Cross multiplication calculator (rule of three)
The fourth proportional by cross multiplication, in direct or inverse proportionality, with the worked calculation.
Result
20
- Calculation
- x = b × c ÷ a = 12 × 5 ÷ 3 = 20
- Rate preserved (b ÷ a)
- 4
- Applied factor
- × 1.66667
Note
If 3 gives 12, then 5 gives 20. Same rate, applied to the new value.
Calculation, assumptions and sources
OverviewRule of three: “if a gives b, what does c give?”, in direct proportionality (x = b·c/a) or inverse proportionality (x = a·b/c; workers and duration, speed and time).
Rule of three: “if a gives b, what does c give?”, in direct proportionality (x = b·c/a) or inverse proportionality (x = a·b/c; workers and duration, speed and time). The formula is written out with the entered values and the result comes with its justification.
Method & formulasdirect: a → b, so c → x = b × c / a
In direct proportionality the ratio is conserved (b/a = x/c); in inverse proportionality the product is conserved (a·b = c·x). Cross multiplication is the same in both cases; only the law the situation follows changes.
- Fourth proportional
- the value x that completes the proportionality table.
- 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: number of workers and job duration, speed and travel time.
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. A degressive tariff, workers who get in each other’s way or a speed that saturates are not. Proportionality is a modelling assumption; it is not checked.
Inverse proportionality“4 workers take 6 days; how long do 8 workers take?” The direct rule would answer 12 days.
“4 workers take 6 days; how long do 8 workers take?” The direct rule would answer 12 days. The situation is inverse: the product workers × days is conserved (4 × 6 = 24), so 8 workers take 3 days.