Rule of three calculator
Solve direct and inverse proportions: find the fourth value X from three known values.
What this calculator does
This calculator solves the rule of three: from three known values (A, B, and C) it finds the fourth value X that completes the proportion "A is to B as C is to X".
It supports both kinds of proportion: direct, when both quantities grow together (more notebooks, more cost), and inverse, when one grows while the other shrinks (more workers, fewer days).
The result updates instantly as you type, and you can always see the formula applied with your own numbers.
Who it is for
- Students solving proportionality problems who want to check the procedure, not just the answer.
- Cooks and bakers scaling recipes up or down to a different number of servings.
- Shop owners converting prices per unit, per dozen, or per weight.
- Anyone planning time or resources: if X people take this long, how long do Y people take?
What information you need
- The two known values that form the first pair of the proportion: A and its corresponding value B.
- The new value C you want to find the unknown X for.
- Whether the relationship is direct (both quantities rise together) or inverse (one rises when the other falls).
How it is calculated
In a direct proportion, the ratio between the quantities stays constant: B ÷ A = X ÷ C. Solving for the unknown, X = (B × C) ÷ A.
In an inverse proportion, the product stays constant: A × B = C × X. Solving for the unknown, X = (A × B) ÷ C.
The divisor (A in the direct case, C in the inverse case) must be non-zero: division by zero is undefined.
The result is rounded to 4 decimal places, rounding ties up (half up), applied once at the end.
Formula
- Direct proportion
X = (B × C) ÷ A- Inverse proportion
X = (A × B) ÷ C
Worked example
If 3 notebooks cost $7.50, how much do 8 notebooks cost? It is a direct proportion: more notebooks, more cost. A = 3, B = 7.50, C = 8.
- Multiply B by C: 7.50 × 8 = 60.
- Divide by A: 60 ÷ 3 = 20.
The 8 notebooks cost $20.00.
How to interpret the result
The result X is in the same units as B: if B is dollars, X is dollars; if B is days, X is days.
In an inverse proportion the result moves the other way: with A = 4 workers taking B = 6 days, C = 8 workers take X = (4 × 6) ÷ 8 = 3 days. Doubling the workers halves the time.
If the result feels illogical (for example, more workers taking more days), you probably picked the wrong type of proportion.
Common mistakes
- Using the direct formula on an inverse problem: the most common mistake, and it produces results backwards from reality.
- Mixing units: if A is in hours and C in minutes, convert to the same unit first.
- Applying the rule of three to relationships that are not proportional, such as tiered rates or volume discounts.
- Placing values in the wrong position: A and B must be the known pair, and C must be the same kind of quantity as A.
Frequently asked questions
How do I know if my problem is direct or inverse?
Ask what happens to the second quantity when the first one increases. If it also increases (more kilos, higher price), it is direct. If it decreases (more machines, fewer hours), it is inverse.
Can I use decimal or negative values?
Yes. Decimals are common (prices, weights, times). Negatives are accepted and follow the sign rules of algebra, although real-life quantities are usually positive.
Why can the divisor not be zero?
Both formulas divide by that value, and division by zero is undefined. Besides, a proportion with zero in that position does not describe a real relationship: there is no way to scale from zero.
How does the rule of three relate to percentages?
A percentage is a direct rule of three with 100 as the reference: "if 100 corresponds to the total, how much corresponds to X?". For those cases, the percentage calculator solves it directly.
Sources
- Last reviewed:
- July 21, 2026
- Calculation version:
- 1.0.0
Important notice
The results of these calculators are informative estimates and may differ from official calculations. They do not constitute legal, tax, or financial advice. Always verify with the competent institutions or a professional.