Calculate rule-of-three problems with value A, value B and the sought value for direct or inverse proportional relationships.
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Inputs
Enter values
Live
Directly proportionalMore of value A means more of value B.
The first known value, for example quantity, persons or distance.
The corresponding value to value A, for example price, time or consumption.
The new value A for which the corresponding value B is sought.
Load example
For illustration only – your own calculation is what counts.
Explanation
How Rule of Three Calculator works
The rule of three uses a known relationship between value A and value B and applies it to a sought value.
For direct proportionality, the value for one unit is calculated first. For inversely proportional problems, the product of value A and value B remains constant.
The important decision before entering values: price per quantity is usually directly proportional, while persons and working time for the same task are usually inversely proportional.
Directly proportionalx = value B / value A x value sought
For direct proportionality, value B is first reduced to one unit of value A.
Inversely proportionalx = value A x value B / value sought
For inversely proportional problems, the product of value A and value B remains constant.
Knowledge
Direct or inversely proportional?
The most important decision in the rule of three is the relationship between the two values.
Directly proportional means: when value A increases, value B increases in the same ratio. Inversely proportional means: when value A increases, value B decreases.
Price and quantity: usually directly proportional.
Consumption and distance: usually directly proportional.
People and working time for the same task: usually inversely proportional.
Practice
Practical examples
For illustration only – your own calculation is what counts.
Price and quantity
Value A
3
Value B
7.5
Sought value
5
Sought value12.5
If 3 kg of apples cost 7.50 EUR, you calculate what 5 kg cost.
Result: The sought value is 12.50.
Working time and people
Value A
2
Value B
12
Sought value
4
Sought value6
If 2 people need 12 hours, the inverse proportion shows how long 4 people need.
Result: The sought value is 6.
Fuel and distance
Value A
100
Value B
6
Sought value
350
Sought value21
If 100 km require 6 litres, you calculate the consumption for 350 km.
Result: The sought value is 21.
Notes
Common mistakes
Confusing proportional and inversely proportional
The problem 'If 4 workers need 6 days, how long do 8 workers need?' is solved proportionally (result: 12 days instead of 3 days).
Check: Does the sought value increase when the given value increases? Yes = proportional. Does it decrease? = inversely proportional. More workers mean less time, so inversely proportional.
Entering 0 as a base value
A value of 0 is used as a base value. Division by 0 is undefined and produces no meaningful result.
All base values must be greater than 0. If a value is 0, there is no proportional relationship that can be transferred.
Applying the rule of three to non-linear relationships
The rule of three is applied to problems that do not have a linear relationship (e.g. area calculation: double the side length gives four times the area).
The rule of three only applies to linear proportionality. For quadratic or exponential relationships, the corresponding formula must be used.
Units not consistent
Value A is in kilograms and the sought value is in grams. The different units lead to a result that is off by a factor of 1000.
Convert all values to the same unit before calculating. The rule of three only works correctly when the compared quantities are in the same units.
Constant conditions merely assumed
More people are assumed to shorten the time proportionally even when workplaces, materials or coordination limit performance.
Use inverse proportion only when the workload and output per person really remain constant under otherwise equal conditions.
FAQ
Frequently asked questions
When is a rule of three directly proportional?
Directly proportional is correct when more of value A also means more of value B.
When is a rule of three inversely proportional?
Inversely proportional is correct when more of value A means less of value B, for example more people and less time.
Why must the values be greater than 0?
The rule of three divides by base values. With 0, the ratio cannot be meaningfully calculated.
What is the unit value?
In the direct rule of three, the unit value is the corresponding value B for exactly one unit of value A.
How do I recognise inversely proportional problems?
Inversely proportional is typical when a larger number reduces the required time or quantity, for example more people for the same job.
Can I use the rule of three with decimal numbers?
Yes. All values can be decimals. Example: If 2.5 kg of flour costs 3.75 EUR, the rule of three calculates the price for 4.2 kg. The formula remains the same, only the numbers become more complex.
Where is the rule of three used in everyday life?
The rule of three appears everywhere in daily life: price comparisons (price per kilo), scaling recipes (for more people), travel planning (fuel costs for longer distances), salary calculation (hourly rate for part-time) and material requirements (paint for a larger area).
What happens with mixed proportionalities?
Sometimes a problem is partly proportional and partly inversely proportional (e.g. price, quantity and quality). The simple rule of three covers only one relationship. For more complex ratios, the problem must be broken down into multiple rule-of-three steps.
Limits
Limitations
The rule of three assumes a linear relationship. Quadratic, exponential or logarithmic relationships cannot be calculated.
The simple rule of three describes the relationship between exactly two quantities. For three or more interdependent quantities, multiple rule-of-three steps or other methods are needed.
The calculator shows exact values without rounding. In practice (e.g. unit quantities), rounding may be necessary.
In reality, relationships are often only approximately proportional. Discounts, volume effects or saturation can violate linearity.
The KMK educational standards define the rule of three as a core competency in the area of proportional relationships. The rule of three is introduced in grades 6-7.
View sourceRetrieved: 06/15/2025 · Verified on: 07/23/2026
Kernlehrplan Mathematik Hauptschule 2022Ministerium für Schule und Bildung NRW
The curriculum covers direct and inverse proportional relationships and the rule-of-three method.
View sourceRetrieved: 07/23/2026 · Verified on: 07/23/2026
Proportionale Zuordnungen und DreisatzaufgabenLandesbildungsserver Baden-Württemberg
The educational material places rule-of-three tasks within proportional relationships and stresses understanding how the quantities relate.
View sourceRetrieved: 07/23/2026 · Verified on: 07/23/2026
Cash discount and pricing questions often use proportional relationships.
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