x = (B × C) ÷ A
How to solve a rule of three
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1
Enter the known pair
Put the pair you already know in Value A and Value B. Example: 3 kg (A) cost $12 (B).
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2
Enter the third value
Put the new value of the same quantity as A in Value C. Example: 5 kg (C).
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3
Choose direct or inverse
Use Direct Proportion when both quantities grow together and Inverse Proportion when one goes up as the other goes down.
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4
Click Calculate
The result X appears instantly. In the example, 5 kg cost $20.
Direct rule of three
In a direct rule of three, if one quantity doubles, the other doubles too. Price and amount, distance and fuel, or a recipe and its ingredients all work this way. The setup is: A is to B as C is to X, so X = B × C ÷ A.
Example: a car uses 8 liters to drive 100 km. How much does it use for 250 km? A = 100, B = 8, C = 250. X = 8 × 250 ÷ 100 = 20 liters. Percentages are another common case: if 200 is 100%, what percentage is 50? A = 200, B = 100, C = 50, and X = 25%.
Inverse rule of three
In an inverse rule of three, when one quantity goes up the other goes down in the same proportion. It shows up with speed and time, number of workers and deadline, or how many taps fill a tank. The formula becomes X = A × B ÷ C.
Example: 4 builders take 6 days to build a wall. How many days do 8 builders take, working at the same pace? A = 4, B = 6, C = 8. X = 4 × 6 ÷ 8 = 3 days. More people working means fewer days, which is why the proportion is inverse.
How to tell direct from inverse
Ask yourself: if I increase the first quantity, does the second go up or down? If it goes up, the proportion is direct; if it goes down, it is inverse. Before trusting the result, do a quick sanity check. More kilos should cost more; more speed should take less time. If the result goes against that logic, you probably picked the wrong type of proportion.
- Direct: price × quantity, distance × fuel, hours worked × pay, recipe × servings.
- Inverse: speed × time, workers × days, taps × time to fill, servings × size of each serving.
Rule of three FAQ
What is the rule of three formula?
For a direct proportion, X = B × C ÷ A. For an inverse proportion, X = A × B ÷ C, where A and B are the known pair and C is the new value.
How do I use the rule of three for percentages?
Put the total in A, 100 in B and the part in C, with a direct proportion. The result X is the percentage the part represents.
What is an inverse rule of three?
It is when the quantities move in opposite directions: if one doubles, the other halves, like speed and travel time.
Does the calculator accept decimals?
Yes. Decimal and negative values are accepted; use the decimal separator your keyboard offers.
Can I use the rule of three to convert units?
Yes, it is a direct proportion. If 1 inch is 2.54 cm, how many centimeters are 12 inches? A = 1, B = 2.54, C = 12, and X = 30.48 cm.
What about the compound rule of three?
With more than two quantities, solve it in stages: do one simple rule of three at a time while keeping the other quantities fixed.
Complete Guide to the Rule of Three and Proportion Calculations
Master proportion calculations with our comprehensive guide to the rule of three. Learn the formulas, applications, and practical examples of proportional reasoning.
What is the Rule of Three?
The rule of three is a fundamental mathematical principle used to find unknown values in proportional relationships. It establishes that if we know three values in a proportion, we can calculate the fourth.
Basic Formula
Types of Proportions
Direct Proportion
When two quantities increase or decrease together. If one doubles, the other also doubles.
Inverse Proportion
When one quantity increases while the other decreases proportionally.
Benefits of Learning the Rule of Three
Mastering proportional reasoning develops essential skills for mathematics and real-world applications
Mathematical Skills
Develops understanding of ratios, proportions, and algebraic relationships
Problem Solving
Enhances analytical thinking and logical reasoning abilities
Logical Thinking
Improves ability to identify patterns and relationships in data
Quick Calculations
Enables fast mental math for everyday proportional problems