Compound Rule of Three Calculator
Free compound rule of three calculator — solve a proportion problem driven by two supplementary variables (each direct or inverse). Works in your browser.
Compound rule of three solves for a single unknown that depends on two supplementary variables at once, each either directly or inversely proportional to the target.
The formula
x = target × f₁ × f₂
where fᵢ = new ÷ known for a direct variable, fᵢ = known ÷ new for an inverse variable.
So every variable contributes a factor — multiply them all together with the known target value, and you have the answer.
A worked example
5 workers build 20 m of wall in 6 hours. How many workers are needed to build 30 m in 9 hours?
- Workers vs hours: inverse — more hours means fewer workers needed. Factor = 6 ÷ 9.
- Workers vs wall length: direct — more wall, more workers. Factor = 30 ÷ 20.
x = 5 × (6 ÷ 9) × (30 ÷ 20) = 5 workers
Picking direct vs inverse
For each supplementary variable, hold everything else fixed and ask: “if this one increases, does the target go up or down?”
If up: direct. If down: inverse. That’s all there is to it.
Worked examples
-
5 workers in 6 hours build 20 m of wall — how many for 30 m in 9 hours?
Unknown value: 5
-
2 printers in 3 days print 6 books. For 10 books in 5 days?
Unknown value: 2
Frequently asked questions
What is the compound rule of three?
A rule of three with two supplementary variables instead of one. Useful when the unknown depends on multiple proportional quantities at once — e.g. workers × hours × output.
How do I pick direct or inverse for each variable?
Hold every other variable fixed and ask: "if I increase this one, does the target go up (direct) or down (inverse)?" Workers vs hours to finish a job: inverse. Workers vs total output: direct.
What's the formula?
x = target × (factor 1) × (factor 2), where each factor is (new ÷ known) for direct variables and (known ÷ new) for inverse variables. Multiply all factors together.
Can I have more than two supplementary variables?
Mathematically yes — every extra variable just adds another factor to the product. This calculator supports two, which covers almost all classroom and real-world problems.
What if I picked the wrong direction for a variable?
You'll get the reciprocal of the right answer for that factor — typically the result is way off in the wrong direction. Flip the relationship and re-run.
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