Cross Multiplication Calculator

Enter two fractions and leave one term as x to solve the proportion, or fill in all four to check whether the ratios are proportional. You get both cross products, the rearrangement used, and a substitution check.

Formula a × d = b × c

Type x (or leave blank) in the term you want to find. Fill in all four and the calculator checks whether the two ratios are equal instead. Fractions like 3/4, mixed numbers like 1 1/2 and percentages like 40% all work.

a b
c d
Try an example
Method

How cross multiplication works

Four steps, and the same four every time. The only thing that changes is which division you do at the end.

Set the two ratios equal

Write the problem as a/b = c/d. The order matters: whatever is being compared in the first ratio must be compared the same way round in the second. If the left fraction is miles per hour, the right one has to be miles per hour too, not hours per mile.

Multiply across the diagonals

Numerator of one side times denominator of the other, both ways: a × d and b × c. Setting them equal gives ad = bc, an equation with no fractions left in it.

Work out the side you know

One of the two products contains the unknown and one does not. Evaluate the one that does not — that number is now the whole of one side of the equation.

Divide to isolate the unknown

The unknown is being multiplied by a known factor, so divide both sides by that factor. Where the unknown sits decides the division: solving for a gives bc/d, for b gives ad/c, for c gives ad/b, and for d gives bc/a.

Substitute back and check

Put your answer into the original proportion and reduce both fractions. If they come out to the same decimal, the answer is right. This catches the most common mistake, which is dividing by the wrong term.

a/b = c/d  ⟹  a × d = b × c

The identity. Multiply both sides of the proportion by b × d and both denominators cancel. Nothing is lost, so the new equation has exactly the same solutions as the original.

b ≠ 0  and  d ≠ 0

The one condition. Both denominators must be non-zero for the fractions to exist. Numerators may be zero — and if either numerator is zero, the other must be too for the proportion to hold.

The mistake to watch for

Cross multiplying down instead of across. a × b = c × d is not the identity and gives a wrong answer that often still looks plausible. The multiplication always crosses the equals sign — that is where the name comes from.

Worked examples

Four cases, four positions for x

The rearrangement differs depending on where the unknown sits. These cover all four.

Unknown in position c

A recipe uses 3 eggs for every 4 cups of flour. How many eggs for 12 cups?

Proportion
3 / 4 = x / 12
Cross multiply
3 × 12 = 4 × x
Known side
36 = 4x
Divide by 4
x = 36 ÷ 4
Check
3/4 = 0.75, 9/12 = 0.75 ✓

x = 9 eggs

Unknown in position a

A map shows 18 cm for 45 km. How many centimetres represent 5 km?

Proportion
x / 5 = 18 / 45
Cross multiply
x × 45 = 5 × 18
Known side
45x = 90
Divide by 45
x = 90 ÷ 45
Check
2/5 = 0.4, 18/45 = 0.4 ✓

x = 2 cm

Unknown in position d

7 identical books weigh 2 kg. What weight corresponds to 21 books?

Proportion
7 / 2 = 21 / x
Cross multiply
7 × x = 2 × 21
Known side
7x = 42
Divide by 7
x = 42 ÷ 7
Check
7/2 = 3.5, 21/6 = 3.5 ✓

x = 6 kg

All four known — a check

Are the ratios 6 : 9 and 10 : 15 proportional?

Proportion to test
6 / 9 = 10 / 15 ?
First product a × d
6 × 15 = 90
Second product b × c
9 × 10 = 90
Compare
90 = 90
Both reduce to
2/3 = 0.666…

Proportional ✓

Reference

Rearranging for each unknown

Every form comes from the same ad = bc. This is the table to memorise if you are doing these by hand.

Solving for Proportion After cross multiplying Rearranged Requires
a x/b = c/d x · d = b · c a = bc / d d ≠ 0
b a/x = c/d a · d = x · c b = ad / c c ≠ 0
c a/b = x/d a · d = b · x c = ad / b b ≠ 0
d a/b = c/x a · x = b · c d = bc / a a ≠ 0
Nothing — a check a/b = c/d ? ad vs bc equal ⟹ proportional b, d ≠ 0
Background

When cross multiplication is the right tool

Cross multiplication solves exactly one shape of problem: two ratios set equal, with one value missing. That shape turns up constantly — unit conversion, recipe scaling, map distances, currency exchange, medication dosing by weight, mixing paint, and any question that starts "if n of these cost m, then…". Recognising the shape is most of the work; the arithmetic afterwards is one multiplication and one division.

