Rule of Three Calculator guide
Cross Multiplication: How and Why It Works
Cross multiplication solves a proportion by multiplying each numerator by the opposite denominator. From a ÷ b = c ÷ d you get a × d = b × c. It works because multiplying both sides of an equation by b × d cancels both denominators, leaving a single equation with no fractions in it.
What cross multiplication is
When two fractions are equal, you can multiply diagonally across the equals sign and the two results will also be equal. Written out: if a ÷ b = c ÷ d, then a × d = b × c.
The name comes from the shape you draw when you do it — two arrows crossing in an X between the fractions.
It is useful because it takes an equation with fractions in it, which is awkward, and turns it into an equation without any, which is not.
Why it is allowed
It can look like a trick, so it is worth seeing that it is just ordinary algebra done quickly.
Start with a ÷ b = c ÷ d. You are allowed to multiply both sides of an equation by the same thing, so multiply both by b × d.
On the left, (a ÷ b) × b × d — the b cancels, leaving a × d. On the right, (c ÷ d) × b × d — the d cancels, leaving c × b.
So a × d = b × c. Cross multiplication is that whole sequence compressed into one move. Knowing that is what stops it feeling arbitrary.
Using it to solve for x
The usual reason you want it is that one of the four values is unknown. Take 3 ÷ 12 = 7 ÷ x.
Cross multiply: 3 × x = 12 × 7, so 3x = 84.
Divide both sides by 3: x = 28.
That is exactly what the calculator on this page is doing, and it shows the cross-multiplication step explicitly so you can compare it against your own working.
The three things that go wrong
- Multiplying straight across instead of diagonally. a × c = b × d is not a valid step and will give you a wrong answer with no warning.
- Using it when the equation is not two equal fractions. Cross multiplication needs a single fraction on each side. If you have a ÷ b + 1 = c ÷ d, you must deal with the +1 first.
- Forgetting to divide at the end. Cross multiplying gets you to 3x = 84, which is not yet an answer. It is easy to write down 84 when the finish line is 28.
A quick note on when to use it
Cross multiplication is the right tool when you have two fractions set equal to each other. That covers proportions, scale problems, unit conversions and similar-triangle questions.
It is not the right tool for adding or comparing fractions. To add 1/2 and 1/3 you need a common denominator, not cross multiplication — those are different jobs, and mixing them up is a common source of confusion when both topics land in the same term.
