Rule of Three Calculator
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The rule of three finds the fourth value of a proportion when you already know the other three. If 3 oranges cost 12, what do 5 cost? If a trip takes 3 hours at 80 km/h, how long does it take at 120? Pick the type — direct, inverse or compound — fill in the fields, and the calculator returns X along with the proportion written out so you can check the reasoning, not just the number.
Direct and inverse: the only decision you have to make
Two quantities are directly proportional when they move together: more oranges, more money; more square meters, more paint. They are inversely proportional when one rises as the other falls: more workers, fewer days; higher speed, less time.
The test takes five seconds. Ask yourself: if I double the first quantity, does the second double, or does it halve? Double means direct. Halve means inverse. That single decision is what separates a right answer from a wrong one, because the arithmetic afterwards is mechanical.
Set the problem out in two rows, one quantity per column, with matching units in each column. Direct: cross-multiply. Inverse: multiply along each row instead, which is the same as flipping one of the fractions. Then check that the answer points the way you expected — bigger or smaller than the value you started from.
- Direct: quantity bought and total price, distance and fuel, area and paint, hours worked and pay.
- Inverse: speed and travel time, number of workers and days, number of taps and filling time.
- Not proportional at all: side and area of a square, compound interest and time, study hours and exam score.
Simple rule of three, worked step by step
Direct example. Three oranges cost 12; what do five cost? Write "3 → 12" above "5 → X". Both quantities grow together, so cross-multiply: 3 × X = 12 × 5, giving 3X = 60 and X = 20. In the calculator that is A = 3, B = 12, C = 5, and the formula applied is X = B × (C ÷ A) = 20.
Inverse example. A trip takes 3 hours at 80 km/h; how long at 120 km/h? Write "80 → 3" above "120 → X". Speed goes up while time goes down, so do not cross-multiply — multiply along the rows: 80 × 3 = 120 × X, giving X = 240 ÷ 120 = 2 hours. In the calculator, choose the inverse mode with A = 80, B = 3, C = 120, and the formula becomes X = B × (A ÷ C) = 2.
Notice that the two modes differ by exactly one thing: whether the ratio is C ÷ A or A ÷ C. Everything else is identical, which is why the calculator reports that ratio as a separate row — it is the multiplier applied to B.
| Problem | Set-up | Type | Answer |
|---|---|---|---|
| 4 servings need 600 g of rice; 6 servings? | 4 → 600 / 6 → X | Direct | X = 900 g |
| 3 hours at 80 km/h; how long at 120 km/h? | 80 → 3 / 120 → X | Inverse | X = 2 h |
| 3 taps fill the tank in 40 min; 5 taps? | 3 → 40 / 5 → X | Inverse | X = 24 min |
| One can of paint covers 60 m²; 140 m²? | 60 → 1 / 140 → X | Direct | X = 2.33 → buy 3 |
| Map scale 1:50,000; 7 cm on paper? | 1 cm → 0.5 km / 7 → X | Direct | X = 3.5 km |
| 6 machines finish in 8 days; 4 machines? | 6 → 8 / 4 → X | Inverse | X = 12 days |
Compound rule of three
In a compound problem, two quantities change at once. The classic: six workers, working 8 hours a day, build a wall in 5 days. How many days would four workers need at 10 hours a day?
Isolate the unknown (days) and compare each other quantity with it separately, never with each other. Workers versus days is inverse: fewer workers, more days. Hours per day versus days is also inverse: longer shifts, fewer days.
Flip the fraction for every inverse quantity. Workers give 6 ÷ 4 = 1.5 instead of 4 ÷ 6, and hours give 8 ÷ 10 = 0.8 instead of 10 ÷ 8. Multiply the known 5 days by both: X = 5 × 1.5 × 0.8 = 6 days. The long check agrees: the wall takes 6 × 8 × 5 = 240 worker-hours, and 4 workers at 10 hours a day supply 40 per day, so 240 ÷ 40 = 6.
The calculator uses one formula for every compound case: X = B × (C ÷ A) × (E ÷ D), treating both quantities as directly proportional. For an inverse quantity you simply enter its pair the other way round — which is precisely the classroom instruction "for an inverse quantity, flip the fraction". For the workers example, enter B = 5, A = 4, C = 6, D = 10, E = 8.
Frequently asked questions
What is the formula for the rule of three?
For a direct proportion written as A is to B as C is to X, the formula is X = B × C ÷ A. For an inverse proportion with the same layout, it becomes X = A × B ÷ C. The calculator applies exactly these: it multiplies B by the ratio C ÷ A in direct mode and by A ÷ C in inverse mode.
How do I know whether a problem is direct or inverse?
Ask whether doubling the first quantity doubles the second or halves it. Doubling means direct, halving means inverse. Buying twice as many oranges costs twice as much (direct); twice as many workers finish in half the time (inverse).
How does the compound rule of three work?
Isolate the unknown quantity, compare every other quantity with it one at a time, decide direct or inverse for each, then multiply the known value by all the ratios. Here the formula is X = B × (C ÷ A) × (E ÷ D); for an inverse quantity, enter its two values in reverse order.
Can the rule of three be used for percentages?
Yes — a percentage is just a direct rule of three with 100 fixed in one column. To find 15% of 260: 100 is to 260 as 15 is to X, giving 39. To find what percentage 39 is of 260: 260 is to 100 as 39 is to X, giving 15.
Educational tool. Examples involving medication, dosage, structural or financial figures are illustrative only and do not replace professional advice. Always convert units before setting up the proportion.
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Also available in Portuguese: Calculadora de Regra de 3