Rule of Three Calculator

A rule of three calculator finds a missing value when two pairs of quantities follow the same proportional relationship. It can solve direct and inverse proportion problems, check an existing proportion and show the formula used for the result.

The method is useful for prices, recipes, percentages, measurements, work rates and journey planning. The arithmetic is simple, but a reliable result depends on choosing the correct relationship, keeping corresponding quantities aligned and using consistent units.

What the calculator can do

The calculator supports four-value proportion layouts labelled A, B, C and D. Any one value may be left empty.

• solve a direct proportion;
• solve an inverse proportion;
• calculate A, B, C or D;
• check a proportion when all four values are entered;
• display the formula and substituted values;
• show a scale-factor comparison;
• add optional labels and units;
• format the result with automatic or fixed decimal precision;
• swap rows or columns without retyping the values.

The calculator uses the full numeric value internally and rounds only the displayed result. This reduces avoidable rounding error.

How to use the Rule of Three Calculator

1. Select Direct proportion or Inverse proportion.
2. Enter the three known values.
3. Leave the value to be calculated empty, or press the X button beside that field.
4. Add labels and units when they make the calculation easier to interpret.
5. Choose automatic precision or a fixed number of decimal places.
6. Press Calculate.
7. Review the formula and verification rather than relying only on the highlighted answer.

If all four fields contain values, the calculator switches to verification mode instead of replacing one of them.

The core proportion formulas

A proportion states that two ratios or two products represent the same relationship.

Direct proportion

For direct proportion, the calculator uses:

A : B = C : D

This can be verified by cross-multiplication:

A × D = B × C

If D is missing:

D = (B × C) ÷ A

The same equation can be rearranged for any other missing value:

• A = (B × C) ÷ D
• B = (A × D) ÷ C
• C = (A × D) ÷ B
• D = (B × C) ÷ A

Inverse proportion

For inverse proportion, the paired products remain constant:

A × B = C × D

If D is missing:

D = (A × B) ÷ C

The rearranged forms are:

• A = (C × D) ÷ B
• B = (C × D) ÷ A
• C = (A × B) ÷ D
• D = (A × B) ÷ C

OpenStax describes direct variation as a relationship in which one quantity is a constant multiple of another, while inverse variation keeps a constant product. These definitions are more dependable than deciding from the size of the numbers alone.

Direct proportion explained

Two quantities are directly proportional when they increase or decrease together at a constant rate. The ratio between corresponding values remains the same.

Common examples include:

• quantity and total price at a fixed unit price;
• servings and ingredient amounts;
• hours worked and pay at a fixed hourly rate;
• distance and time at a constant speed;
• material quantity and covered area under the same conditions.

A unit-rate check reaches the same answer: £12 ÷ 3 = £4 per notebook, then 5 × £4 = £20.

Inverse proportion explained

Two quantities are inversely proportional when one increases while the other decreases so that their product remains constant.

Typical examples include:

• workers and completion time for a fully divisible task;
• speed and journey time over a fixed distance;
• machines and production time under equal performance;
• flow rate and filling time for a fixed volume;
• people and individual share of a fixed amount.

The mathematical result assumes equal productivity and no coordination delay. Real work may not scale perfectly because tools, space and task order can limit parallel work.

How to recognise the correct relationship

Ask what should happen to the second quantity when the first quantity increases.

• If both increase or both decrease at the same rate, test a direct proportion.
• If one increases while the other decreases and the paired product stays constant, test an inverse proportion.
• If neither condition holds, a rule of three may not be appropriate.

A useful second check is to calculate either the ratio or the product:

• direct relationship: compare B ÷ A with D ÷ C;
• inverse relationship: compare A × B with C × D.

The fact that one answer becomes smaller does not by itself prove inverse proportion.

Price calculation example

Six kilograms of fruit cost £16.80. What is the price of nine kilograms at the same rate?

Place the quantities in consistent columns:

• A = 6 kg;
• B = £16.80;
• C = 9 kg;
• D = unknown.

The relationship is direct:

D = (16.80 × 9) ÷ 6 = £25.20

The result is valid only if the unit price remains £2.80 per kilogram. Delivery charges, minimum-order fees or volume discounts must be handled separately.

