Multiplying Fractions — Rules & Worked Examples

To multiply fractions, multiply the numerators, multiply the denominators, and simplify the result if possible. You do not need a common denominator. For example, $\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}$.

$$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$

This rule assumes $b \ne 0$ and $d \ne 0$. In plain language, fraction multiplication often means "take a fraction of another fraction."

Why Multiplying Fractions Means "Of"

The fastest intuition is to read multiplication as "of." For example, $\frac{2}{3} \times \frac{3}{4}$ means "two-thirds of three-fourths."

If you start with $\frac{3}{4}$ of a whole and then take $\frac{2}{3}$ of that amount, the result must be smaller than $\frac{3}{4}$. That is exactly what the multiplication rule gives.

Worked Example: $\frac{2}{3} \times \frac{3}{4}$

Find

$$\frac{2}{3} \times \frac{3}{4}$$

Step 1: multiply the numerators.

$$2 \times 3 = 6$$

Step 2: multiply the denominators.

$$3 \times 4 = 12$$

So

$$\frac{2}{3} \times \frac{3}{4} = \frac{6}{12}$$

Now simplify:

$$\frac{6}{12} = \frac{1}{2}$$

So $\frac{2}{3}$ of $\frac{3}{4}$ is $\frac{1}{2}$. The answer makes sense because you are taking part of a quantity that is already less than $1$.

You can also notice that the $3$ in the numerator and denominator cancels before multiplying, which gives the same result more quickly:

$$\frac{2}{3} \times \frac{3}{4} = \frac{2}{1} \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$$

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

How To Multiply A Fraction By A Whole Number

If one factor is a whole number, write it over $1$ first.

For example,

$$3 \times \frac{5}{8} = \frac{3}{1} \times \frac{5}{8} = \frac{15}{8}$$

If you want a mixed number,

$$\frac{15}{8} = 1 \frac{7}{8}$$

Common Mistakes When Multiplying Fractions

Using Addition Rules By Mistake

Students sometimes write

$$\frac{2}{3} \times \frac{3}{4} = \frac{2+3}{3+4}$$

That is not the rule. For multiplication, multiply top by top and bottom by bottom.

Looking For A Common Denominator First

You need a common denominator when you add or subtract fractions, not when you multiply them. For multiplication, you can go straight to numerator times numerator and denominator times denominator.

Forgetting To Simplify

$\frac{6}{12}$ and $\frac{1}{2}$ represent the same value, but $\frac{1}{2}$ is the simpler final answer.

Canceling In The Wrong Situation

Canceling common factors works in products like

$$\frac{2}{3} \times \frac{3}{4}$$

It does not work across addition, such as

$$\frac{2}{3} + \frac{3}{4}$$

because addition follows a different rule.

When You Use Multiplying Fractions

Multiplying fractions shows up whenever you need part of a part. That happens in recipes, scale models, probability with dependent steps, and measurement conversions.

For example, if a recipe uses $\frac{3}{4}$ cup of milk and you want $\frac{2}{3}$ of the recipe, you need $\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}$ cup of milk.

Try A Similar Problem

Try $\frac{5}{6} \times \frac{2}{5}$. Simplify before multiplying if you can, then check whether your answer makes sense: because both positive fractions are less than $1$, the product should also be less than either factor. If you want another case right after this one, explore dividing fractions next and compare how the rule changes.