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Rule of Three Calculator
Solve any direct or inverse proportion in seconds — fill in three values and we find the fourth.
What is the rule of three?
The rule of three is a classic technique for solving proportions. It works whenever you know three values that are linked by a constant ratio and you want to find the fourth. It comes in two flavors: direct, when both quantities grow or shrink together (more flour, more cookies), and inverse, when one grows as the other shrinks (more workers, less time). Despite the name, it isn't a rule about three numbers — it's a method that uses three known values to find one unknown.
How to use this calculator
Choose Direct or Inverse at the top, then fill in A, B and C. Read the relationship out loud as "A is to B as C is to X" — that mental check tells you whether the proportion really is direct or inverse before you trust the answer. The result and the worked-out steps appear instantly. You don't need to clear anything to switch modes; just toggle between Direct and Inverse and the answer updates.
When is it direct, when is it inverse?
Ask yourself: if A doubles, does B double too? If yes, the proportion is direct — use it for recipe scaling, unit conversions, prices per item, distance at a constant speed, and percentages. If doubling A makes B drop to half, the proportion is inverse — use it for travel time at higher speed, splitting work among more people, or pressure and volume of a gas. Some everyday examples are surprisingly inverse, so it's worth pausing to check.
Common examples
| Example | Type |
|---|---|
| If 4 cups of flour make 20 cookies, how many cookies do 7 cups make? | Direct |
| If 1 USD is worth 5.10 BRL, how many reais are 250 USD? | Direct |
| If a trip takes 3 hours at 80 km/h, how long does it take at 120 km/h? | Inverse |
| If 6 painters finish a wall in 8 days, how long do 12 painters take? | Inverse |
| If 250 g of ground beef costs $3.50, how much does 750 g cost? | Direct |
Frequently asked questions
What's the difference between direct and inverse proportion?
In a direct proportion, the two quantities move together: if one doubles, the other also doubles. In an inverse proportion, they move in opposite directions: if one doubles, the other halves. The math is different — direct uses (B×C)/A, inverse uses (A×B)/C — so picking the right type matters.
Can I use decimals or negative numbers?
Decimals work in any locale (use either a dot or a comma, depending on your browser). Negative numbers are accepted too, though most real-world proportions involve positive quantities. The only forbidden value is zero in the denominator — A in direct mode or C in inverse mode — because dividing by zero has no answer.
What if I need a compound rule of three (more than two variables)?
This calculator solves the simple version with one unknown linking two ratios. Compound problems — for example, workers, hours and days together — chain several proportions back-to-back. You can solve them by applying the simple rule of three step by step, fixing one variable at a time.
Why is currency conversion always direct?
Because the exchange rate is constant for a given moment: doubling the dollars you exchange always doubles the reais you receive. The relationship grows linearly, which is the definition of a direct proportion.
Why is travel time inverse to speed?
Because the distance stays the same. If you go twice as fast, you take half as long; the product of speed and time is constant. Whenever the product of two quantities is fixed, the relationship between them is inverse.
Is the rule of three the same as cross multiplication?
Yes — for direct proportions. Cross multiplication is the operational shortcut: write A/B = C/X, multiply across, and solve for X. The rule of three is the broader concept, including the inverse case where you cross-multiply differently.