Rule of Three

The rule of three solves a proportion when three values are known and the fourth is missing. It is useful for scaling a price, a recipe, a map distance or a simplified work time. Unlike a percentage, it looks for a concrete quantity in a presumed proportional relationship, so the answer remains expressed in the original unit instead of only as a rate. That makes it practical for everyday checks where the unit still matters.

Formula used

A / B = C / D, therefore D = B × C / A

For a direct proportion, the engine uses result = B × C ÷ A. For an inverse proportion, used for example with people and simplified duration, it uses result = A × B ÷ C. The relationship type must be chosen before interpreting the result.

Worked example and result reading

Situation

Example matching the default values: A = 2, B = 10 and C = 5 in direct proportion. The result is 10 × 5 ÷ 2 = 25.

Interpretation

The result is the value that preserves the selected proportion. If 2 units correspond to 10, then 5 units correspond to 25 when the ratio stays constant. If the relationship is not linear, the calculation is only an approximation.

Detailed calculation guide

Direct proportion

When more of A leads to more of the result, direct calculation fits. This is the case for a constant unit price or an ingredient amount scaled to more servings.

Inverse proportion

When more of C reduces the result, inverse mode can help. It fits simplified effort-sharing examples, but not cases with coordination, fatigue or capacity limits.

Same unit

A and C must describe the same quantity: items with items, people with people, centimeters with centimeters. Mixing units produces a result that cannot be read properly.

Compared with ratio

A ratio describes the relationship between two quantities. The rule of three uses that relationship to find a missing value in a practical situation.

Prices and quantities

The calculation is reliable if the unit price does not change. When volume discounts, fixed costs or free shipping appear, a simple proportion is not enough.

Quick check

Before accepting the result, check whether it increases or decreases in the expected direction. This catches many A, B and C order mistakes.

Key takeaways

The rule of three finds a fourth proportional value.
Direct and inverse modes answer different questions.
Units must stay consistent across compared values.
A non-linear relationship needs a richer model.

Decision checklist

The proportion type is selected before calculating.
A and C use the same unit.
B is the value associated with A.
The result moves in the expected direction.
Tiers, fixed costs and real-world limits are checked separately.

Result checks before use

Same unit family

Check that A and C describe the same thing: two quantities, two distances, two numbers of people or two durations. B and D must also share the final unit. Mixing grams and kilograms, hours and minutes or prices and quantities breaks the proportion before the formula is applied.

Direct or inverse relationship

Before calculating, ask whether the result should rise or fall. If more quantity means more price, the proportion is direct. If more workers reduce project duration, the proportion is inverse. This choice determines the formula and prevents inverted results.

Cross multiplication

After the calculation, verify consistency. In a direct proportion, A × D should equal B × C. In an inverse proportion, C × D should equal A × B. If the gap is large, review values, units and rounding.

Proportional context

The rule of three assumes a linear relationship. It works for constant unit prices, recipes, scales and simple dosages. It becomes fragile with fixed fees, subscriptions, thresholds, progressive discounts or declining rates because the value no longer changes strictly in the same ratio.

Scenarios to compare

Unit price

If 2 items cost €10, then 5 items cost €25 at a constant price.

Recipe

250 g for 4 servings becomes 375 g for 6 servings.

Map

If 2 cm represent 5 km, then 8 cm represent 20 km.

Shared work

If 4 people finish in 12 days, 6 people give 8 days in a simplified inverse model.

Common mistakes to avoid

Using direct proportion when the relationship is inverse.
Swapping the known value B with the value to recover.
Comparing different units without conversion.
Applying the formula to a price with volume discounts.
Rounding too early in a recipe or quote.

What to know before using the result

The rule of three assumes a stable proportion. It ignores price tiers, fixed fees, discounts, diminishing returns, professional rounding and stock limits. A simplified inverse relation does not always describe real work.

Frequently asked questions

What is the rule of three formula?

For direct proportion: result = B × C ÷ A.

When should I use inverse proportion?

When increasing one quantity reduces the other, such as simplified shared work time.

How is it different from a percentage?

A percentage expresses a part out of 100; the rule of three recovers a missing value.

Why does my result look wrong?

Check the proportion direction, units and order of the three entered values.

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