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.
- One formula solves every case: a × d = b × c. Rearrange to isolate whichever variable is missing.
- Always verify after solving - plug the answer back in and check that both cross-products are equal.
- Direct proportion (a/b = c/d) and inverse proportion (a × b = c × d) are different relationships. This calculator solves direct proportions only.
- Most proportion errors happen at setup, not calculation - confirm that the quantities on each side of the equation are in the same order.
- Proportions are behind everyday tasks: scaling recipes, reading maps, converting units, and working with similar triangles.
How to use our proportions 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.
Propotions explained to a beginner
Think of a road map. The legend says 1 cm = 10 km. You measure 4 cm between two cities and want to know the real distance. You know the scale stays constant, so: 1 cm to 10 km is the same relationship as 4 cm to ? km. Write it as a proportion: 1/10 = 4/?. Cross-multiply: 1 × ? = 10 × 4, so ? = 40 km.
That's all a proportion is - a statement that two scale relationships are equal. One side of the equation is your known scale (1 cm = 10 km). The other side uses the same scale with a new pair of numbers, one of which is unknown. Cross-multiplication is just a shortcut for eliminating the fractions so you can solve in one step.
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 for calculating proportions
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 |
In practice I find that most proportion errors come from setup, not arithmetic. When I work through a problem with proportions, I always write out the proportion explicitly with units on both sides before cross-multiplying - something like "250 g / 4 servings = x g / 10 servings".
That extra second of labeling makes it obvious if the quantities are in the wrong positions before any calculations happen.
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
Proportions appear in any situation where a known scale or rate is applied to a new quantity. The six domains below cover the most common setups - each one is a direct proportion where two ratios stay equal as the numbers change.
| 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 for solving proportions
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.
My personal method: if doubling one quantity should double the other (more workers, more bricks), it's direct. If doubling one quantity should halve the other (more workers, fewer days needed), it's inverse. Getting that question right before writing any equation saves a lot of frustration.
FAQs about proportion calculation
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).
Quiz: how well do you know proportions?
1. What does a proportion state?
2. Solve: 5:8 = c:24. What is c?
3. In the proportion a/b = c/d, what does cross-multiplication give you?
4. From the real-world applications table, which domain uses the proportion "1 cm : 5 km = 3.5 cm : d km"?
5. For "3 workers take 4 days; how long for 6 workers?", why is writing 3/4 = 6/d a mistake?