You can find a fraction of a whole number. A fraction of a fraction works the same way: $\frac{2}{3} \times \frac{3}{4}$ means $\frac{2}{3}$ of $\frac{3}{4}$.
Key idea
Multiply the numerators and multiply the denominators: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$. Then simplify. No common denominator is needed.
Why it works
Shade $\frac{3}{4}$ of a square with 4 columns. Then shade $\frac{2}{3}$ of that part with 3 rows. The square is now cut into $3 \times 4 = 12$ equal pieces, which is why the denominators multiply. The double-shaded part is $2 \times 3 = 6$ of them, which is why the numerators multiply. So $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$.
Worked examples
Worked example: Half of a half
Find $\frac{1}{2} \times \frac{1}{2}$.
- Half of one half is a quarter of the whole.
- $\frac{1 \times 1}{2 \times 2} = \frac{1}{4}$.
Answer: $\frac{1}{4}$
Worked example: Multiply and simplify
Find $\frac{2}{3} \times \frac{3}{4}$.
- Numerators: $2 \times 3 = 6$. Denominators: $3 \times 4 = 12$.
- $\frac{6}{12} = \frac{1}{2}$.
Answer: $\frac{1}{2}$
Worked example: A story
There is $\frac{3}{5}$ of a pan of lasagna left. Leo eats $\frac{1}{2}$ of what is left. What part of the whole pan does he eat?
- $\frac{1}{2}$ of $\frac{3}{5}$ is $\frac{1}{2} \times \frac{3}{5}$.
- $\frac{1 \times 3}{2 \times 5} = \frac{3}{10}$.
Answer: $\frac{3}{10}$ of the pan
Common mistake
Finding a common denominator first and keeping it: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{2}$. That treats multiplying like adding. Check: half of a half must be smaller than a half.