About

Cross multiplication converts a proportion ab = cd into a linear equation a ⋅ d = b ⋅ c, then isolates the unknown through division. The method works because multiplying both sides of an equation by the same non-zero quantity preserves equality. A miscalculated proportion propagates errors downstream - scaling recipes, converting currencies, dosing medication, or sizing structural members all depend on correct ratio arithmetic. Manual computation invites sign errors and decimal-point slips, particularly when one operand is negative or irrational.

This calculator accepts any three known values in a four-term proportion, detects the unknown automatically, and returns the result with configurable decimal precision. It also generates a verification proof: both cross products are computed independently so you can confirm a ⋅ d = b ⋅ c holds. Note: the tool assumes all denominators are non-zero. Division by zero is trapped and reported explicitly.

Formulas

Cross multiplication is derived from the fundamental property of proportions. Given the equality of two ratios:

ab = cd

Multiplying both sides by b ⋅ d yields the cross product identity:

a ⋅ d = b ⋅ c

Isolating each variable produces four solution formulas:

a = b ⋅ cd

b = a ⋅ dc

c = a ⋅ db

d = b ⋅ ca

Where a = numerator of the first ratio, b = denominator of the first ratio, c = numerator of the second ratio, d = denominator of the second ratio. The constraint b ≠ 0 and d ≠ 0 must hold for the proportion to be defined. Additionally, the divisor in the solution formula must be non-zero.

Reference Data

Scenario	Proportion Setup	Unknown	Formula	Example	Result
Scale a recipe	34 = x12	x	x = 3 ⋅ 124	36 ÷ 4	9
Map scale distance	150000 = 3.5x	x	x = 50000 ⋅ 3.5	175000 cm	175000
Currency conversion	10.92 = 250x	x €	x = 0.92 ⋅ 250	230	230
Drug dosage (mg/kg)	51 = x72	x mg	x = 5 ⋅ 72	360	360
Photo aspect ratio	169 = 1920x	x px	x = 9 ⋅ 192016	17280 ÷ 16	1080
Gear ratio	4015 = x6	x teeth	x = 40 ⋅ 615	240 ÷ 15	16
Dilution ratio	110 = x500	x mL	x = 50010	500 ÷ 10	50
Similar triangles	58 = 15x	x	x = 8 ⋅ 155	120 ÷ 5	24
Percentage proportion	25100 = x360	x	x = 25 ⋅ 360100	9000 ÷ 100	90
Speed/time proportion	602 = 150x	x hr	x = 2 ⋅ 15060	300 ÷ 60	5
Concrete mix ratio	13 = 4x	x bags	x = 3 ⋅ 4	12	12
Exchange rate	1110.5 = 50x	x ¥	x = 110.5 ⋅ 50	5525	5525

Frequently Asked Questions

Division by zero is undefined. If you enter values where the divisor would be zero (e.g., solving for d when a = 0), the calculator reports an error. However, note a subtle case: if a = 0 and c = 0, the proportion 0/b = 0/d holds for any non-zero d, making it indeterminate rather than undefined. The tool flags both conditions explicitly.

Yes. The algebraic identity a ⋅ d = b ⋅ c is valid over the entire real number field R, including negatives and non-integers. Enter −3.7 exactly as you would any number. The calculator preserves sign through all arithmetic steps and displays it in the step-by-step breakdown.

The calculator rounds the final answer to your chosen decimal places (default: 4). The verification then recomputes both cross products using the rounded value. Due to floating-point representation in IEEE 754, the two products may differ by a small epsilon (typically < 10⁻¹⁰). The tool uses a tolerance of 10⁻⁹ to confirm equality and displays both products so you can inspect rounding impact.

Cross multiplication is unit-agnostic. It operates on the numeric values only. You must ensure that the units are consistent within each ratio position. For example, 60 km / 1 hr = x km / 3 hr is valid because km occupies both numerators and hr both denominators. Mixing km in a numerator with miles in the other numerator produces a meaningless result.

This tool solves a single four-term proportion at a time. For a chain of equal ratios, decompose it into pairwise proportions: solve a/b = c/d first, then use the result in c/d = e/f. Each step is an independent cross multiplication.

Two similar triangles have proportional sides. If triangle 1 has sides a and b, and triangle 2 has corresponding sides c and d, the ratio a/b = c/d encodes the similarity condition. Cross multiplying recovers the area relationship: the rectangle a × d has the same area as rectangle b × c. This geometric invariant is why the method appears in Euclid's Elements (Book V, Definition 5).