Rule of three
Direct and inverse proportion solver with step-by-step explanation and a calculation history.
The more, the more – both quantities grow together (e.g. more bricks = more kg).
Tip: the field with a question mark is the value being found – leave it empty. You can edit the quantity labels (bricks, kg…). Click another field to move the question mark.
Fill in three values and leave the fourth (with the question mark) empty.
What the rule of three is
The rule of three is a simple way to find a fourth value from three known values when a proportional relationship holds between two quantities. It's written in two rows: the first a known pair, the second a value and the unknown number to find.
Direct proportion – “the more, the more”
Both quantities grow or shrink together. If 1 brick weighs 57 kg, how much do 150 bricks weigh? More bricks = more kilograms. We cross-multiply:
a₁ ∶ b₁ = a₂ ∶ b₂ ⟹ b₂ = (a₂ × b₁) ÷ a₁
Example: b₂ = (150 × 57) ÷ 1 = 8,550 kg.
Inverse proportion – “the more, the less”
When one quantity grows, the other shrinks. If a car covers a route in 40 minutes at 70 km/h, how long will it take at 90 km/h? Higher speed = shorter time. We multiply straight across:
a₁ × b₁ = a₂ × b₂ ⟹ b₂ = (a₁ × b₁) ÷ a₂
Example: b₂ = (70 × 40) ÷ 90 ≈ 31.1 minutes.
How to tell which proportion applies
Ask: “if the first quantity increases, does the second increase too, or decrease?” If they grow together (more work = more pay, more people = more food), it's direct. If it decreases (more workers = less time, higher speed = shorter time), it's inverse. Switch the toggle above and the visualisation and formula change.
The rule of three assumes pure proportionality. In reality this doesn't always hold (twice as many workers won't finish a job in exactly half the time) – treat the result as a mathematical model.