Multiplying Fractions — Rules & Worked Examples

Take two-thirds of three-quarters of a pizza and you end up with exactly half. That "fraction of a fraction" is what multiplying fractions does — and unlike adding them, you never need a common denominator.

When this method applies

Multiplying fractions is the move whenever you need part of a part: recipes scaled down, scale models, probability with dependent steps, and measurement conversions. The trigger phrase is "of" — "$\frac{2}{3}$ of $\frac{3}{4}$" is $\frac{2}{3} \times \frac{3}{4}$. The rule is:

valid for $b \ne 0$ and $d \ne 0$. Reading multiplication as "of" also predicts the size of the answer: taking $\frac{2}{3}$ of $\frac{3}{4}$ must be smaller than $\frac{3}{4}$.

The steps

1. Multiply the numerators — the top numbers together.
2. Multiply the denominators — the bottom numbers together.
3. Simplify the answer — reduce if the numerator and denominator share a common factor.
4. Rewrite whole numbers if needed — turn any whole number into a fraction over $1$ before multiplying.

Full worked example: $\frac{2}{3} \times \frac{3}{4}$

Step 1 — numerators:

Step 2 — denominators:

So

Step 3 — simplify:

So $\frac{2}{3}$ of $\frac{3}{4}$ is $\frac{1}{2}$, which makes sense: you took part of a quantity already less than $1$.

You can also cancel the shared $3$ before multiplying for the same result, faster:

This shortcut is valid because you are canceling common factors across multiplication, not across addition or subtraction.

Step 4 in action: a fraction times a whole number

Write the whole number over $1$ first:

For a mixed number,

Where each step trips students up, and how to check

• Step 1 — using addition rules by mistake. Writing $\frac{2}{3} \times \frac{3}{4} = \frac{2+3}{3+4}$ is wrong; multiply top by top and bottom by bottom. Self-check: did you add anywhere? You should not have.
• Before Step 1 — hunting for a common denominator. You need one to add or subtract, not to multiply. Go straight to numerator times numerator.
• Step 3 — forgetting to simplify. $\frac{6}{12}$ and $\frac{1}{2}$ are equal, but $\frac{1}{2}$ is the finished answer.
• The shortcut — canceling in the wrong situation. Canceling works in products like $\frac{2}{3} \times \frac{3}{4}$, but not across addition such as $\frac{2}{3} + \frac{3}{4}$, which follows a different rule.

A reliable final check: since both factors here are positive and less than $1$, the product should be less than either factor.

Try it

Compute $\frac{5}{6} \times \frac{2}{5}$. Simplify before multiplying if you can, then verify the size: both positive fractions are less than $1$, so the product should be smaller than both. As a real-world version, a recipe that uses $\frac{3}{4}$ cup of milk needs $\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}$ cup when you make $\frac{2}{3}$ of it.

FAQ

Wondering why multiplication needs no common denominator but addition does? Because addition combines parts of the same whole (so the pieces must be the same size), while multiplication takes a fraction of another quantity — a fundamentally different operation.