Multiplying Two Fractions

Learn to multiply fractions and simplify using cancellation

1. Understanding Fraction × Fraction

Key Insight

$\frac{1}{3}$ of $\frac{1}{2}$ means taking one third of one half. The result, one sixth, is smaller than both fractions. This always happens when both fractions are less than 1.

Example: $\frac{1}{3}$ of $\frac{1}{2}$ = $\frac{1}{3} \times \frac{1}{2}$

Start with

\frac{1}{2}

of a pizza

↓

Take

\frac{1}{3}

of that half =

\frac{1}{6}

Visualise: Siti had $\frac{1}{2}$ of a pizza. She gave $\frac{1}{3}$ of it to her friend.

The whole pizza is divided into 6 parts (2 × 3 = 6). Her friend got 1 part = $\frac{1}{6}$ of the pizza.

Important: Order Doesn't Matter!

$\frac{1}{3}$ of $\frac{1}{2}$ is the same as $\frac{1}{2}$ of $\frac{1}{3}$

Both equal $\frac{1}{6}$

2. Multiplying Fractions: The Method

The Rule

$\frac{2}{5} \times \frac{3}{7} = \frac{2 \times 3}{5 \times 7} = \frac{6}{35}$

Multiply the numerators together. Multiply the denominators together.

Example: Find $\frac{1}{3} \times \frac{5}{4}$

$\frac{1}{3} \times \frac{5}{4} = \frac{1 \times 5}{3 \times 4} = \frac{5}{12}$

Answer: $\frac{5}{12}$

Example: Find the product of $\frac{3}{4}$ and $\frac{8}{7}$

Multiply, then simplify

$\frac{3}{4} \times \frac{8}{7} = \frac{3 \times 8}{4 \times 7} = \frac{24}{28}$

Simplify: $\frac{24}{28} = \frac{24 \div 4}{28 \div 4} = \frac{6}{7}$

Answer: $\frac{6}{7}$

3. Cancellation (Simplifying Before Multiplying)

Why Cancellation?

Cancelling (simplifying before multiplying) makes the numbers smaller and easier to work with. If you cancel all the common factors, the answer is already in its simplest form. Always check.

How to Cancel

Look for a common factor between any numerator and any denominator. Divide both by that common factor.

$\frac{3}{4} \times \frac{8}{7}$ using cancellation

Step 1: Look for common factors

8 (numerator of second fraction) and 4 (denominator of first fraction) share factor 4.

Step 2: Cancel (divide both by 4)

8 ÷ 4 = 2 and 4 ÷ 4 = 1

$\displaystyle \frac{3}{\underset{1}{\cancel{4}\vphantom{\big|}}} \times \frac{\overset{2}{\cancel{8}\vphantom{\big|}}}{7} = \frac{3}{1} \times \frac{2}{7}$

Step 3: Multiply the simplified fractions

$\frac{3 \times 2}{1 \times 7} = \frac{6}{7}$

Answer: $\frac{6}{7}$ (same answer, easier calculation!)

$\frac{5}{6} \times \frac{9}{10}$

Find common factors:

5 and 10 share factor 5 (5 ÷ 5 = 1, 10 ÷ 5 = 2)
9 and 6 share factor 3 (9 ÷ 3 = 3, 6 ÷ 3 = 2)

$\displaystyle \frac{\overset{1}{\cancel{5}\vphantom{\big|}}}{\underset{2}{\cancel{6}\vphantom{\big|}}} \times \frac{\overset{3}{\cancel{9}\vphantom{\big|}}}{\underset{2}{\cancel{10}\vphantom{\big|}}} = \frac{1 \times 3}{2 \times 2} = \frac{3}{4}$

Answer: $\frac{3}{4}$

Tip: You can cancel diagonally!

A numerator of the first fraction can cancel with a denominator of the second fraction (or vice versa).

