Proportion

A proportion is an equation that shows two ratios are equal. For example,

\frac{1}{2} = \frac{3}{6}

is a proportion because both ratios have the same value.

A proportion is a mathematical statement asserting that two ratios are equivalent. It takes the form $\frac{a}{b} = \frac{c}{d}$ , where $b \neq 0$ and $d \neq 0$ . The values $a$ and $d$ are called the extremes, while $b$ and $c$ are called the means. A key property of any proportion is that the cross products are equal: $a \times d = b \times c$ .

Key Formula

\frac{a}{b} = \frac{c}{d} \quad \Longrightarrow \quad a \times d = b \times c

Where:

$a$ = the first term (an extreme)
$b$ = the second term (a mean), must not be zero
$c$ = the third term (a mean)
$d$ = the fourth term (an extreme), must not be zero

Worked Example

Problem: If 3 notebooks cost $12, how much do 7 notebooks cost? Set up and solve a proportion.

Step 1: Write a ratio for the known relationship and set it equal to a ratio with the unknown.

\frac{3}{12} = \frac{7}{x}

Step 2: Cross-multiply to eliminate the fractions.

3 \times x = 12 \times 7

Step 3: Simplify the right side.

3x = 84

Step 4: Divide both sides by 3 to solve for

x

x = \frac{84}{3} = 28

Answer: 7 notebooks cost $28.

Why It Matters

Proportions come up constantly in everyday situations—scaling a recipe, converting units, reading maps, and calculating sale prices. In science, proportions help you work with concentrations, speed, and density. Once you can solve a proportion, you have a reliable tool for finding unknown quantities whenever two things are related by a constant ratio.

Common Mistakes

Mistake: Setting up the two ratios with mismatched units (e.g., notebooks/cost on one side and cost/notebooks on the other).

Correction: Keep the same quantity on top in both ratios. If the first ratio is notebooks over cost, the second ratio must also be notebooks over cost.

Mistake: Multiplying straight across instead of cross-multiplying (computing

a \times c

and

b \times d

Correction: Cross-multiplication means you multiply diagonally:

a \times d = b \times c

. This is what lets you solve for the unknown.

Related Terms

Ratio — The building block of every proportion
Proportional — Describes quantities that form a proportion
Direct Variation — A relationship where two quantities stay in proportion