Cross Multiplication

Cross multiplication is a quick way to solve a proportion, which is an equation where one fraction equals another fraction. If

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

and $b \ne 0$ and $d \ne 0$, then you can rewrite it as

$$ad = bc$$

In plain language, if two fractions are equal, the diagonal products match. This only works when you truly have a fraction equals fraction setup.

What Cross Multiplication Means

Cross multiplication applies to a proportion:

$$\frac{\text{something}}{\text{something}} = \frac{\text{something}}{\text{something}}$$

You multiply diagonally across the equal sign. In

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

the cross-products are $ad$ and $bc$.

This is not a separate magic trick. It comes from multiplying both sides by $bd$, which clears both denominators when $b \ne 0$ and $d \ne 0$.

Why Cross Multiplication Works

If two fractions represent the same value, they describe the same comparison in two different forms.

For example,

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

because both fractions reduce to the same ratio. Cross multiplication confirms that:

$$2 \cdot 6 = 12 \quad \text{and} \quad 3 \cdot 4 = 12$$

Matching cross-products are a quick check that two nonzero-denominator fractions are equal.

Cross Multiplication Example

Solve

$$\frac{x}{5} = \frac{12}{15}$$

Cross multiply the diagonals:

$$15x = 5 \cdot 12$$

so

$$15x = 60$$

Now divide both sides by $15$:

$$x = 4$$

Check the answer in the original proportion:

$$\frac{4}{5} = \frac{12}{15}$$

Both sides simplify to $\frac{4}{5}$, so the solution is correct.

When You Can Use Cross Multiplication

Use cross multiplication when both sides are fractions and those fractions are set equal to each other. You also need every denominator involved to be nonzero.

For example, it is valid in

$$\frac{x+1}{4} = \frac{3}{10}$$

because it is a proportion.

But not every equation with a fraction needs this method. In

$$\frac{x}{5} = 7$$

there is only one fraction, so the simpler move is to multiply both sides by $5$ and get $x = 35$.

Common Cross Multiplication Mistakes

One common mistake is using cross multiplication when the equation is not a proportion. The method is for fraction equals fraction, not for every equation that happens to contain a fraction.

Another mistake is forgetting denominator restrictions. If a denominator could be $0$, that value must be excluded. For example, in

$$\frac{x}{x-2} = \frac{3}{4}$$

you must state $x \ne 2$ before solving.

A third mistake is multiplying straight across instead of diagonally. In

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

the cross-products are $ad$ and $bc$, not $ab$ and $cd$.

Where Cross Multiplication Is Used

Cross multiplication appears in proportions, similar figures, scale drawings, unit conversions, and rate problems. It is useful when one unknown sits inside a ratio and you want to preserve the same comparison.

It is also a fast way to check whether two fractions are equivalent, as long as the denominators are nonzero.

Try A Similar Problem

Try your own version with

$$\frac{y}{8} = \frac{9}{12}$$

Solve for $y$, then substitute your answer back into the original proportion. If you want to explore another case, try a proportion with a variable in the denominator and state the restriction before solving.