Rule of Three Calculator
The rule of three is probably the most useful math operation in everyday life — we use it to calculate percentages, adjust recipes, scale budgets, convert currencies, calculate task times, and much more. It consists of solving a proportion when you know 3 values and need the fourth. There are two types: direct (if one quantity grows, the other grows proportionally: more kilos of meat = more money) and inverse (if one grows, the other decreases: more workers = less time). This calculator solves both using the correct formulas: X = (B x C) / A for direct and X = (A x B) / C for inverse.
When to use this calculator
- Adjust a recipe from 4 servings to 6 people (direct).
- Calculate how much X kg of a product costs knowing the price of Y kg (direct).
- Figure out how long 6 people take on a task that 4 people complete in 6 hours (inverse).
- Convert currencies with a fixed exchange rate (direct).
- Calculate a percentage (every percentage is a rule of three).
Example: scaling a cookie recipe from 12 to 18 cookies
- Context: the recipe makes 12 cookies and needs 500 g of flour. How much flour for 18?
- It's direct: more cookies → more flour.
- Setup:
12 cookies → 500 g flour.18 cookies → X g flour. - Formula:
X = (B x C) / A = (500 x 18) / 12. - Calculation:
(500 x 18) / 12 = 9000 / 12 = 750 g. - Verification: if we double the recipe (24 cookies), we'd need 1000 g. Proportional.
How it works
2 min readDirect Rule of Three
When two quantities are directly proportional: if one grows, the other grows in the same proportion.
A → B
C → X
X = (B x C) / AIntuition: the ratio B/A is constant. If A is multiplied by k, B must also be multiplied by k.
Examples of Directly Proportional Quantities
Step-by-Step Example
If 3 kg of chicken costs $12, how much do 5 kg cost?
3 kg → $12
5 kg → X
X = (12 x 5) / 3
X = 60 / 3
X = $20Inverse Rule of Three
When two quantities are inversely proportional: if one grows, the other decreases.
A → B
C → X
X = (A x B) / CIntuition: the product A x B is constant. If A is multiplied by k, B is divided by k.
Examples of Inversely Proportional Quantities
Step-by-Step Example
If 4 people paint a house in 6 hours, how long do 8 people take?
4 people → 6 hours
8 people → X
X = (4 x 6) / 8
X = 24 / 8
X = 3 hoursWith double the people, it takes half the time.
How Do I Know If It's Direct or Inverse?
Think of the extreme case:
1. Ask: if A increases a lot, what happens to B?
2. If B also increases → direct.
3. If B decreases → inverse.
Quick Reference Table
| Situation | Type |
|---|---|
| Kg of apples and price | Direct |
| Study hours and learning | Direct (normally) |
| Workers and task time | Inverse |
| Car speed and travel time | Inverse |
| Number of builders and meters built | Direct |
| Builders and days for the same project | Inverse |
Application in Percentages
Every percentage calculation is a rule of three with base 100:
100 → total value
X% → partial valueExample: What is 15% of 2000?
100 → 2000
15 → X
X = (2000 x 15) / 100 = 300Scaling Recipes
Recipe for 4 people: 200 g of rice. For 6 people:
4 people → 200 g
6 people → X
X = (200 x 6) / 4 = 300 gCommon Mistakes
1. Using direct when it's inverse: 'if 6 workers take 4 days, 12 workers take 8 days' — wrong! The inverse says 2 days.
2. Cross-multiplying incorrectly: in A → B, C → X, the direct formula is X = B x C / A, not A x C / B.
3. Mixing units: if A is in minutes and C is in hours, convert first.
4. Using rule of three where it doesn't apply: not all relationships are linear. For example, fuel consumption vs speed is NOT linear.
5. Confusing proportion with percentage: '20% of the class are women' is solved with a rule of three (base 100).
Frequently asked questions
How do I know if I should use direct or inverse rule of three?
Think of the extreme case. If A increases 10 times, what happens to B? If B also increases 10 times, it's direct. If B is divided by 10, it's inverse. Clear direct examples: price-quantity, time-salary, recipe-servings. Clear inverse examples: workers-task time, speed-travel time (fixed distance).
Is the rule of three the same as a proportion?
Very related. A proportion is the equality of two ratios: a/b = c/d. The rule of three is the procedure for solving a proportion when one of the 4 values is unknown. Every rule of three is based on a proportion, and every proportion is solved with a rule of three.
Can the rule of three be used for percentages?
Absolutely yes. It's the mathematical basis of every percentage calculation. 'What is 15% of 2000?' is solved with: 100 → 2000, 15 → X, so X = (15 x 2000) / 100 = 300. All discounts, increases, variations, and commissions are rules of three disguised as percentages.
What is the compound rule of three for?
For when there are more than two quantities involved. Example: workers + hours/day + total days. It extends the logic of the simple rule by applying multiple direct and inverse relationships in sequence.
When should I NOT use the rule of three?
When the relationship is non-linear. Examples: fuel consumption vs speed (parabolic), compound interest (exponential), air resistance vs speed (quadratic), population growth (exponential). The rule of three assumes pure proportionality. If doubling the input doesn't exactly double the output, don't apply it.
How do I convert between currencies with the rule of three?
Simple direct. If 1 USD = 0.92 EUR, then X EUR = (0.92 x 500) / 1 = 460 EUR to convert 500 USD. To go from EUR to USD: reverse the operation. Always make sure to use the correct exchange rate for your transaction type.
Does the rule of three work with decimals and negatives?
With decimals yes, no problem: 3.5 kg → $7.50, X kg → $20.00 → X = (3.5 x 20.00) / 7.50 = 9.33 kg. With negatives: mathematically yes, but in real contexts it rarely makes sense (negative kilos or time don't exist). If negatives appear, you're probably modeling differences or variations, not absolute quantities.
How do I adjust a recipe from 4 to 5 people?
Direct rule of three. If the original recipe uses 200 g of rice for 4, for 5 people: X = (200 x 5) / 4 = 250 g. For each ingredient do the same calculation. You can also compute the scale factor 5/4 = 1.25 and multiply each ingredient by that number.