Cross-Multiplication Explained: The Fastest Way to Solve Proportions

What Is Cross-Multiplication?

Cross-multiplication is a shortcut method for solving proportions. When you have:

A / B = C / D

Cross-multiplication tells us that A × D = B × C. This works because multiplying both sides of the equation by B × D eliminates the fractions.

Step-by-Step Examples

Example 1: Solving for the Last Value

Problem: 3/4 = 9/x

1. Cross-multiply: 3 × x = 4 × 9
2. Simplify: 3x = 36
3. Divide: x = 12

Verification: 3/4 = 0.75 and 9/12 = 0.75 ✓

Example 2: Solving for the First Value

Problem: x/7 = 6/14

1. Cross-multiply: x × 14 = 7 × 6
2. Simplify: 14x = 42
3. Divide: x = 3

Verification: 3/7 ≈ 0.4286 and 6/14 ≈ 0.4286 ✓

Example 3: With Decimals

Problem: 2.5/x = 5/8

1. Cross-multiply: 2.5 × 8 = x × 5
2. Simplify: 20 = 5x
3. Divide: x = 4

Verification: 2.5/4 = 0.625 and 5/8 = 0.625 ✓

Why Does Cross-Multiplication Work?

The mathematical proof is straightforward:

Starting with: A/B = C/D

Multiply both sides by B × D:

• Left side: (A/B) × B × D = A × D
• Right side: (C/D) × B × D = B × C

Therefore: A × D = B × C

This is why cross-multiplication always gives us a valid equation to solve.

Common Mistakes to Avoid

1. Wrong diagonal — Always multiply the numerator of one fraction by the denominator of the other
2. Forgetting to check for zero — If any denominator is zero, the proportion is undefined
3. Not simplifying first — Sometimes simplifying the known fraction first makes calculation easier
4. Sign errors with negatives — Be careful when cross-multiplying with negative numbers

Practice Problems

Try these on our Proportions Calculator to verify your answers:

1. 5/8 = 15/x (answer: x = 24)
2. x/12 = 7/28 (answer: x = 3)
3. 1.5/3 = x/10 (answer: x = 5)
4. 9/x = 3/7 (answer: x = 21)

Wrapping Up

Cross-multiplication is the most efficient technique for solving proportion problems. Once you internalize the pattern — multiply diagonals and solve — you can handle any proportion in seconds.

Use our Proportions Calculator to practice and verify your work!

---

Related Articles

• Getting Started with Proportions: A Complete Guide — New to proportions? Start with our beginner-friendly introduction
• Proportions in Finance: Master Ratios for Smarter Money Decisions — Apply cross-multiplication to real financial calculations
• Proportions in Cooking: How to Scale Any Recipe Like a Pro — Use the technique to scale recipes perfectly
• How Architects and Engineers Use Proportions Every Day — See cross-multiplication in construction and design