Cross multiplication (rule of three)
direct: a → b, so c → x = b × c / a
The most used calculation in the world, done properly: “if a gives b. What does c give?” in direct proportionality (x = b·c/a), and its treacherous twin in inverse proportionality (x = a·b/c, workers and duration, speed and time), which most rule-of-three calculators wrongly treat as direct. The situation is chosen explicitly, the formula is instantiated before your eyes, and every result comes with the sentence that justifies it.
20
- Calculation
- x = b × c ÷ a = 12 × 5 ÷ 3 = 20
If 3 gives 12, then 5 gives 20. Same rate, applied to the new value.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasdirect: a → b, so c → x = b × c / a
direct: a → b, so c → x = b × c / a
inverse: a × b = c × x, so x = a × b / c
In direct proportionality, the ratio is conserved (b/a = x/c): doubling one doubles the other. In inverse proportionality, the product is conserved (a·b = c·x): doubling one halves the other. Cross multiplication is the same algebraic move in both cases, what changes is the law the situation follows, and recognising it is on you: the tool asks which one, it does not guess.
- Fourth proportional
- · the value x that completes the proportionality table, the one cross multiplication computes.
- Direct situation
- · both quantities grow together at a constant rate: price and quantity, distance and duration at fixed speed, recipe doses.
- Inverse situation
- · one grows as the other shrinks, at constant product: more workers = shorter job, more speed = shorter trip.
Validity domainThe calculation assumes the situation IS proportional (directly or inversely) over the range considered.
The calculation assumes the situation IS proportional (directly or inversely) over the range considered. Many are not: a degressive tariff, a worksite where workers get in each other’s way, a speed that saturates. Proportionality is a modelling assumption, not an automatic property; the tool applies it exactly, it does not verify it.
Common pitfall: applying the direct rule to an inverse situation“4 workers take 6 days; how long do 8 workers take?
“4 workers take 6 days; how long do 8 workers take?” The direct rule would answer 12 days, twice the workers. Twice the time, absurd. The situation is inverse: the product workers × days is conserved (4 × 6 = 24), so 8 workers take 3 days. Cross multiplication does not protect you from this trap: it computes correctly within the law you give it. Recognising the law is the step that thinks.