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Rule of Three Calculator

Find the missing value in a proportion, with every step of the cross-multiplication shown.

Relationship

Direct: both quantities go up or down together, e.g. more items, higher price. X = B × C ÷ A.

X (goes with C)

18

Step by step

  1. Write down which values go together
    4 → 12
    6 → X
  2. Cross-multiply: A × X = B × C
    4 × X = 12 × 6
  3. Divide to get X on its own
    X = (12 × 6) ÷ 4 = 72 ÷ 4
  4. Result
    X = 18

How to use

  1. Choose Simple for one pair of values (A goes with B, C goes with X) or Compound when three or more quantities change at once.
  2. For a simple rule of three, pick Direct or Inverse. Ask yourself: if A gets bigger, does B get bigger too (direct) or smaller (inverse)?
  3. Type A, B and C. Decimals can be written with a point or a comma.
  4. Read X in the box next to C. The steps below show the cross-multiplication with your own numbers, so you can copy the working into homework or check a spreadsheet.

What the rule of three does

The rule of three solves one of the most common problems in everyday math: you know that one amount goes with another, and you want to know what goes with a third amount. Four tickets cost $48, so what do 7 tickets cost? A recipe for 6 people uses 450 g of pasta, so how much for 10? A map scale says 2 cm is 15 km, so how far is 9 cm? In each case three numbers are known and one is missing — hence the name.

Write the two known values as a pair (A goes with B), put the new value C under A, and leave X under B. The calculator fills in X as you type, with no button to press.

Direct proportion: cross-multiplication

When both quantities grow at the same rate, the ratio between them stays fixed, so A/B = C/X. Multiplying across the diagonals — cross-multiplication — gives:

A × X = B × C, so X = B × C ÷ A

With the default values, 4 → 12 and 6 → X: 4 × X = 12 × 6 = 72, so X = 72 ÷ 4 = 18. If 4 kg of apples cost $12, 6 kg cost $18.

Percentages are a special case of the same idea. “What is 15% of 80?” is 100 → 15, 80 → X, which gives 15 × 80 ÷ 100 = 12. For percentage changes, discounts or finding the original price, the percentage calculator does it in one go.

Inverse proportion: when more means less

Not every relationship works that way. If 4 workers need 12 days to paint a building, 6 workers will not need 18 days — they’ll need fewer. Here the product stays fixed rather than the ratio: 4 workers × 12 days = 48 worker-days of effort, and that’s the same whoever does it. So:

A × B = C × X, so X = A × B ÷ C

With the same numbers, X = 4 × 12 ÷ 6 = 8 days. Switch the calculator between Direct and Inverse and you’ll see the answer change from 18 to 8 — a quick reminder that choosing the wrong type gives a confidently wrong result.

Typical inverse situations are speed and travel time, the number of people sharing a fixed budget, pipes filling a tank, and machines working on a fixed job.

Compound rule of three

Sometimes several things change at once. Suppose 6 workers build 30 m of wall in 10 days, and you want to know how long 8 workers would take to build 60 m. Compare each other quantity with the one you’re solving for (days), one at a time:

  • More workers means fewer days, so workers are inverse: multiply by 6/8.
  • More metres means more days, so length is direct: multiply by 60/30.

X = 10 × 6/8 × 60/30 = 3,600 ÷ 240 = 15 days. That’s the default example in Compound mode. You can add a third quantity, such as hours worked per day (inverse) or the hardness of the material (direct).

Checking your answer

A quick sanity check catches most mistakes. In a direct problem, if C is bigger than A, X should be bigger than B. In an inverse problem it’s the other way round. If the result points in the wrong direction, the relationship type is probably wrong. For money questions that grow over time, like savings with interest, a straight proportion is not enough; the compound interest calculator handles that growth properly.

Frequently asked questions

What is the rule of three?

It's a quick way to find a missing value when two quantities are proportional. If A corresponds to B, and you know a new value C, the rule of three gives the X that corresponds to C. For a direct proportion X = B × C ÷ A; for an inverse proportion X = A × B ÷ C.

How do I know whether to use direct or inverse?

Imagine doubling the first quantity. If the second one also doubles (twice the flour, twice the cookies), it's direct. If it halves (twice the speed, half the travel time), it's inverse. Most everyday cases — prices, recipes, unit conversions, map scales — are direct. Workers and days, speed and time, or taps filling a tank are the classic inverse cases.

Is the rule of three the same as cross-multiplication?

Cross-multiplication is the step that solves it. Writing A/B = C/X and multiplying across the diagonal gives A × X = B × C, and dividing by A leaves X. The calculator shows exactly that step with your numbers.

What is a compound rule of three?

It's the same idea with more than two quantities. For example: 6 workers build 30 m of wall in 10 days; how many days do 8 workers need for 60 m? Each other quantity scales the answer by its own ratio — inverse for workers, direct for metres — giving 10 × 6/8 × 60/30 = 15 days.

Why does the calculator say a value can't be zero?

The value you divide by can't be zero. In a direct rule of three that's A; in an inverse one it's C; in a compound one it's the bottom of each ratio. A zero there usually means the problem has no meaningful answer, like asking how long zero workers take.

Can I use decimals and negative numbers?

Yes. Type decimals with a point or a comma, and large numbers with or without thousands separators (1,250 or 1250). Negative numbers work mathematically, though real-world proportions almost always use positive amounts.

How many decimal places does the result show?

Up to four. A result like 6.6667 is rounded; if you need more precision, the last step shows the exact division (for example 20 ÷ 3) so you can carry it out yourself.

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