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Multiplying
Fractions

Learn to multiply fractions, simplify products, and interpret the result using visual area models.

Pre-Algebra

What Is a Fraction?

A fraction like \(\frac{a}{b}\) represents \(a\) parts out of \(b\) equal parts of a whole. The top number is the numerator (how many parts you have) and the bottom number is the denominator (how many equal parts make a whole).

3 out of 4 = \(\frac{3}{4}\)

What Does Multiplying Fractions Mean?

Multiplying fractions answers the question: "What is a fraction of another fraction?" For example, \(\frac{1}{2} \times \frac{3}{4}\) asks: what is half of three-quarters?

"Of" Means Multiply
\(\frac{1}{2}\) of \(\frac{3}{4}\) is the same as \(\frac{1}{2} \times \frac{3}{4}\).
Smaller Result
Multiplying two proper fractions always gives a product smaller than either factor.

The Area Model

Draw a rectangle, split it one way for the first fraction and the other way for the second. The overlap shows the product.

\(\frac{1}{2}\) \(\frac{3}{4}\) Overlap \(\frac{3}{8}\)

The Core Rule

$$\frac{\color{#818cf8}{a}}{\color{#818cf8}{b}} \times \frac{\color{#fbbf24}{c}}{\color{#fbbf24}{d}} = \frac{\color{#4ade80}{a \times c}}{\color{#4ade80}{b \times d}}$$

To multiply two fractions, multiply the numerators together and multiply the denominators together:

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

Simplify Before or After?

You can simplify (cancel common factors) either before you multiply or after. Simplifying before keeps numbers small and saves work.

Cross-Cancel First
\(\frac{4}{9} \times \frac{3}{8}\): divide 4 and 8 by 4, divide 3 and 9 by 3 to get \(\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}\).
Simplify After
\(\frac{4}{9} \times \frac{3}{8} = \frac{12}{72}\). Then reduce: \(\frac{12}{72} = \frac{1}{6}\). Same answer, bigger numbers.

Multiplying vs Adding Fractions

Multiplying Adding
Rule Straight across: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\) Need common denominator first
Denominator Multiply the denominators Keep (or find) a common denominator
Result size Product is smaller (proper fractions) Sum is larger
Meaning "A fraction of a fraction" "Combining parts together"

How to Multiply Fractions

1
Convert mixed numbers to improper fractions if needed (e.g., \(2\frac{1}{3} = \frac{7}{3}\)).
2
Cross-cancel any common factors between a numerator and a denominator.
3
Multiply numerators together for the new numerator.
4
Multiply denominators together for the new denominator.
5
Simplify the result. Convert back to a mixed number if the problem asks for it.

Example 1: \(\frac{2}{3} \times \frac{5}{7}\)

\(2\) \(3\) \(\times\) \(5\) \(7\) = \(2 \times 5 = 10\) \(3 \times 7 = 21\)
\(\frac{10}{21}\)

Example 2: \(\frac{3}{8} \times \frac{4}{9}\)

\(3\) \(8\) \(\times\) \(4\) \(9\) \(\div 3\) \(\div 4\) → \(1\) \(2\) \(\times\) \(1\) \(3\) =
\(\frac{1}{6}\)
 3 and 9 share factor 3 4 and 8 share factor 4

Example 3: \(1\frac{1}{2} \times \frac{2}{5}\)

1
Convert: \(1\frac{1}{2} = \frac{3}{2}\).
2
Cross-cancel: The 2 in \(\frac{3}{\color{#f87171}{2}}\) and the 2 in \(\frac{\color{#f87171}{2}}{5}\) share a factor of 2. Cancel to get \(\frac{3}{1} \times \frac{1}{5}\).
3
Multiply: \(\frac{3 \times 1}{1 \times 5} = \frac{3}{5}\).

Fractions in Real Life

Problem: A recipe calls for \(\frac{3}{4}\) cup of sugar. You want to make half the recipe. How much sugar do you need?

\(\frac{3}{4}\) cup \(\times \frac{1}{2}\) → \(\frac{3}{8}\) cup

Multiplying a Fraction by a Whole Number

A whole number is just a fraction with denominator 1. So \(5 \times \frac{2}{3}\) becomes \(\frac{5}{1} \times \frac{2}{3}\).

1
Rewrite the whole number: \(5 = \frac{5}{1}\).
2
Multiply straight across: \(\frac{5 \times 2}{1 \times 3} = \frac{10}{3}\).
3
Convert: \(\frac{10}{3} = 3\frac{1}{3}\).

Why Does the Product Get Smaller?

When both factors are proper fractions (between 0 and 1), you are taking a part of a part. Each fraction shrinks the other.

0 \(\frac{1}{2}\) 1 \(\frac{3}{4}\) \(\frac{3}{8}\) \(\times \frac{1}{2}\)

Common Mistakes

✕ Adding Denominators
Writing \(\frac{2}{3} \times \frac{1}{4} = \frac{2}{7}\). The denominators should be multiplied, not added: \(3 \times 4 = 12\).
✕ Cross-Multiplying
Confusing fraction multiplication with solving proportions. Cross-multiplication is for equations, not for multiplying.
✕ Forgetting to Simplify
Leaving \(\frac{6}{8}\) instead of reducing to \(\frac{3}{4}\). Always check if the numerator and denominator share a common factor.

Summary of Fraction Multiplication

Scenario Example Result
Fraction × Fraction \(\frac{2}{3} \times \frac{5}{7}\) \(\frac{10}{21}\)
With cross-cancel \(\frac{3}{8} \times \frac{4}{9}\) \(\frac{1}{6}\)
Mixed number \(1\frac{1}{2} \times \frac{2}{5}\) \(\frac{3}{5}\)
Whole number × Fraction \(5 \times \frac{2}{3}\) \(3\frac{1}{3}\)
Context (recipe) \(\frac{1}{2} \times \frac{3}{4}\) cup \(\frac{3}{8}\) cup

Question 1 of 5

What is \(\frac{3}{5} \times \frac{2}{7}\)?

Question 2 of 5

Simplify: \(\frac{4}{9} \times \frac{3}{10}\)

Question 3 of 5

What is \(2\frac{1}{3} \times \frac{3}{7}\)?

Question 4 of 5

A garden is \(\frac{3}{4}\) acre. Maria plants flowers on \(\frac{2}{3}\) of the garden. How many acres of flowers does she plant?

Question 5 of 5

When you multiply two proper fractions (both between 0 and 1), the product is: