Rule of Three Calculator — Direct and Inverse Proportions
The rule of three solves proportions: given three known values, it finds the fourth. If A relates to B, then C relates to X — and the calculator finds X instantly, showing the step-by-step solution. It handles both direct proportions (both quantities grow together, like ingredients in a recipe) and inverse proportions (one grows while the other shrinks, like workers versus time to finish a job).
This is one of the most used pieces of everyday math: converting currencies, scaling recipes, computing percentages, adjusting doses by weight, estimating travel times, and pricing by quantity all reduce to a rule of three. Understanding whether the relationship is direct or inverse is the only real decision — the calculator takes care of the arithmetic and lays out each step so the logic stays clear.
How it works
Direct proportion: A/B = C/X, so X = (B × C) ÷ A. Inverse proportion: A × B = C × X, so X = (A × B) ÷ C. Use direct when both quantities increase together, and inverse when one increases as the other decreases.
Use cases
- Scaling a recipe: if 300 g of flour makes 12 cookies, how much for 20?
- Converting units or currencies from a known reference rate
- Calculating percentages, discounts, and proportional price adjustments
- Estimating time: if 3 machines take 8 hours, how long do 4 machines take? (inverse)
- Adjusting proportional doses or dilutions in labs and cleaning products
- Solving school proportion exercises with a step-by-step explanation
Frequently asked questions
What is the rule of three and how does it work?
The rule of three is a method for finding an unknown value in a proportion when three values are known. You arrange the values as A is to B as C is to X, cross-multiply, and solve: X = (B × C) ÷ A for direct proportions. For example, if 5 kg costs $20, then 8 kg costs X = (20 × 8) ÷ 5 = $32. It works because proportional quantities keep a constant ratio.
What is the difference between direct and inverse proportion?
In a direct proportion, both quantities move in the same direction: more kilograms, more cost. In an inverse proportion, they move in opposite directions: more workers, less time to finish the same job. The formulas differ — direct uses X = (B × C) ÷ A while inverse uses X = (A × B) ÷ C — so identifying the type correctly is the key step before calculating.
How do I know if a problem is inverse proportion?
Ask what happens to the second quantity when the first one increases. If it decreases, the proportion is inverse. Classic examples: speed versus travel time (faster means less time), number of workers versus days to finish (more workers means fewer days), and flow rate versus filling time. If both quantities rise together, like quantity and price, the proportion is direct.
Can I calculate percentages with the rule of three?
Yes — percentage problems are direct proportions where one pair is the total and 100%. To find what percent 30 is of 120, set 120 → 100% and 30 → X, giving X = (30 × 100) ÷ 120 = 25%. The same setup answers questions like discounts, tax amounts, and grade percentages, which is why the rule of three is often taught right alongside percentages.
What is a compound rule of three?
The compound rule of three involves more than two quantities, such as workers, hours per day, and days to complete a job. Each quantity is classified as directly or inversely proportional to the unknown, and the ratios are multiplied together accordingly. This calculator focuses on the simple rule of three with two quantities, which covers the vast majority of everyday problems.