut into $3 \times 4 = 12$ pieces of the same size. So each share is $\frac{1}{12}$ of the pan. Dividing by 4 is the same as taking $\frac{1}{4}$ of it: $\frac{1}{4} \times \frac{1}{3} = \frac{1}{12}$.
Worked examples
Worked example: Share a third
Find $\frac{1}{3} \div 4$.
- Cut $\frac{1}{3}$ into 4 equal parts.
- The whole would have $3 \times 4 = 12$ such parts, so each is $\frac{1}{12}$.
- Check: $\frac{1}{12} \times 4 = \frac{4}{12} = \frac{1}{3}$.
Answer: $\frac{1}{12}$
Worked example: A story
Five friends share $\frac{1}{2}$ pound of trail mix equally. How much does each get?
- $\frac{1}{2} \div 5$.
- $\frac{1}{2 \times 5} = \frac{1}{10}$.
- Check: $5 \times \frac{1}{10} = \frac{1}{2}$.
Answer: $\frac{1}{10}$ pound
Common mistake
Answering with a bigger fraction, such as $\frac{1}{3} \div 4 = \frac{4}{3}$. Sharing makes each part smaller than what was shared. Check by multiplying back: $\frac{4}{3} \times 4$ is far more than $\frac{1}{3}$.