Cross multiplication is the standard move for any equation of the form “fraction = fraction”. Multiply each numerator by the other side’s denominator, set the two products equal, and the fractions disappear. This calculator does that with expressions as well as numbers, so it can solve proportions where x sits in a numerator, a denominator or both — and it checks the answers against the original denominators so you never keep an illegal solution.
How to use the cross multiplication calculator
- Fill in the four boxes of A/B = C/D. Each box takes a number or an expression in x, such as
x + 2,3,2x − 1orx/4. - Leave out x entirely to compare two plain fractions.
- Read the solution on the tape. The steps show the cross-multiplied equation, its expansion, the solving method and a check of each answer in the original fractions.
The cross multiplication rule
It works because multiplying both sides of the equation by B·D cancels each denominator. The condition matters: if an answer makes B or D zero, the original equation was never defined there.
Depending on where x appears, the cross-multiplied equation is:
- linear, when x appears only on one side of each product, such as (x + 2)/3 = (x − 1)/2;
- quadratic, when x appears in both factors of a product, such as (x + 1)/x = 3/(x − 2);
- an identity or a contradiction, when the x terms cancel.
Worked examples
Linear. Solve (x + 2)/3 = (x − 1)/2. Cross-multiply: 2(x + 2) = 3(x − 1), so 2x + 4 = 3x − 3 and x = 7. Check: 9/3 = 3 and 6/2 = 3 ✓.
Quadratic. Solve (x + 1)/x = 3/(x − 2). Cross-multiply: (x + 1)(x − 2) = 3x, so x2 − 4x − 2 = 0. The quadratic formula gives x = 2 ± √6, about 4.449 and −0.449. Neither makes a denominator zero, so both stand.
Extraneous. Solve x/(x − 2) = 2/(x − 2). Cross-multiplying leads to (x − 2)2 = 0, so x = 2 — but that makes both denominators zero. There is no solution.
Comparing fractions. Is 3/4 equal to 5/7? Cross products: 3 × 7 = 21 and 4 × 5 = 20. They differ, and since 21 > 20, 3/4 is the larger fraction.
When cross multiplication is the right tool
Cross multiplication applies only when each side is a single fraction. If a side is a sum, such as 1/x + 1/2 = 3/4, combine it into one fraction first or multiply every term by the least common denominator. Also be careful with comparisons when a denominator is negative: multiplying an inequality by a negative number reverses it, which is why the calculator checks the signs of the denominators before deciding which fraction is larger.
Typical uses:
- scaling a recipe or a map (1 inch : 50 miles = 3.5 inches : x);
- unit conversions set up as equal ratios;
- similar triangles, where corresponding sides form equal fractions;
- solving rational equations in algebra class.
Related calculators
For a quick missing-number proportion with plain numbers, the ratio calculator and the solve for x in fractions calculator are streamlined. To order several fractions at once, use the comparing fractions calculator. The proportion calculator handles direct and inverse variation, and the quadratic formula calculator explains the quadratic step in more depth.
Frequently asked questions
What is cross multiplication?
For a/b = c/d, multiplying both sides by b and d gives a·d = b·c. You multiply each numerator by the opposite denominator, which turns a fraction equation into one without fractions.
Can x appear in a denominator?
Yes. Cross-multiplying (x + 1)/x = 3/(x − 2) gives (x + 1)(x − 2) = 3x, a quadratic. The calculator solves it exactly and then discards any answer that would make a denominator zero.
What is an extraneous solution?
A value that solves the cross-multiplied equation but not the original, because it makes a denominator zero. For x/(x − 2) = 2/(x − 2), cross-multiplying leads to x = 2, which is not allowed, so there is no solution.
How do cross products compare fractions?
For a/b and c/d with positive denominators, a/b > c/d exactly when a·d > b·c. For 3/4 and 5/7, 3·7 = 21 and 4·5 = 20, so 3/4 is larger.