Cross Multiplication Calculator

Solve proportions using cross multiplication: a/b = c/d. Find the unknown variable step-by-step.

Enter Proportion Values

Enter three known values. The unknown will be calculated.

3 4
=
6 ?

Result

Unknown Value
8
d = 8
a3
b4
c6
d8
Cross Product (a*d)24
Cross Product (b*c)24

Step-by-Step Solution

a/b = c/d => a*d = b*c

Understanding Cross Multiplication

Cross multiplication is a fundamental algebraic technique used to solve proportions -- equations that state two ratios are equal. Given a proportion a/b = c/d, cross multiplication transforms it into a*d = b*c, eliminating fractions and making it easy to solve for any unknown variable.

This method works because if two fractions are equal, their cross products must also be equal. It is one of the most frequently used techniques in algebra, geometry, chemistry (stoichiometry), physics, and everyday problem-solving involving rates, ratios, and scaling.

Cross Multiplication Formulas

Basic Cross Multiplication

If a/b = c/d, then cross multiplying gives:

a * d = b * c

Solve for a

Given b, c, and d, find a:

a = (b * c) / d

Solve for b

Given a, c, and d, find b:

b = (a * d) / c

Solve for c

Given a, b, and d, find c:

c = (a * d) / b

Solve for d

Given a, b, and c, find d:

d = (b * c) / a

Verification

After solving, verify by checking:

a/b = c/d (both sides equal)

How Cross Multiplication Works

Step-by-Step Process

  1. Write the proportion as a/b = c/d, identifying the unknown variable.
  2. Cross multiply: multiply the numerator of each fraction by the denominator of the other fraction.
  3. Set the two cross products equal: a*d = b*c.
  4. Isolate the unknown by dividing both sides by the coefficient of the unknown.
  5. Verify by substituting back into the original proportion.

Example Problem

Solve: 3/4 = x/12. Cross multiply: 3 * 12 = 4 * x, giving 36 = 4x. Divide both sides by 4: x = 9. Verify: 3/4 = 9/12 = 0.75. Both sides are equal, confirming x = 9.

Practical Applications

Cross multiplication is used in countless real-world situations. Map reading (scale ratios), recipe scaling, unit conversions, similar triangles in geometry, mixture problems in chemistry, and financial calculations (exchange rates, interest proportions) all rely on proportional reasoning and cross multiplication.

Common Mistakes to Avoid

  • Make sure you cross multiply correctly: numerator of one side times denominator of the other.
  • Check that the denominator is not zero before dividing.
  • Ensure units are consistent on both sides of the proportion.
  • Do not confuse cross multiplication with simple multiplication of fractions.
  • Always verify your answer by substituting back into the original proportion.

Why Cross Multiplication Works

The mathematical proof is straightforward. Starting with a/b = c/d, multiply both sides by b*d: (a/b)*b*d = (c/d)*b*d. Simplifying gives a*d = b*c. This algebraic manipulation is valid as long as b and d are non-zero, which is required since they are denominators of fractions.