Rule of Three Calculator (Direct + Inverse Proportion)
Solve direct and inverse proportions. Worked examples for school + everyday use: cooking ratios, work-time scaling, currency conversion.
Based on 2 references↓X (unknown)
50
Formula
3 / 30 = 5 / x → x = (30 × 5) / 3 = 50Proporção direta: quando uma grandeza aumenta, a outra aumenta na mesma proporção.
Rule of three is the workhorse arithmetic tool for everyday proportional reasoning — scaling a recipe, converting currencies on the fly, figuring out how long a job takes with more or fewer people, calculating a fuel-efficiency comparison. It is taught early in BR schools as a formal topic ("regra de três") and is the most-searched math operation in PT-BR after percentage. The two flavours — direct and inverse — confuse many adults: when the second quantity grows with the first, use direct; when it shrinks, use inverse.
Direct example: 3 kg of meat costs $30, so 5 kg costs $50. The price grows with the weight, both in the same direction. Inverse example: 4 painters finish a wall in 6 days, so 8 painters finish in 3 days. Time shrinks as workforce grows. The trap is mistaking inverse for direct (or vice versa) — many real-world problems involve hidden inverse relationships (more workers → less time, faster speed → less time, larger denominator → smaller share).
The math behind rule of three
Direct proportion: a / b = c / x → x = (b × c) / a. The two ratios are equal because the quantities scale together. If 3 kg of meat costs $30, the rate is $10/kg; 5 kg at $10/kg = $50.
Inverse proportion: a × b = c × x → x = (a × b) / c. The product is constant because as one grows, the other shrinks. If 4 workers × 6 days = 24 worker-days of total work, then 8 workers need 24/8 = 3 days.
Choosing direct vs inverse: ask "if I double the input, does the output double (direct) or halve (inverse)?" Pace, throughput, and workforce-to-time relationships are usually inverse. Cost, fuel, and ingredient relationships are usually direct.
Practical examples
Direct: scaling a recipe
Setup: Recipe for 4 servings uses 200g of flour. How much for 6 servings?
4 / 200 = 6 / x → x = (200 × 6) / 4 = **300g**
Takeaway: Direct proportion: more servings, more flour, same direction.
Inverse: workers and days
Setup: 5 workers complete a project in 12 days. How long for 8 workers?
5 × 12 = 8 × x → x = 60 / 8 = **7.5 days**
Takeaway: Inverse proportion: more workers, fewer days, opposite direction.
Direct: currency conversion
Setup: R$ 5 = $1 (rate). How much is R$ 230 in dollars?
5 / 1 = 230 / x → x = (1 × 230) / 5 = **$46**
Takeaway: FX is direct: more reais, more dollars at a fixed rate.
Inverse: speed and travel time
Setup: At 60 km/h, the trip takes 4 hours. How long at 80 km/h?
60 × 4 = 80 × x → x = 240 / 80 = **3 hours**
Takeaway: Inverse: faster speed, less time, opposite direction.
Frequently asked questions
When is rule of three useful in real life?▾
Anytime two quantities are proportional. Cooking (scaling recipes), shopping (unit price comparisons), travel (speed/time), workforce planning (people/duration), unit conversions (currency, weight, volume), discounts and tax adjustments.
How do I know if it is direct or inverse?▾
Ask: "if I double the first quantity, what happens to the second?" Doubles too → direct. Halves → inverse. Cost-per-item, fuel-for-distance, ingredients-per-serving are direct. Workers-to-time, speed-to-duration, denominator-to-share are inverse.
Why does the calculator say "regra de três"?▾
It's the Brazilian Portuguese name for the same operation. BR schools teach this as formal content; English speakers more often see it labeled "proportion" or "cross-multiplication". The math is identical.
Can I use rule of three for percentages?▾
Yes — many percentage problems are direct rule of three. "What is 15% of 200?" → 100/200 = 15/x → x = 30. We have a dedicated Percentage Calculator with shortcuts for the most common modes.
Why does my answer look wrong?▾
The most common error is using direct when the relationship is inverse (or vice versa). If the answer feels backwards, switch the mode toggle. Second most common: input swap — make sure A pairs with B (the known ratio) and C is the new value you're scaling.
Sources & references
Cross-check every number in this calculator against the primary sources below.
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