4. Practice Problems

Practice 1: Find $\frac{2}{5} \times \frac{9}{5}$

ShowHide solution

$\frac{2}{5} \times \frac{9}{5} = \frac{2 \times 9}{5 \times 5} = \frac{18}{25}$

Answer: $\frac{18}{25}$

Practice 2: Find the product of $\frac{7}{6}$ and $\frac{2}{3}$

ShowHide solution

Using cancellation: 6 and 2 share factor 2

$\displaystyle \frac{7}{\underset{3}{\cancel{6}\vphantom{\big|}}} \times \frac{\overset{1}{\cancel{2}\vphantom{\big|}}}{3} = \frac{7 \times 1}{3 \times 3} = \frac{7}{9}$

Answer: $\frac{7}{9}$

Practice 3: $\frac{4}{9} \times \frac{3}{8}$ (Use cancellation)

ShowHide solution

Cancellation opportunities:

4 and 8 share factor 4: 4 ÷ 4 = 1, 8 ÷ 4 = 2
3 and 9 share factor 3: 3 ÷ 3 = 1, 9 ÷ 3 = 3

$\displaystyle \frac{\overset{1}{\cancel{4}\vphantom{\big|}}}{\underset{3}{\cancel{9}\vphantom{\big|}}} \times \frac{\overset{1}{\cancel{3}\vphantom{\big|}}}{\underset{2}{\cancel{8}\vphantom{\big|}}} = \frac{1 \times 1}{3 \times 2} = \frac{1}{6}$

Answer: $\frac{1}{6}$

Practice 4: $\frac{5}{12} \times \frac{8}{15}$

ShowHide solution

Cancellation:

5 and 15 share factor 5: 5 ÷ 5 = 1, 15 ÷ 5 = 3
8 and 12 share factor 4: 8 ÷ 4 = 2, 12 ÷ 4 = 3

$\displaystyle \frac{\overset{1}{\cancel{5}\vphantom{\big|}}}{\underset{3}{\cancel{12}\vphantom{\big|}}} \times \frac{\overset{2}{\cancel{8}\vphantom{\big|}}}{\underset{3}{\cancel{15}\vphantom{\big|}}} = \frac{1 \times 2}{3 \times 3} = \frac{2}{9}$

Answer: $\frac{2}{9}$

Key Takeaways

Fraction × Fraction: Multiply numerators, multiply denominators
For example: $\frac{2}{5} \times \frac{3}{7} = \frac{2 \times 3}{5 \times 7} = \frac{6}{35}$
Result is smaller: When multiplying two proper fractions, the result is smaller than both
Cancellation: Look for common factors between ANY numerator and ANY denominator
Cancel first: Simplifying before multiplying makes calculations easier
Always simplify: Express the final answer in its simplest form

Multiplying Two Fractions

1. Understanding Fraction × Fraction

Key Insight

Example: 13\frac{1}{3}31​ of 12\frac{1}{2}21​ = 13×12\frac{1}{3} \times \frac{1}{2}31​×21​

Important: Order Doesn't Matter!

2. Multiplying Fractions: The Method

The Rule

Example: Find 13×54\frac{1}{3} \times \frac{5}{4}31​×45​

Example: Find the product of 34\frac{3}{4}43​ and 87\frac{8}{7}78​

Multiply, then simplify

3. Cancellation (Simplifying Before Multiplying)

Why Cancellation?

How to Cancel

34×87\frac{3}{4} \times \frac{8}{7}43​×78​ using cancellation

56×910\frac{5}{6} \times \frac{9}{10}65​×109​

Tip: You can cancel diagonally!

4. Practice Problems

Practice 1: Find 25×95\frac{2}{5} \times \frac{9}{5}52​×59​

Practice 2: Find the product of 76\frac{7}{6}67​ and 23\frac{2}{3}32​

Practice 3: 49×38\frac{4}{9} \times \frac{3}{8}94​×83​ (Use cancellation)

Practice 4: 512×815\frac{5}{12} \times \frac{8}{15}125​×158​

Key Takeaways

Example: $\frac{1}{3}$ of $\frac{1}{2}$ = $\frac{1}{3} \times \frac{1}{2}$

Example: Find $\frac{1}{3} \times \frac{5}{4}$

Example: Find the product of $\frac{3}{4}$ and $\frac{8}{7}$

$\frac{3}{4} \times \frac{8}{7}$ using cancellation

$\frac{5}{6} \times \frac{9}{10}$

Practice 1: Find $\frac{2}{5} \times \frac{9}{5}$

Practice 2: Find the product of $\frac{7}{6}$ and $\frac{2}{3}$

Practice 3: $\frac{4}{9} \times \frac{3}{8}$ (Use cancellation)

Practice 4: $\frac{5}{12} \times \frac{8}{15}$