It is not the right tool for adding or subtracting fractions, and it does not help with a single fraction on its own. If you have 2/3 + 1/4 you need a common denominator, not a cross product. The test is whether there is an equals sign between two fractions: if there is, cross multiply; if there is not, there is nothing to cross.

Why the two products are equal in a true proportion

A ratio a/b is a single number — the quotient. Saying a/b = c/d says those two quotients are the same number, call it k. Then a = kb and c = kd. Substituting into the cross products gives ad = kbd and bc = bkd, which are the same product with the factors written in a different order. That is the whole proof, and it also explains why the equality fails the moment the two ratios differ: the k on each side is no longer the same number.

Reading the size of the gap

When two ratios are not proportional, the difference between the cross products tells you which way round they fail. If ad > bc then a/b > c/d, so the first ratio is the larger one — which makes cross multiplication a fast way to compare two fractions without converting either to a decimal. This calculator reports the gap for exactly that reason.

Related routes to the same answer

Cross multiplication is not the only way through. You can scale one fraction to match the other's denominator, which is what the equivalent ratio calculator does. You can find the constant of proportionality first and multiply, which is the approach on the direct proportion calculator. You can reduce both ratios to lowest terms and compare, using the simplify ratio calculator. All three give the same result; cross multiplication is simply the fewest steps when exactly one value is unknown.

Questions

Cross multiplication — questions

The points that most often trip people up, answered directly.

What is cross multiplication?
Cross multiplication is the shortcut for clearing the denominators from a proportion. If a/b = c/d, then multiplying both sides by b × d gives a × d = b × c — the numerator of each fraction multiplied by the denominator of the other. The two results are called the cross products, and in a true proportion they are always equal.
How do you cross multiply to solve for x?
Write the proportion so the unknown is one of the four terms, cross multiply to get a single multiplication statement, then divide to isolate x. For 3/4 = x/12 you get 3 × 12 = 4 × x, so 36 = 4x and x = 9. Which division you do at the end depends on where x sits — this calculator picks the right rearrangement for you and shows it.
Why does cross multiplication work?
It is just multiplying both sides of an equation by the same thing. Starting from a/b = c/d, multiply both sides by b and you get a = bc/d; multiply by d as well and you get ad = bc. Because you did the identical operation to both sides, the equality is preserved. The one requirement is that b and d are not zero, since you cannot divide by zero in the first place.
How do I know if two ratios are proportional?
Cross multiply and compare. If a × d equals b × c the ratios are proportional; if the two products differ, they are not. Leave all four fields filled in on this calculator and it reports the two products side by side along with the verdict and the size of the gap.
Can I cross multiply with fractions, decimals or negatives?
Yes. Every field here accepts decimals (2.5), fractions (3/4), mixed numbers (1 1/2), percentages (40%) and negative values. Negative numbers behave normally under cross multiplication — the identity ad = bc holds regardless of sign. For proportions where the terms are themselves fractions, the dedicated Proportion Calculator with Fractions lays the inputs out as numerators and denominators.
Is cross multiplication the same as finding a common denominator?
They achieve the same thing but in different orders. Finding a common denominator rewrites both fractions so you can compare numerators directly. Cross multiplication skips the rewriting and jumps straight to the comparison, which is why it is faster for solving equations. For adding or subtracting fractions you still need the common denominator — cross multiplication only helps when two fractions are set equal to each other.
What happens if a denominator is zero?
The proportion has no meaning. A fraction with a zero denominator is undefined, so there is nothing to cross multiply. This calculator flags the field rather than returning a number. A zero numerator is fine — 0/5 = 0/9 is a perfectly valid proportion.
Can cross multiplication be used on inequalities?
Only with care. If you cross multiply a/b < c/d you must know the signs of b and d, because multiplying an inequality by a negative number reverses it. With both denominators positive the inequality direction is preserved. This is why cross multiplication is taught for equations rather than inequalities.