Multiplying Fractions Formula

The Formula

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

When to use: \frac{2}{3} \times \frac{3}{4} means 'two-thirds of three-quarters.' Take \frac{3}{4} of something, then take \frac{2}{3} of that result.

Quick Example

\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}

Notation

\frac{a}{b} \times \frac{c}{d} — multiply numerators and denominators straight across

What This Formula Means

Multiplying two fractions by multiplying the numerators together and the denominators together.

\frac{2}{3} \times \frac{3}{4} means 'two-thirds of three-quarters.' Take \frac{3}{4} of something, then take \frac{2}{3} of that result.

Formal View

\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d} where b, d \neq 0

Worked Examples

Example 1

easy
Multiply \frac{2}{3} \times \frac{5}{7}.

Solution

  1. 1
    Multiply the numerators: 2 \times 5 = 10.
  2. 2
    Multiply the denominators: 3 \times 7 = 21.
  3. 3
    The product is \frac{10}{21}, which is already in simplest form since \gcd(10, 21) = 1.

Answer

\frac{10}{21}
To multiply fractions, multiply numerator by numerator and denominator by denominator. No common denominator is needed, unlike addition.

Example 2

medium
Compute \frac{4}{9} \times \frac{3}{8}.

Common Mistakes

  • Cross-multiplying instead of multiplying straight across
  • Not simplifying before or after multiplying
  • Expecting the product to be larger than the original fractions

Why This Formula Matters

Used for scaling, area calculations, probability, and finding a fraction of a quantity.

Frequently Asked Questions

What is the Multiplying Fractions formula?

Multiplying two fractions by multiplying the numerators together and the denominators together.

How do you use the Multiplying Fractions formula?

\frac{2}{3} \times \frac{3}{4} means 'two-thirds of three-quarters.' Take \frac{3}{4} of something, then take \frac{2}{3} of that result.

What do the symbols mean in the Multiplying Fractions formula?

\frac{a}{b} \times \frac{c}{d} — multiply numerators and denominators straight across

Why is the Multiplying Fractions formula important in Math?

Used for scaling, area calculations, probability, and finding a fraction of a quantity.

What do students get wrong about Multiplying Fractions?

Students expect the product to be larger, but multiplying by a fraction less than 1 makes the result smaller.

What should I learn before the Multiplying Fractions formula?

Before studying the Multiplying Fractions formula, you should understand: fractions, multiplication.

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