Solving Proportions with Cross-Multiplication

A proportion is an equation that says two ratios (or fractions) are equal. It looks like this:

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

When you have a proportion with an unknown value (usually represented by a variable like $x$), you can easily find that missing number using a method called cross-multiplication.

What is Cross-Multiplication?

The cross-multiplication rule states that if two fractions are equal, their diagonal products are also equal.

If $\frac{a}{b} = \frac{c}{d}$, then $a \times d = b \times c$.

How to Solve a Proportion

Let's solve the proportion: $\frac{3}{x} = \frac{9}{12}$

Step 1: Cross-multiply. Multiply the numerator of the first fraction by the denominator of the second fraction. Then, multiply the denominator of the first fraction by the numerator of the second fraction. Set these two products equal to each other: $3 \times 12 = x \times 9$

Step 2: Simplify the equation. $36 = 9x$

Step 3: Solve for $x$. Divide both sides by 9 to isolate $x$: $x = \frac{36}{9}$ $x = 4$

Solving with the Unknown in the Numerator

Let's look at another example: $\frac{x}{5} = \frac{8}{20}$

1. Cross-multiply: $x \times 20 = 5 \times 8$
2. Simplify: $20x = 40$
3. Solve for $x$: $x = \frac{40}{20} = 2$

Proportions Written with Colons

Sometimes proportions are written using colons instead of fractions, like this: If $2:5 = x:30$, find $x$.

The easiest way to solve this is to rewrite the ratios as fractions first: $\frac{2}{5} = \frac{x}{30}$

Now, cross-multiply: $2 \times 30 = 5 \times x$ $60 = 5x$ $x = \frac{60}{5} = 12$

Alternative Method: Equivalent Fractions

You don't always have to cross-multiply. If you can easily spot the relationship between the numbers, you can use equivalent fractions.

Take $\frac{x}{5} = \frac{8}{20}$.

Ask yourself: "How do I get from 20 to 5?" You divide by 4. Since you divided the denominator by 4, do the exact same thing to the numerator: $8 \div 4 = 2$ So, $x = 2$. Both methods give you the exact same correct answer!