Multiplying Fractions

Multiplying fractions is actually simpler than adding or subtracting them. There is no need to find a common denominator. You multiply straight across: numerator times numerator, denominator times denominator. The skill that makes multiplication faster and easier is cross-canceling, which lets you simplify before you multiply so you work with smaller numbers.

The Basic Rule

Formula:

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

Multiply the numerators together to get the new numerator. Multiply the denominators together to get the new denominator. Simplify if needed.

Example 1: Multiply $\frac{2}{3} \times \frac{4}{5}$

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

Since 8 and 15 share no common factors, $\frac{8}{15}$ is already simplified.

Answer: $\frac{8}{15}$

Example 2: Multiply $\frac{3}{4} \times \frac{2}{9}$

$\frac{3}{4} \times \frac{2}{9} = \frac{3 \times 2}{4 \times 9} = \frac{6}{36} = \frac{1}{6}$

We simplified by dividing both 6 and 36 by 6.

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

Cross-Canceling (Simplify Before You Multiply)

Cross-canceling means dividing a numerator and a denominator by a common factor before multiplying. This keeps the numbers small and often eliminates the need to simplify at the end.

How it works: Look diagonally across the multiplication sign. If any numerator shares a common factor with any denominator, divide both by that factor.

Example 3: Multiply $\frac{3}{4} \times \frac{2}{9}$ using cross-canceling

Look at the numbers: 3 (numerator of the first) and 9 (denominator of the second) are both divisible by 3. Also, 2 (numerator of the second) and 4 (denominator of the first) are both divisible by 2.

$\frac{\cancel{3}^{1}}{\cancel{4}_{2}} \times \frac{\cancel{2}^{1}}{\cancel{9}_{3}} = \frac{1 \times 1}{2 \times 3} = \frac{1}{6}$

We got $\frac{1}{6}$ directly, no simplification needed at the end.

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

Example 4: Multiply $\frac{5}{8} \times \frac{4}{15}$

Cross-cancel: 5 and 15 share a factor of 5. And 4 and 8 share a factor of 4.

$\frac{\cancel{5}^{1}}{\cancel{8}_{2}} \times \frac{\cancel{4}^{1}}{\cancel{15}_{3}} = \frac{1 \times 1}{2 \times 3} = \frac{1}{6}$

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

Multiplying a Fraction by a Whole Number

Any whole number can be written as a fraction with 1 as the denominator. For example, $6 = \frac{6}{1}$.

Example 5: Multiply $\frac{3}{4} \times 6$

$\frac{3}{4} \times \frac{6}{1} = \frac{3 \times 6}{4 \times 1} = \frac{18}{4} = \frac{9}{2} = 4\frac{1}{2}$

Answer: $4\frac{1}{2}$

Multiplying Mixed Numbers

To multiply mixed numbers, first convert them to improper fractions, then multiply using the standard rule.

Converting a mixed number to an improper fraction:

$a\frac{b}{c} = \frac{(a \times c) + b}{c}$

Example 6: Multiply $2\frac{1}{3} \times 1\frac{1}{2}$

Step 1: Convert to improper fractions:

$2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{7}{3}$

$1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2}$

Step 2: Multiply:

$\frac{7}{3} \times \frac{3}{2}$

Cross-cancel: 3 in the denominator and 3 in the numerator cancel:

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

Answer: $3\frac{1}{2}$

Example 7: Multiply $3\frac{1}{4} \times 2\frac{2}{5}$

Step 1: Convert:

$3\frac{1}{4} = \frac{13}{4} \qquad 2\frac{2}{5} = \frac{12}{5}$

Step 2: Multiply. Cross-cancel: 12 and 4 share a factor of 4:

$\frac{13}{\cancel{4}_{1}} \times \frac{\cancel{12}^{3}}{5} = \frac{13 \times 3}{1 \times 5} = \frac{39}{5} = 7\frac{4}{5}$

Answer: $7\frac{4}{5}$

Quick Reference: Fraction of Common Amounts

This table is useful for mental math when you need a fraction of a standard quantity:

Real-World Application: Nursing

A doctor orders $\frac{3}{4}$ of the standard dose of a medication. The standard dose is $\frac{2}{3}$ of a milligram. How many milligrams should the nurse administer?

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

Cross-cancel: 3 in the numerator of the first fraction and 3 in the denominator of the second:

$\frac{\cancel{3}^{1}}{4} \times \frac{2}{\cancel{3}_{1}} = \frac{1 \times 2}{4 \times 1} = \frac{2}{4} = \frac{1}{2} \text{ mg}$

The nurse should administer $\frac{1}{2}$ mg. In a clinical setting, the nurse would double-check this calculation and might verify it against a dosage chart or with a colleague, since medication errors can have serious consequences.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Multiply straight across: numerator times numerator, denominator times denominator
• Cross-cancel first to simplify your work and avoid large numbers
• Whole numbers become fractions over 1 (e.g., $5 = \frac{5}{1}$)
• Mixed numbers must be converted to improper fractions before multiplying
• Always simplify your final answer to lowest terms

Return to Arithmetic for more foundational math topics.