# Cross Multiplication Calculator Use our free cross multiplication calculator to solve equations with fractions. Shows step-by-step work for proportions and fraction comparisons. --- - **Canonical URL:** https://dothecalculation.com/calculators/cross-multiplication-calculator - **Category:** Math calculators - **Publisher:** Do The Calculation (https://dothecalculation.com) - **Cost:** Free, no account or sign-up required - **Privacy:** Runs entirely in the browser; inputs are never sent to a server - **Methodology:** https://dothecalculation.com/methodology --- ## Solve proportions with cross multiplication Clear denominators in equal fractions, isolate unknown variables, and understand why cross products solve proportional equations. - A/B = C/D solving - Cross-product mapping - Variable isolation steps ## Ratios, proportions, and equivalent fractions A ratio compares two quantities using division. A proportion is an equation stating that two ratios are equal. When we write A/B = C/D, we are asserting that the fractions are equivalent and therefore represent the same multiplicative relationship. Cross multiplication follows from this equality. By multiplying both sides by the product of the denominators, we eliminate the fractions and obtain a simpler multiplication equation. ## Why cross multiplication works Start with the proportion: $$\frac{A}{B} = \frac{C}{D}$$ Multiply both sides by BD: $$BD \times \frac{A}{B} = BD \times \frac{C}{D}$$ Cancellation leaves: $$AD = BC$$ This is the cross-product equality. Once the denominators are cleared, solving for any unknown becomes a basic one-step or two-step algebra problem. ## Solving the four unknown-position cases If x appears in the top-left position, x/B = C/D, then x = BC/D. If x appears in the bottom-left, A/x = C/D, then x = AD/C. If x appears in the top-right, A/B = x/D, then x = AD/B. If x appears in the bottom-right, A/B = C/x, then x = BC/A. These formulas all come from the same cross-product identity AD = BC. ## Real-world uses of proportions Cross multiplication appears in unit conversions, percentage problems, recipe scaling, map reading, dilution chemistry, and geometric similarity. Anytime two ratios are equivalent, cross multiplication provides a direct route to the missing quantity. ## Four worked examples, one in each unknown position **Unknown as a numerator.** x/8 = 15/20. Cross multiply: 20x = 120, so x = **6**. Check by simplifying the right side to 3/4 and confirming 6/8 reduces to 3/4. **Unknown as a denominator.** 7/x = 21/9. Cross multiply: 63 = 21x, so x = **3**. Note the unknown moves to the other side here — a common slip is to divide by the wrong factor and report 21/63. **Unknown on the right.** 5/12 = x/48. Cross multiply: 240 = 12x, so x = **20**. Because 48 is 4 × 12, you can also scale directly: 5 × 4 = 20, which is a useful check. **Unknown appearing twice.** 4/x = x/9 gives x² = 36, so x = **6** (the geometric mean of 4 and 9). This form has two algebraic solutions, +6 and −6; in a length, concentration, or price context the negative root is discarded because the quantity cannot be negative. Every one of these is the same single step — multiply the diagonals, set them equal, divide by whatever multiplies the unknown. The four cases feel different only because the unknown moves. ## Setting up a proportion so the units cancel Most proportion errors happen before any multiplication, at the setup. The rule that prevents them: write the same kind of quantity in the same position on both sides. If miles are on top on the left, miles must be on top on the right. A car travels 150 miles on 12 litres; how far on 30 litres? Write 150 miles / 12 L = x miles / 30 L. Cross multiplying gives 12x = 4,500, so x = **375 miles**. Had you written 150/12 = 30/x, you would be asking a different and meaningless question, and the arithmetic would still produce a number — which is why the answer looking plausible is not evidence the setup was right. Attach the units as you write and read the result back as a sentence. "375 miles on 30 litres" is checkable against intuition: 30 litres is two and a half times 12, and 375 is two and a half times 150. If the multiplier on one side does not match the multiplier on the other, the setup was wrong. ## Where cross multiplication does not apply Cross multiplication is only valid for an equation of two ratios, a/b = c/d. It is not a general rule for handling fractions, and applying it outside that shape is one of the most persistent errors in early algebra. It does not work on addition. To solve 1/3 + 1/x = 5/6 you must combine the left side over a common denominator first; there are no diagonals to multiply while an addition sign sits between the fractions. It also does not apply to a single fraction with nothing to equal — 3/4 alone is a value, not an equation. Both denominators must be non-zero, since