Rule of Three Calculator — Direct & Inverse Proportion
The rule of three finds an unknown value x when you know three related values — a, b, and c. It covers two cases: direct proportion (both quantities rise and fall together, like price and quantity) and inverse proportion (one rises while the other falls, like workers and days). Enter your three values, pick the type, and get the answer instantly.
The rule of three finds an unknown value x from three known values a, b, and c. Direct proportion: x = (b × c) / a — use when both quantities grow together (e.g. more items → higher cost). Inverse proportion: x = (a × b) / c — use when one grows as the other shrinks (e.g. more workers → fewer days). Example: if 3 kg cost $9, then 5 kg cost x = (9 × 5) / 3 = $15.
When to use this calculator
- Scaling a recipe: if 4 cups of flour make 24 cookies, how many cups make 60 cookies?
- Workforce planning: if 6 workers finish a job in 10 days, how many days will 15 workers need?
- Unit pricing: if 3 liters of paint cost $18, what do 7 liters cost?
- Speed and distance: if a car covers 120 km in 2 hours, how far will it go in 5 hours?
Worked Example — Direct Proportion (unit pricing)
- Problem: 3 kg of apples cost $7.50. How much do 8 kg cost?
- Set proportion type to Direct (price rises with quantity).
- Enter a = 3, b = 7.50, c = 8.
- Formula: x = (b × c) / a = (7.50 × 8) / 3 = 60 / 3 = 20.
How it works
2 min readWhat Is the Rule of Three?
The rule of three finds an unknown quantity x given three values — a, b, and c — that form a proportion:
a → b
c → x = ?"a corresponds to b, so c corresponds to x." The formula depends on the type of proportion.
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Direct vs Inverse Proportion at a Glance
| Situation | Type | Formula | Example |
|---|---|---|---|
| More items → higher cost | Direct | x = (b × c) / a | 3 kg = $9, so 5 kg = $15 |
| More workers → fewer days | Inverse | x = (a × b) / c | 6 workers × 10 days = 9 workers × x days |
| More speed → less travel time | Inverse | x = (a × b) / c | 60 km/h in 4h → 80 km/h in 3h |
| Scaling a recipe up | Direct | x = (b × c) / a | 2 cups → 12 cookies; 5 cups → 30 cookies |
| Currency conversion | Direct | x = (b × c) / a | 1 USD = 1.09 EUR; 250 USD = 272.50 EUR |
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How It Is Calculated
Direct Proportion
Both quantities move in the same direction — if one doubles, so does the other.
Formula: x = (b × c) / a
| a | b | c | x = (b × c) / a |
|---|---|---|---|
| 1 | 5 | 2 | 10 |
| 2 | 6 | 3 | 9 |
| 4 | 100 | 10 | 250 |
| 3 | 7.50 | 8 | 20 |
| 5 | 25 | 12 | 60 |
When to use Direct: price and quantity, distance and fuel, ingredient scaling, currency conversion.
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Inverse Proportion
Both quantities move in opposite directions — if one doubles, the other halves. The product a × b = c × x stays constant.
Formula: x = (a × b) / c
| a | b | c | x = (a × b) / c | Constant a×b |
|---|---|---|---|---|
| 6 | 10 | 15 | 4 | 60 |
| 8 | 15 | 12 | 10 | 120 |
| 4 | 9 | 6 | 6 | 36 |
| 10 | 6 | 4 | 15 | 60 |
| 5 | 12 | 3 | 20 | 60 |
When to use Inverse: workers and time, speed and travel time, pressure and volume (Boyle's Law).
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How to Set Up the Problem
1. Identify which two quantities are related (e.g., kilograms and price).
2. Write the pair you know fully: a and b.
3. Write the quantity whose pair you want to find: c.
4. Ask: if c is larger than a, will x be larger too? Yes → Direct. Will x be smaller? → Inverse.
5. Enter a, b, c and read x.
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Common Mistake
The most frequent error is choosing the wrong proportion type. Quick check: multiply a × b. If that product should equal c × x, it is inverse. If a/b should equal c/x (same ratio), it is direct.
Frequently asked questions
What is the rule of three and when do I use it?
The rule of three is a shortcut for solving proportions: given three values in a ratio (a, b, c), find the fourth (x). Use it whenever two quantities vary proportionally — pricing, scaling recipes, unit conversions, workforce problems, speed and distance, and hundreds of everyday calculations.
How do I know whether my problem is direct or inverse?
Ask: if the first quantity increases, does the second increase too? If yes → direct (more items → higher total cost). If the second decreases when the first increases → inverse (more workers → fewer days needed). Still unsure? Check whether the product a × b = c × x should stay constant; if so, it is inverse.
What do a, b, c, and x represent?
'a' and 'b' are a pair of corresponding known values (e.g., 3 kg costs $9). 'c' is a second known quantity of the same kind as 'a' (e.g., 5 kg). 'x' is the unknown — the value that corresponds to c under the same proportional relationship (e.g., the cost of 5 kg = $15).
Can I use this calculator for unit conversions?
Yes — unit conversion is always a direct proportion. To convert 75 miles to kilometers using 1 mile = 1.60934 km: set a = 1, b = 1.60934, c = 75. The result x = 120.7 km. The same method works for currency, weight, volume, and any other linear conversion.
Can I use decimals?
Decimals work perfectly — enter them with a period (e.g., 3.5). For fractions, divide first (¾ → 0.75) then enter the decimal. The formula handles any real numbers as long as 'a' is not zero.
What happens if I enter zero for 'a'?
The formula divides by 'a' (direct) or by 'c' (inverse), so neither can be zero. If you enter 0 for 'a', the calculator returns a division-by-zero error. Make sure both 'a' and 'c' represent non-zero quantities.
Is this the same as cross-multiplication?
Yes. In a direct proportion a/b = c/x, cross-multiplying gives a × x = b × c, so x = (b × c) / a — exactly the formula this calculator uses. Cross-multiplication and the rule of three are equivalent methods.
What is the difference between simple and compound rule of three?
The simple rule of three (what this calculator solves) involves exactly two proportional quantities. The compound rule of three involves three or more quantities varying simultaneously — for example, if both the number of workers and the hours per day change, you need two successive simple rule-of-three steps.
Worked example: if I drive 120 km in 2 hours, how far in 5 hours?
Direct proportion (more time → more distance). Set a = 2 (hours), b = 120 (km), c = 5 (hours). x = (120 × 5) / 2 = 300 km. Check: 300 / 5 = 60 km/h, same speed as 120 / 2 = 60 km/h. Correct.
Worked example: 8 pumps drain a tank in 15 hours. How long with 12 pumps?
More pumps → less time → inverse proportion. Set a = 8, b = 15, c = 12. x = (8 × 15) / 12 = 120 / 12 = 10 hours. Check: 8 × 15 = 120 = 12 × 10. The constant product confirms it is inverse.