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 can cross-multiply to get $ad = bc$ and then solve the simpler equation.

That method only works when you really have one ratio equal to another ratio. If the quantities are not proportional, or if the order of the quantities changes, cross-multiplication can give the wrong result.

What a proportion is

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

A ratio compares two quantities in a fixed order. For example, $2:3$ compares the first quantity to the second quantity. A proportion says that one ratio equals another ratio:

$$2:3 = 4:6$$

You can also write the same idea as fractions:

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

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

Why cross-multiplication works

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 behind cross-multiplication. It is not a separate trick. It comes from multiplying both sides of an equal-fraction equation by the same nonzero quantity.

Solve a proportion step by step

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:

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

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

If you prefer ratio notation, this same problem is

$$3:5 = 12:20$$

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

Common mistakes when solving proportions

• No equality, no proportion. If the problem does not say two ratios are equal, cross-multiplication may not apply. For example, adding fractions 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.

• Check your answer in the original equation. A quick substitution usually catches arithmetic mistakes faster than redoing the whole problem.

When proportions are the right model

Proportions show up whenever one ratio stays constant. Common examples include equivalent fractions, map scales, similar triangles, recipes that are 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.

Quick check before you cross-multiply

Before using cross-multiplication, ask:

1. Do I really have one ratio equal to another ratio?
2. Are the quantities in the same order on both sides?
3. Are the denominators nonzero?
4. Does the situation actually stay proportional?

Those four checks prevent most beginner errors.

Try a similar problem

Try solving

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

Then check your answer by substituting it back into the proportion. If you want to go one step further, explore another case with units, such as a map scale or a similar-triangles problem, and see whether the same structure still holds.