Cross Multiplication

Cross multiplication is a shortcut for solving proportions — equations where two fractions are set equal to each other. You multiply each numerator by the denominator on the opposite side, then solve the resulting equation.

Cross multiplication is a technique applied to a proportion of the form $\frac{a}{b} = \frac{c}{d}$ , where $b \neq 0$ and $d \neq 0$ . By multiplying the numerator of each fraction by the denominator of the other, the proportion is converted into the equation $a \cdot d = b \cdot c$ . This eliminates the fractions and produces a simpler equation that can be solved using standard algebraic methods.

Key Formula

\frac{a}{b} = \frac{c}{d} \implies a \cdot d = b \cdot c

Where:

$a$ = the numerator of the first fraction
$b$ = the denominator of the first fraction (cannot be 0)
$c$ = the numerator of the second fraction
$d$ = the denominator of the second fraction (cannot be 0)

Worked Example

Problem: Solve for x:

\frac{3}{8} = \frac{x}{20}

Step 1: Identify the two diagonals. Multiply the numerator of the first fraction by the denominator of the second, and the denominator of the first fraction by the numerator of the second.

3 \times 20 = 8 \times x

Step 2: Carry out the multiplication on each side.

60 = 8x

Step 3: Divide both sides by 8 to isolate x.

x = \frac{60}{8} = 7.5

Answer:

x = 7.5

Visualization

Why It Matters

Cross multiplication is one of the most frequently used tools in middle-school math and beyond. Anytime you need to find a missing value in a proportion — scaling a recipe, converting units, working with maps, or calculating percentages — cross multiplication gives you a fast, reliable way to set up and solve the equation.

Common Mistakes

Mistake: Using cross multiplication when two fractions are being added or subtracted, not set equal.

Correction: Cross multiplication only works on proportions (two fractions connected by an equals sign). For addition or subtraction of fractions, you need a common denominator instead.

Mistake: Multiplying straight across (numerator × numerator and denominator × denominator) instead of diagonally.

Correction: Remember the 'cross' in cross multiplication: each numerator gets multiplied by the opposite denominator. Think of drawing an X between the two fractions.

Related Terms

Proportion — The type of equation cross multiplication solves
Ratio — Proportions compare two equal ratios
Equation — Cross multiplication converts a proportion into a simpler equation