Proportions — How to Solve Cross-Multiplication

A proportion is an equation that says two ratios are equal. To solve a proportion such as $\frac{a}{b} = \frac{c}{d}$, with $b \ne 0$ and $d \ne 0$, you cross-multiply to get $ad = bc$ and then solve the simpler equation.

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

Each symbol here is one term of a ratio. A ratio compares two quantities in a fixed order: $2:3$ compares the first quantity to the second. A proportion says one ratio equals another:

$$2:3 = 4:6$$

which is the same statement written as fractions:

$$\frac{2}{3} = \frac{4}{6}$$

These forms mean the same thing as long as the second term in each ratio is not zero.

Why cross-multiplication works

Cross-multiplication is not a separate trick. It comes straight from the equation. Start with

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

where $b \ne 0$ and $d \ne 0$. Multiply both sides by $bd$:

$$bd \cdot \frac{a}{b} = bd \cdot \frac{c}{d}$$

The denominators cancel, so you get

$$ad = bc$$

That is the whole idea. You are multiplying both sides of an equal-fraction equation by the same nonzero quantity, which preserves equality.

Worked example: solve a proportion

Solve

$$\frac{3}{5} = \frac{x}{20}$$

Cross-multiply:

$$3 \times 20 = 5x$$

so

$$60 = 5x$$

Divide both sides by $5$:

$$x = 12$$

Check the result by substituting back:

$$\frac{12}{20} = \frac{3}{5}$$

because both fractions simplify to the same value. So the missing number is $12$.

In ratio notation this same problem is

$$3:5 = 12:20$$

The order matters. Swapping one side to $20:12$ would describe a different ratio.

Practice it yourself

Solve

$$\frac{7}{9} = \frac{x}{27}$$

Cross-multiply, solve for $x$, then check your value by substituting it back into the proportion the way the worked example does. You should find the two fractions reduce to the same thing. For a second pass, set up a proportion with units, such as a map scale or a similar-triangles problem, and confirm the same structure still holds.

Calculation traps to watch for

• No equality, no proportion. If the problem does not say two ratios are equal, cross-multiplication may not apply. Adding fractions, for instance, is not a proportion.

• Keep the quantity order fixed. $2:3$ and $3:2$ are different ratios. If one side compares miles to hours, the other side must also compare miles to hours.

• Check the denominator condition. The fraction form of a proportion only makes sense when the denominators are not zero.

• Verify in the original equation. A quick substitution catches arithmetic slips faster than redoing the whole problem.

Before you cross-multiply, run four checks: Do I really have one ratio equal to another ratio? Are the quantities in the same order on both sides? Are the denominators nonzero? Does the situation actually stay proportional? Those four checks prevent most beginner errors.

When proportions are the right model

Proportions show up whenever one ratio stays constant: equivalent fractions, map scales, similar triangles, recipes scaled up or down, and price problems where the cost changes at a constant rate. That condition matters. If the relationship is not proportional, a proportion model can give the wrong answer even when the algebra is correct.