Multiplying Fractions by Fractions

Multiplying a fraction by another fraction is just like finding "a part of a part." When you see a math problem asking you to find $\frac{2}{5}$ of $\frac{3}{4}$, the word "of" usually means you need to multiply!

The Basic Rule

Unlike adding or subtracting, multiplying fractions is very straightforward because you do not need to find a common denominator. Just follow these two simple steps:

1. Multiply the numerators (the top numbers) together to get your new numerator.
2. Multiply the denominators (the bottom numbers) together to get your new denominator.

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

Example 1: Straightforward Multiplication

Let's solve $\frac{2}{3} \times \frac{4}{5}$:

• Multiply the numerators: $2 \times 4 = 8$
• Multiply the denominators: $3 \times 5 = 15$

Put them together to get your answer:

$\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$

Since $8$ and $15$ share no common factors other than $1$, the fraction $\frac{8}{15}$ is already in its simplest form.

Example 2: Simplifying Your Answer

Sometimes, you will need to simplify the final fraction. Let's look at $\frac{3}{4} \times \frac{2}{7}$:

• Multiply the numerators: $3 \times 2 = 6$
• Multiply the denominators: $4 \times 7 = 28$

$\frac{3}{4} \times \frac{2}{7} = \frac{6}{28}$

Both $6$ and $28$ are even numbers, which means they can both be divided by $2$ to simplify the fraction.

$\frac{6 \div 2}{28 \div 2} = \frac{3}{14}$

Pro Tip: Simplify Before You Multiply

To make the math easier, you can "cross-simplify" before you multiply. Look at the numbers diagonally from each other.

In the problem $\frac{3}{4} \times \frac{2}{7}$:

• Notice that the $2$ (top right) and the $4$ (bottom left) can both be divided by $2$.
• The $2$ becomes a $1$, and the $4$ becomes a $2$.
• The new problem is $\frac{3}{2} \times \frac{1}{7}$.

Now multiply: $\frac{3 \times 1}{2 \times 7} = \frac{3}{14}$

You get the exact same answer, but with smaller, easier numbers to multiply!