CodeKitHubWhat Is This Tool?
The rule of three is a classic technique for solving proportions: given that A relates to B the same way C relates to an unknown X, it finds X by cross multiplication. In direct proportion, X = (B × C) ÷ A. In inverse proportion — where increasing one quantity decreases the other — X = (A × B) ÷ C instead.
It's one of the most widely taught pieces of practical math because it turns everyday 'if this, then what' questions into a single formula: unit prices, recipe scaling, map distances, speed and time, and currency conversion all reduce to a rule of three.
Why Use It?
- Solves both direct and inverse proportions — pick the mode that matches your problem.
- Shows the cross-multiplication step, not just the answer, so you can check the method.
- Instant results as you type — no page reload, no sign-up.
- Useful for shopping comparisons, recipe scaling, map scales, speed/time problems and more.
- Runs entirely in your browser — nothing you enter is sent anywhere.
How to Use
- Choose direct or inverse proportion depending on how your two quantities relate.
- Enter A and B, the known pair (for example, 3 items cost $12).
- Enter C, the known value of the new situation (for example, 5 items).
- Read off X, the answer, calculated instantly by cross multiplication.
Example
Input
Direct proportion: 3 items cost $12 (A=3, B=12). How much do 5 items cost (C=5)?Output
X = $20X = (B × C) ÷ A = (12 × 5) ÷ 3 = 60 ÷ 3 = 20. As the number of items increases, the price increases proportionally, so this is direct proportion.
Direct vs. inverse proportion at a glance
| Type | Formula for X | Example |
|---|---|---|
| Direct | X = (B × C) ÷ A | 3 apples cost $6; 5 apples cost X |
| Inverse | X = (A × B) ÷ C | 4 painters take 6 days; 3 painters take X days |
Related tools
The rule of three is a form of percentage and ratio math, so the percentage calculator handles the closely related question of 'what percent is X of Y'. For converting the resulting quantities between measurement systems, the unit converter is a natural companion.
Frequently Asked Questions
What is the rule of three?
The rule of three (also called cross multiplication) solves a proportion of the form A:B = C:X for the unknown X, using three known values. It's called 'of three' because you need three known numbers to find the fourth. In direct proportion, X = (B × C) ÷ A; in inverse proportion, X = (A × B) ÷ C.
What's the difference between direct and inverse proportion?
In direct proportion, both quantities move in the same direction: if one doubles, so does the other (more items cost more money). In inverse proportion, they move in opposite directions: if one doubles, the other halves (more workers finish a job in less time). Picking the wrong mode gives a completely wrong answer, so identify which kind of relationship you have before calculating.
Can you give real-life examples of the rule of three?
Unit price shopping: if a 500 g bag costs $2.50, how much does a 750 g bag cost (direct)? Recipe scaling: if a recipe for 4 people needs 200 g of flour, how much for 6 people (direct)? Map scale: if 1 cm on a map equals 5 km, how many km does 8 cm represent (direct)? Work rate: if 4 workers finish a job in 6 days, how long would 3 workers take (inverse)?
How do I know if a problem is direct or inverse proportion?
Ask what happens to the second quantity if the first one increases. If it also increases (more hours worked, more pay earned), it's direct. If it decreases (more machines, less time needed to finish), it's inverse. Speed, time and distance problems are a common source of inverse proportion: at a higher speed, a fixed distance takes less time.
Is my data stored anywhere?
No. The calculation runs entirely in your browser with JavaScript; nothing you enter is sent to a server, stored or logged.