Ratio & Proportion Calculator

Last updated: 2026-05-17

What is a proportion?

A proportion is an equation stating that two ratios are equal: A:B = C:D. This means A relates to B in exactly the same way that C relates to D. Given any three of the four values, the fourth is uniquely determined.

The fundamental relationship

A proportion can be written as a fraction equation:

$$\frac{A}{B} = \frac{C}{D}$$

Cross-multiplying gives the equivalent statement:

$$A \times D = B \times C$$

This cross-multiplication identity is the basis for solving any missing term:

• Find A: $A = \dfrac{B \times C}{D}$
• Find B: $B = \dfrac{A \times D}{C}$
• Find C: $C = \dfrac{A \times D}{B}$
• Find D: $D = \dfrac{B \times C}{A}$

Ratios vs. proportions

A ratio compares two quantities — for example, a recipe uses 3 cups of flour to 2 cups of sugar (ratio 3:2). A proportion is the statement that two such ratios are equal: doubling the recipe keeps the ratio at 3:2 — meaning 6:4 = 3:2.

Ratios describe a relationship between two quantities; proportions state that two ratios are equivalent.

Direct and inverse proportions

This calculator handles direct proportion — as one quantity increases, the other increases proportionally (same direction). If A doubles, C doubles when B and D are fixed.

Inverse proportion is different: as one quantity increases, the other decreases (opposite direction). For example, "4 workers finish a job in 6 days; how long for 8 workers?" This is an inverse proportion and requires a different formula ($A \times B = C \times D$ where the roles of the variables swap).

Applications

Scaling recipes — a recipe serving 4 uses 300 g of flour. The amount for 7 people follows from 300:4 = ?:7, giving $300 \times 7 / 4 = 525$ g.

Maps and scale drawings — a map scale of 1:50,000 means 1 cm on paper equals 50,000 cm (500 m) on the ground. Two cities 4.6 cm apart on the map are $4.6 \times 50{,}000 = 230{,}000$ cm = 2.3 km apart in reality.

Similar triangles — corresponding sides of similar triangles are proportional. If triangle A has sides 3, 4, 5 and triangle B has a corresponding side of 9, the other sides are $9 \times 4/3 = 12$ and $9 \times 5/3 = 15$.

Unit conversion — converting 45 miles to kilometers with the ratio 1 mile = 1.609 km: $45 / 1 = ? / 1.609$, so $? = 45 \times 1.609 = 72.4$ km.

Mixing and dilution — a cleaning solution diluted 1:20 (1 part concentrate to 20 parts water) needs, for 500 ml of solution, a concentrate amount of $500 / 21 \approx 23.8$ ml from 1:20 = ?:500.

Financial scaling — a business generating €180,000 revenue with 12 staff requires, at the same ratio, $12 \times 270{,}000 / 180{,}000 = 18$ staff for €270,000 of revenue ($12 / 180{,}000 = ? / 270{,}000$).

Setting up the proportion correctly

The two ratios must compare the same kinds of quantities in the same order. For "3 apples cost $2, how much do 7 apples cost?", the proportion is $3:2 = 7:?$ (apples:dollars on both sides). Mixing units, or reversing the order within one ratio while leaving the other unchanged, is a common source of error.

The divisor in the calculation cannot be zero: a proportion where B, C, or D is zero is undefined (division by zero), which signals that the problem setup needs to be reconsidered.

Frequently Asked Questions (FAQ)

What is a proportion?

A proportion states that two ratios are equal: A:B = C:D, or equivalently A/B = C/D. This means the relationship between A and B is the same as the relationship between C and D. Proportions are used in scaling recipes, maps, similar triangles, and unit conversion.

What is cross-multiplication?

Cross-multiplication is a shortcut to solve proportions. If A/B = C/D, then A × D = B × C. To find any missing term, isolate it using basic algebra. For example, if 2/4 = 3/D, then D = (4 × 3) ÷ 2 = 6.

What is the difference between a ratio and a proportion?

A ratio compares two quantities (e.g., 3:4). A proportion is an equation stating that two ratios are equal (e.g., 3:4 = 6:8). Ratios describe a relationship; proportions state that two such relationships are equivalent.