Cross Multiplication

Cross multiplication is a quick way to solve a proportion, which is an equation where one fraction equals another fraction. The whole method is captured by one move: 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. The symbols $a$ and $d$ form one diagonal, $b$ and $c$ form the other, and the two cross-products $ad$ and $bc$ are equal. This only works when you truly have a fraction-equals-fraction setup.

Why The Diagonal Products Match

Cross multiplication is not a separate magic trick. It comes from multiplying both sides of

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

by $bd$, which clears both denominators when $b \ne 0$ and $d \ne 0$. Once both denominators are gone, what remains is $ad = bc$.

You can see this directly with equal fractions. Because

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

both fractions reduce to the same ratio, and cross multiplication confirms it:

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

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

Worked 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.

Practice, And When The Method Applies

Solve

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

for $y$, then substitute your answer back into the original proportion to verify. For a tougher case, try a proportion with a variable in the denominator and state the restriction before solving.

Use cross multiplication only when both sides are fractions set equal to each other and every denominator involved is 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 it. In

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

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

Calculation Traps To Avoid

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 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 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.