Rule of Three Calculator

What the rule of three actually is

The rule of three is the oldest practical tool in commercial arithmetic. Medieval Italian merchants called it the *regola del tre*, the "golden rule" of the counting house, because every everyday problem — from weighing cloth to splitting profits — reduced to cross-multiplying three known numbers to find a fourth.

In modern terms, a rule-of-three calculadora takes three values that sit in a proportion and returns the missing one. The classical layout is four boxes arranged in two rows, with an arrow showing the direction of the relationship. It survives because it matches how most adults actually reason: "if this costs that, how much do I need for this other quantity?"

Direct vs inverse proportion

The single biggest mistake students and shop-floor staff make is confusing direct with inverse proportion. The rule looks identical on paper, but the maths flips.

Step-by-step method

Every rule-of-three problem follows the same six steps.

• Write what you know as two paired columns — same units in each column, one known pair and one pair with the unknown.
• Label the columns so quantities of the same type sit above each other (litres above litres, pounds above pounds).
• Decide direction — draw a downward arrow on the known side; on the unknown side, same direction means direct, opposite means inverse.
• Cross-multiply — direct: diagonals equal; inverse: verticals equal.
• Solve for x algebraically.
• Sanity check — does the result sit inside the sensible range?

Six worked examples from everyday life

The rule turns up in more places than most people notice.

Rule of three with percentages

Percentages are just proportions where one side is 100. The rule of three handles them in one line.

• What is 18 % of £450? — 100 / 450 = 18 / x, so x = (450 × 18) / 100 = £81.
• £90 is what percentage of £600? — 600 / 100 = 90 / x, so x = (90 × 100) / 600 = 15 %.
• £70 discount represents what full price if 14 % off? — 14 / 70 = 100 / x, so x = (70 × 100) / 14 = £500 original price.
• Markup of 22 % on cost price £85 — 100 / 85 = 22 / x, so x = (85 × 22) / 100 = £18.70 markup; sale price £103.70.

Compound (double) rule of three

When three quantities interact at once — say, people, hours and output — the rule extends to a compound version. The idea is to hold one factor constant at a time.

Example: 4 workers paint 120 m² of wall in 3 days. How many days for 6 workers to paint 200 m²? Hold workers fixed first: more area means more days; direct. Then hold area fixed: more workers means fewer days; inverse.

Compact formula: x = (3 × 4 × 200) / (6 × 120) = 2,400 / 720 = 3.33 days ≈ 3 days 8 hours.

UK education context

The rule of three appears in the English Key Stage 2 maths curriculum as "ratio and proportion" and is then reinforced at Key Stage 3 (Year 7–9). At GCSE it shows up in foundation and higher papers under "ratio, proportion and rates of change", and at A-level the idea widens into direct and inverse variation with constants of proportionality. Adult numeracy courses and Functional Skills Level 2 assessments rely heavily on the rule for real-world tasks such as scaling recipes, reading maps and checking pay slips.

Common mistakes and how to avoid them

Nine out of ten rule-of-three errors trace back to four causes.

• Mixing units — mixing 750 ml with 1 L without converting. Always convert to the same unit before cross-multiplying.
• Wrong direction — treating an inverse problem as direct (or vice versa). Always do the doubling sanity check before you multiply.
• Decimal drift — losing a decimal point when working mentally. Use a scratchpad for anything beyond two-digit figures.
• Forgetting context — calculating 0.12 painters when the answer has to be a whole person. Round to the nearest sensible unit and explain why.

Rule of three in personal finance

A surprising share of money decisions are rule-of-three problems in disguise. Three cases UK households deal with often.

• Currency conversion — £1 = R$ 6.40 today. £350 → x = 350 × 6.40 = R$ 2,240. Direct.
• Shared rent — a £2,400-per-month house split by 4 people: 2,400 / 4 = £600/person. Inverse in the sense of "more housemates, smaller share".
• Simple monthly interest — £500 loan at 2 % per month. After 5 months: 1 month → £10; 5 months → x = 50. Direct.
• Council tax instalments — annual bill of £1,836 split into 10 monthly payments: 1836 / 10 = £183.60 per month. Direct.
• Energy bill estimate — 600 kWh over 30 days at 28 p/kWh. Daily: 600 × 0.28 / 30 = £5.60/day. Direct.

The golden rule in classical arithmetic

Fibonacci's *Liber Abaci* (1202) dedicates an entire chapter to the rule, calling it the operation "by which all problems of trade are solved". Tudor-era merchant manuals in England reprinted the same layout word for word. The rule still lives in modern spreadsheet formulas — every time you multiply a ratio by a total to rescale, you are doing cross-multiplication.

Why does it endure? Because it matches the way the human mind actually anchors numbers: to a known reference point, then scaling. Psychometric research on numerical reasoning tests repeatedly shows that adults who apply the rule of three deliberately outperform those who compute percentages from scratch, even when the underlying maths is identical.

Related calculators

• **Percentage calculadora** — quick increases, decreases and VAT.
• **Bhaskara calculadora** — when the proportion contains a squared unknown.
• **Compound interest** — money that grows in geometric proportion.
• **Length converter** — preparing inputs in the same unit before applying the rule.

Classroom checklist to avoid silly mistakes

Five habits that earn marks on GCSE or Functional Skills papers.

• Re-read the question: does it ask for minutes, pounds, grams or a percentage?
• Convert every input to the same unit before cross-multiplying.
• State direct or inverse explicitly in a sentence before you write algebra.
• Set up the proportion with the same quantity vertically aligned.
• Sanity-check — if the answer is 0.13 painters or 17,000 days, something is wrong.

How the calculator works under the bonnet

The calculator detects whether you have selected direct or inverse proportion and applies the matching algebraic rearrangement in JavaScript running entirely in your browser. No values are logged on our servers. Results are rounded to sensible decimal places based on the input magnitude. See our editorial policy and corrections policy for how we review mathematical content.