Cross Multiplication Calculator
Solve for any unknown variable x in proportions (A/B = C/D) with step-by-step cross product proofs.
Proportion Equation
Solution & Analysis
Step 2: 100 = 5x
Step 3: x = 100 ÷ 5 = 20
Real-World Applications of Cross Multiplication
Cross multiplication is an essential mathematical shortcut used whenever two ratios share an equivalent proportional relationship. From scaling culinary recipes for large parties to converting foreign currency exchange rates and calculating architectural blueprint distances, cross multiplication simplifies fraction algebra into a single multiplication and division step.
For related problem solving and complete calculations, you can also explore our Ratio Calculator and Fraction Calculator.
The Cross Product Identity
Mathematically, multiplying both sides of $\frac{A}{B} = \frac{C}{D}$ by the common denominator $(B \times D)$ cancels out the fractions completely, producing:
Frequently Asked Questions
How does cross multiplication work to solve for x?
In any proportion A / B = C / D, the product of the means equals the product of the extremes: A × D = B × C. If one value is unknown (x), you multiply the two diagonal known numbers together and divide by the remaining number opposite to x.
What is the difference between direct and inverse proportions?
In a direct proportion (A/B = C/D), as one quantity increases, the other increases proportionally at constant ratio (e.g. recipe ingredients). In an inverse proportion (A × B = C × D), as one quantity increases, the other decreases (e.g. more construction workers finish a job in fewer days).
Can cross multiplication be used with negative numbers and decimals?
Yes. Cross multiplication is a fundamental algebraic identity valid across all real numbers (positive, negative, fractions, and decimals), provided that neither denominator is zero.
Why can't the denominator be zero?
Division by zero is undefined in mathematics. If either B or D equals 0, the proportion represents an impossible ratio and cannot be solved.
When should I use inverse proportion instead?
Use inverse proportion when an increase in one variable causes a proportional decrease in the other. A classic example is worker productivity: if 4 painters take 10 hours to paint a house (4 × 10 = 40 worker-hours), then 8 painters will complete the same house in 5 hours (8 × 5 = 40).