Quick answer
To solve a proportion a:b = c:d for the missing value, use cross-multiplication: a × d = b × c. Isolate the unknown by dividing both sides by its coefficient. Example: 3:4 = 9:d → 3d = 36 → d = 12.
How to use this calculator
Select which variable you want to solve for (A, B, C, or D). Enter the three known values. The calculator solves for the missing value using cross-multiplication and verifies the result.
What is a proportion?
A proportion is an equation stating that two ratios are equal:
$$\frac{a}{b} = \frac{c}{d} \iff a \times d = b \times c$$
A proportion describes a scaling relationship: if a/b = c/d, then multiplying a and b by the same factor k gives c and d. Proportions appear wherever quantities scale together — recipes, maps, unit conversions, similar triangles, and financial ratios.
Cross-multiplication method
Given a/b = c/d, multiply both sides by b × d to eliminate the denominators:
a × d = b × c
To solve for each variable:
$$d = \frac{b \times c}{a}$$
$$a = \frac{b \times c}{d}$$
The other two: b = a×d/c and c = a×d/b. In every case, the missing variable equals the product of the two values on the opposite diagonal, divided by the remaining known value.
Worked examples
Example 1: find d given 3/4 = 9/d
$$\frac{3}{4} = \frac{9}{d}: \quad d = \frac{4 \times 9}{3} = 12$$
Cross-multiply: 3d = 4 × 9 = 36. Divide by 3: d = 12. Verify: 3 × 12 = 36 = 4 × 9 ✓.
Example 2: find c given 5/8 = c/24
$$\frac{5}{8} = \frac{c}{24}: \quad c = \frac{5 \times 24}{8} = 15$$
Cross-multiply: 8c = 5 × 24 = 120. Divide by 8: c = 15. Verify: 5 × 24 = 120 = 8 × 15 ✓.
Example 3: recipe scaling
A recipe uses 250 g flour for 4 servings. For 10 servings: 250/4 = x/10 → 4x = 2500 → x = 625 g.
Example 4: map scale
1 cm on a map = 5 km in reality. Distance on map = 3.5 cm. Real distance: 1/5 = 3.5/d → d = 17.5 km.
| Proportion | Missing | Calculation | Answer |
|---|---|---|---|
| 2:5 = 8:? | d | d = 5×8/2 | 20 |
| ?:3 = 10:15 | a | a = 3×10/15 | 2 |
| 7:b = 14:6 | b | b = 7×6/14 | 3 |
| 4:9 = c:27 | c | c = 4×27/9 | 12 |
Direct vs inverse proportion
This calculator solves direct proportions where a/b = c/d — as one quantity increases, the other increases proportionally:
$$y = kx \implies \frac{y_1}{x_1} = \frac{y_2}{x_2}$$
Inverse proportion means as one quantity increases, the other decreases: a × b = c × d (constant product). For example, 4 workers take 6 days; 8 workers take 3 days. Inverse proportion is a different relationship and is not solved by this calculator.
Real-world applications
| Domain | Example proportion |
|---|---|
| Cooking | 250 g : 4 servings = x g : 10 servings |
| Maps | 1 cm : 5 km = 3.5 cm : d km |
| Finance | £3 interest : £100 = £x : £850 |
| Speed | 60 km : 1 hr = 450 km : h hr |
| Geometry | 3 cm : 5 cm = 9 cm : x cm (similar triangles) |
| Chemistry | 2 mol H₂ : 1 mol O₂ = x mol H₂ : 5 mol O₂ |
Common mistakes
Setting up the proportion backwards
Proportion direction matters. For "3 workers take 4 days; how long for 6 workers?" the correct proportion is not 3/4 = 6/d (that's inverse proportion). Recognise whether the quantities scale in the same direction (direct) or opposite directions (inverse) before writing the proportion.
Cross-multiplying the wrong pair
In a/b = c/d, the cross-multiplication is a×d = b×c (diagonals). A common error is to multiply a×c or b×d instead.
Frequently asked questions
How do you solve a proportion?
Cross-multiply: a × d = b × c. Isolate the unknown by dividing by its coefficient.
What is a proportion?
An equation stating two ratios are equal: a/b = c/d. Describes quantities that scale together.
What is the difference between ratio and proportion?
A ratio (a:b) compares two quantities. A proportion is an equation equating two ratios (a:b = c:d).