division by zero is undefined; if solving produces a value that makes an original denominator zero, that root is extraneous and must be rejected. And where the proportion involves measurements, remember that cross multiplication assumes a strictly linear relationship. Doubling a recipe's ingredients works; doubling a cake's baking time does not, because the underlying relationship is not proportional even though both quantities can be written as ratios. ## Frequently asked questions ### What is cross multiplication? It is the process of multiplying the numerator of one fraction by the denominator of the other when two fractions are equal in a proportion. ### When can I use cross multiplication? Only when two fractions are set equal in a proportion, such as A/B = C/D. ### Why can’t I use cross multiplication for addition of fractions? Because fraction addition requires a common denominator. Cross multiplication is for clearing denominators in equations of equal ratios, not combining unlike fractions. ### What happens if a denominator is zero? The proportion is undefined. Division by zero is not permitted, so denominators must be non-zero. ### How do I solve x/5 = 12/20? Cross multiply to get 20x = 60, then divide by 20 to get x = 3. ### What are means and extremes in a proportion? In A/B = C/D, A and D are the extremes, while B and C are the means. ### Is cross multiplication the same as equivalent fractions? It is a consequence of equivalent fractions. Equivalent fractions satisfy the same cross-product equality. ### Can proportions involve decimals? Yes. Decimals can be used directly or converted into fractions before solving. ### How do proportions appear in percentage problems? A classic form is Part/Whole = Percent/100, which can be solved by cross multiplication. ### What is the fastest way to check a solved proportion? Substitute the solved value back into the original ratios and verify that both sides are numerically equal. ## Related concepts - **Equivalent fractions** — Different fractional expressions representing the same rational value. - **Means and extremes** — The inside and outside terms of a proportion used in the cross-product identity. - **Unit conversion** — A common application of proportional reasoning across measurement systems. ## Related guides - [Cross Multiplication: A Beginner's Guide for Students & Teachers](https://dothecalculation.com/blog/math/how-to-cross-multiply-step-by-step-guide) — Master cross multiplication with the "butterfly method," step-by-step worked examples, practice questions with answers, common mistakes, and real-world applications. - [Cross Multiplication Word Problems: 12 Real-World Examples with Step-by-Step Solutions](https://dothecalculation.com/blog/math/cross-multiplication-word-problems) — The hard part of a proportion problem is rarely the cross multiplication itself — it is translating the words into a fraction = fraction setup. 12 fully worked examples across recipes, maps, dosages, chemistry, sports, and more. - [Ratio and Proportion Guide: Simplify, Solve, Split Totals](https://dothecalculation.com/blog/math/ratio-and-proportion) — Learn how to simplify ratios, solve proportions, and split totals with worked examples tied to the live DTC ratio calculator. ## Related calculators - [Ratio and Proportion Calculator](https://dothecalculation.com/calculators/ratio-proportion-calculator) — Simplify ratios, solve proportions, and divide quantities by ratio instantly with this free step-by-step ratio calculator. - [System of Linear & Nonlinear Equations Solver](https://dothecalculation.com/calculators/system-of-equations-calculator) — Solve a system of two linear equations via Cramer’s Rule, or a line-and-parabola nonlinear system via substitution, with clear steps and an intersection graph. - [Linear Programming Solver (Graphical Method)](https://dothecalculation.com/calculators/linear-programming-calculator) — Optimize a 2-variable objective function subject to linear inequality constraints using the graphical corner-point method with this free solver. - [Arithmetic & Geometric Sequence Solver](https://dothecalculation.com/calculators/number-sequence-calculator) — Generate terms, calculate general formulas, and sum series for arithmetic and geometric progressions with this free solver. - [Modular Arithmetic & Congruence Solver](https://dothecalculation.com/calculators/modular-arithmetic-calculator) — Perform modular addition, multiplication, and exponentiation, and solve linear modular congruence equations with clear steps. - [Derivative & Limit Solver](https://dothecalculation.com/calculators/derivative-solver) — Compute first and second derivatives of algebraic functions, find limits, and solve tangent line equations with clear steps. --- _This calculator is for educational and planning purposes. 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