Recipe scaling example

A recipe for four people uses 300 grams of flour. For ten people:

D = (300 × 10) ÷ 4 = 750 grams

This is a reasonable proportional estimate for flour. It does not prove that cooking time, seasoning or baking-vessel size should also be multiplied by 2.5. Each part of a real recipe may need a separate decision.

Speed and journey-time example

A fixed journey takes five hours at an average speed of 60 mph. At 75 mph:

D = (60 × 5) ÷ 75 = 4 hours

This is an inverse relationship because the distance remains fixed. The estimate assumes the stated average speed can be maintained for the entire journey and does not account for traffic, stops or route changes.

Calculate percentages with the rule of three

A percentage can be treated as a direct proportion based on 100.

To find what percentage 45 represents of 250:

250 : 100 = 45 : D

D = (100 × 45) ÷ 250 = 18%

The positions must stay consistent: the total corresponds to 100%, and the partial amount corresponds to the unknown percentage.

Labels, units and decimal input

Optional labels help identify the meaning of each column. For a shopping example, the left column might be labelled “Weight” with the unit “kg”, while the right column is “Cost” with the unit “£”.

Comparable values must use the same unit. Convert 2 kg to 2,000 g before comparing it with 5,000 g, or convert both values to kilograms.

The English version normally uses a decimal point, such as 2.5 or 16.80. The calculator interprets the local number format and lets users choose the number of displayed decimal places.

How proportion verification works

When all four values are entered, the calculator checks whether they satisfy the selected relationship.

For direct proportion:

A × D should equal B × C.

For inverse proportion:

A × B should equal C × D.

The tool also calculates an expected value for comparison where possible. A small difference may result from values that were rounded before entry.

Zero, negative numbers and numeric limits

Zero can be valid in some proportional situations, but it cannot be used as a divisor. If the selected formula requires division by zero, the calculator returns an error rather than an invented result.

Negative values are mathematically possible, but their real-world meaning must be considered. A negative temperature change may be meaningful; a negative number of products, workers or recipe servings usually is not.

Very large or very small values can also exceed normal browser-number precision. The calculator reports values within the supported numeric range, but highly specialised scientific work may require arbitrary-precision software.

When the rule of three should not be used

A simple proportion is inappropriate when the rate is not constant.

• a taxi fare with a fixed starting charge;
• shipping with a base fee plus a per-kilogram fee;
• quantity discounts or tiered pricing;
• progressive tax rates;
• compound interest;
• workers with different productivity;
• tasks that cannot be divided evenly;
• acceleration or changing average speed;
• equipment operating near a capacity limit;
• recipes in which time or heat does not scale with quantity.

In these cases, separate the fixed and variable components or use a model designed for the actual relationship.

A reliable calculation workflow

1. Write down what each quantity represents.
2. Convert comparable values to consistent units.
3. Decide whether the relationship is direct, inverse or not proportional.
4. Place the same type of quantity in the same column.
5. Enter three known values and leave one field empty.
6. Review the displayed formula.
7. Check the result with a unit rate, constant ratio or constant product.
8. Consider fixed fees, thresholds and real-world limits.
9. Round only at the final step.
10. State the assumptions when the result is used for planning.

Common mistakes

• selecting inverse proportion merely because the result should be smaller;
• placing quantity in the left column on one row and in the right column on another;
• mixing kilograms and grams without conversion;
• including fixed charges in a constant-rate calculation;
• assuming twice as many workers always halve the time;
• rounding every intermediate step;
• leaving more than one field empty;
• treating a mathematical estimate as a guaranteed real-world outcome.

Frequently asked questions

Mathematical and technical sources

• OpenStax College Algebra 2e: Modeling Using Variation
• OpenStax College Algebra 2e: Key Concepts for Direct and Inverse Variation
• Math Is Fun: Proportions
• Math Is Fun: Directly and Inversely Proportional
• Khan Academy: Direct and Inverse Proportion
• Schema.org: FAQPage structured-data type

Sources reviewed: 23 June 2026.