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Rule of Three Calculator

Solve direct and inverse rule of three problems, find the missing term of any proportion, and scale a result when several quantities change at once. Every answer shows its formula and steps.

Detailed optionsStep-by-step guideWorked examples
Direct, inverse and compound proportions. Every step shown.
01 / THE CLASSIC RULE OF THREE

If A gives B, what does C give?

Enter three known values. The fourth value, X, updates as you type.

corresponds to (X)20

Example: 3 kg of apples cost 12. How much do 5 kg cost? X = 12 × 5 ÷ 3 = 20.

X in a direct proportion20
X = 12 × 5 ÷ 3 = 20
Unit value: 4 per 1 unit of A
Quick reference · same proportion, different values of C
C2.55102550
X102040100200
02 / SOLVE ANY PROPORTION

A : B = C : D — find the missing term.

Fill any three boxes and leave one empty. Works for recipes, scale drawings, maps and exchange rates.

Cross-multiplication: A × D = B × C. A map scale of 2 cm : 3 km means 9 km is 2 × 9 ÷ 3 = 6 cm on the map.

Missing C found6
C = 2 × 9 ÷ 3 = 6
2 : 3 = 6 : 9Both ratios = 0.666667
03 / MORE THAN TWO QUANTITIES

Compound rule of three

Scale a known result when several quantities change at once. Mark each one as direct or inverse.

Ask for each quantity: if only this changed, would the result go up (direct) or down (inverse)? More workers → fewer days is inverse; more walls → more days is direct.

Days after the changes6 days
X = 6 × 5/10 × 4/2 = 6
  • Workers: 5 → 10 · inverse× 0.5
  • Walls: 2 → 4 · direct× 2
YOUR ANIMATED GUIDE

Follow the rule of three, step by step.

These steps use your current inputs. Choose a step or play the walkthrough.

STEP 1 / 4
A : B = C : X
1234

Set up the proportion

In a direct proportion both quantities rise or fall together, so the ratio stays the same: A is to B as C is to X.

3 → 12 · 5 → X
Explore at your own pace

How to use this rule of three calculator

The rule of three finds a fourth value when three values of a proportion are known. If 3 kg cost 12, then 5 kg cost 12 × 5 ÷ 3 = 20. This page covers the classic direct and inverse rule of three, a four-term proportion solver for A : B = C : D, and a compound rule of three for problems where several quantities change at once. Every answer shows the substituted formula and a step-by-step guide.

What is the rule of three?

The rule of three is a shortcut for solving proportions. You know that A corresponds to B and want to know what C corresponds to. Because the relationship is proportional, the ratio between the pairs stays the same, so the unknown X can be found by multiplying two values and dividing by the third. It is used for prices, recipes, unit conversions, speeds, dosage tables, maps and scale models.

Direct versus inverse proportion

In a direct proportion both quantities move in the same direction: twice the weight costs twice as much, so X = B × C ÷ A. In an inverse proportion they move in opposite directions: twice as many workers need half the time, so the product stays constant and X = A × B ÷ C. Before calculating, ask whether increasing the first quantity should increase or decrease the answer. Choosing the wrong type is the most common mistake.

The compound rule of three

Some problems involve more than two related quantities, such as workers, hours per day and the size of a job. The compound rule of three treats each changing quantity separately. A direct quantity multiplies the known result by new ÷ old; an inverse quantity multiplies it by old ÷ new. Multiplying all of these factors together gives the scaled result, as long as each relationship is truly proportional.

Rule of Three formulas, explained

Direct rule of three

X = (B × C) ÷ A

A → B and C → X. Requires A ≠ 0. The unit value B ÷ A is the same for both pairs.

Inverse rule of three

X = (A × B) ÷ C

The product A × B is constant. Requires C ≠ 0.

Four-term proportion

A : B = C : D ⇔ A × D = B × C

Cross-multiply, then divide by the term next to the unknown. B and D must not be zero.

Compound rule of three

X = known × Π(new ÷ old) for direct × Π(old ÷ new) for inverse

Multiply one factor for every quantity that changes.

PUT THE FORMULA TO WORK

Worked examples you can try.

Load an example into the calculator, then change one value to explore.

EXAMPLE 01

Price for a different weight

3 kg of apples cost 12. Direct proportion: 5 kg cost 12 × 5 ÷ 3 = 20.

EXAMPLE 02

More workers, fewer days

4 workers paint a house in 6 days. Inverse proportion: 8 workers need 4 × 6 ÷ 8 = 3 days.

EXAMPLE 03

Scale a recipe

A recipe uses 250 g of flour for 4 people. For 10 people: 250 : 4 = C : 10 gives C = 625 g. Load it into the proportion solver.

EXAMPLE 04

Compound: workers and walls

5 workers build 2 walls in 6 days. 10 workers building 4 walls: 6 × 5/10 × 4/2 = 6 days.

MAKE THE NUMBERS MAKE SENSE

Common mistakes to avoid

  • Using a direct proportion when the quantities move in opposite directions, such as workers and time.
  • Mixing units, for example grams in one pair and kilograms in the other.
  • Applying the rule of three to relationships that are not proportional, such as tax brackets, fixed fees or speeds with a fixed delay.
A LITTLE MORE CLARITY

Frequently asked questions

What is the rule of three formula?

For a direct proportion where A corresponds to B, the value for C is X = B × C ÷ A. For an inverse proportion, X = A × B ÷ C.

Does the calculator support inverse proportions?

Yes. Choose Inverse proportion when one quantity rises as the other falls, such as workers and the days needed to finish a job.

How do I know whether a problem is direct or inverse?

Imagine doubling the first quantity. If the answer should also double, use direct. If the answer should halve, use inverse. Speed and travel time, workers and days, or pipes and filling time are typical inverse pairs.

Can the values be decimals or negative numbers?

Yes. The calculator accepts decimals and negative numbers. Zero is rejected wherever it would be a divisor, because a proportion cannot be scaled from zero.

What is the difference between the rule of three and a proportion?

They are the same idea. A proportion states that two ratios are equal, A : B = C : D. The rule of three is the method of finding the missing fourth term by cross-multiplication.

When should I use the compound rule of three?

Use it when more than one quantity changes between the known situation and the new one, for example the number of machines and the number of hours both change. Each factor must be proportional to the result on its own.

How are answers rounded?

Calculations keep full floating-point precision and show up to six decimal places. Simple fractions are also shown as an exact value where possible.

Methodology & assumptions

Calculations run in your browser using the formulas and assumptions explained on this page. Intermediate values retain full floating-point precision; displayed amounts are rounded. Undefined operations such as division by zero are rejected, every candidate solution is substituted back into the original equation, and numerical answers are refined to about 15 significant digits. Sharing an input link places those inputs in the URL.

Further reading:Cross-multiplication and the rule of three
KEEP EXPLORING

A little more perspective.

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