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Home › Math › Ratios, Proportions & Rates › Solving Proportions with Cross-Multiplication

Solving Proportions with Cross-Multiplication

Master the cross-multiplication technique for solving proportions, and apply it to multi-step real-world word problems.

Lesson 4 of 10 Ratios, Proportions & Rates Intermediate ⏱ 9 min read
🔥 Why This Matters

Cross-multiplication is the universal key to solving "if this, then how much of that?" questions. How many cups of flour for 36 cookies when the recipe makes 24? How long to drive 312 miles at the same speed you drove 130 miles in 2 hours? Every time you scale something — a recipe, a budget, a distance estimate — cross-multiplication turns a proportion into a one-step equation you can solve in seconds.

🎯 What You'll Learn
  • Apply the cross-multiplication property to convert a proportion into a linear equation
  • Solve for an unknown whether it appears in the numerator or denominator
  • Verify a solution by substituting back into the original proportion
📖 Key Vocabulary
Cross-MultiplicationMultiplying each numerator by the opposite denominator: if a/b = c/d, then a×d = b×c. Cross-ProductThe result of cross-multiplying — a×d and b×c. Equal in any true proportion. Unknown VariableThe value you're solving for — usually x — placed in one of the four positions of the proportion. VerificationSubstituting the answer back into both ratios to confirm they are equal.
Key Concept

For any proportion \(\dfrac{a}{b} = \dfrac{c}{d}\), the cross-multiplication property states:

\[ a \cdot d = b \cdot c \]

This works because multiplying both sides of the proportion by the product \(bd\) eliminates both fractions simultaneously, leaving a simple multiplication equation. Steps to solve:

  1. Write the proportion with \(x\) in one position.
  2. Cross-multiply to get a single equation.
  3. Divide both sides to isolate \(x\).
  4. Check by substituting back.

The Cross-Multiply Arrows

Cross-Multiplication — a/b = c/d

a
b
=
c
d

Green diagonal: a × d  |  Blue diagonal: b × c  →  Set equal and solve.

Worked Example 1 — Basic: Unknown in the Numerator

Solve: \(\dfrac{x}{15} = \dfrac{4}{5}\)

  1. Cross-multiply: \(x \times 5 = 15 \times 4\)
  2. Simplify: \(5x = 60\)
  3. Divide: \(x = 12\)
\[ x = 12 \]

Check: \(\dfrac{12}{15} = 0.8\) and \(\dfrac{4}{5} = 0.8\). ✓

Worked Example 2 — Intermediate: Unknown in the Denominator

Solve: \(\dfrac{7}{x} = \dfrac{21}{45}\)

  1. Cross-multiply: \(7 \times 45 = x \times 21\)
  2. Simplify: \(315 = 21x\)
  3. Divide: \(x = 315 \div 21 = 15\)
\[ x = 15 \]

Check: \(\dfrac{7}{15} \approx 0.467\) and \(\dfrac{21}{45} \approx 0.467\). ✓

Worked Example 3 — Real World: Fuel for a Road Trip

A car travels 150 miles on 5 gallons of gas. How many gallons are needed for a 390-mile road trip? Let \(x\) = gallons needed.

\[ \frac{150 \text{ mi}}{5 \text{ gal}} = \frac{390 \text{ mi}}{x \text{ gal}} \]
  1. Cross-multiply: \(150x = 5 \times 390 = 1950\)
  2. Divide: \(x = 1950 \div 150 = 13\)

The road trip requires 13 gallons. Check: \(150/5 = 30\) mpg; \(390/13 = 30\) mpg ✓.

✏️ Quick Check

Solve each proportion — show your cross-multiplication step:

  1. \(\dfrac{x}{9} = \dfrac{8}{12}\)
  2. \(\dfrac{3}{x} = \dfrac{9}{24}\)
  3. If 6 workers complete a job in 8 days, how many days for 4 workers at the same rate?
▶ Show Answers
  1. \(12x = 9 \times 8 = 72\) → x = 6. Check: 6/9 = 8/12 = 2/3 ✓
  2. \(9x = 3 \times 24 = 72\) → x = 8. Check: 3/8 = 9/24 = 3/8 ✓
  3. \(\frac{6}{8} = \frac{4}{x}\) → \(6x = 32\) → x = 5.33 days. (Fewer workers → more days, so set it up as workers/days or use inverse proportion reasoning.)
⚠️ Common Mistakes
  • Multiplying across instead of diagonally: a/b = c/d does NOT mean a×c = b×d. Cross-multiply diagonally: a×d = b×c.
  • Forgetting to divide after cross-multiplying: After getting 5x = 60, students sometimes write x = 60 instead of x = 12. Always complete the solve by dividing.
  • Not checking the answer: A quick substitution back into both ratios confirms you haven't made an arithmetic slip — takes 10 seconds and saves wrong answers.
✅ Key Takeaways
  • Cross-multiply diagonally: a/b = c/d → a×d = b×c. This converts the proportion to a one-step equation.
  • Always divide to finish: after cross-multiplying you get a×d = b×c; isolate x by dividing both sides.
  • x can be anywhere — numerator or denominator — the process is identical.
  • Always verify by substituting your answer back in and confirming both sides are equal.
💼 Career Connection — Architecture & Construction

Architects scale blueprints constantly. A drawing at 1/4 inch = 1 foot means a room that measures 3.75 inches on paper is actually 15 feet in reality — solved in seconds with a proportion. When contractors order materials for a 250-unit apartment building based on a 4-unit prototype, every material quantity is a solved proportion. Errors here translate directly to cost overruns — sometimes millions of dollars.

Calculator Connection

The Proportion Solver handles cross-multiplication automatically — enter any three of the four values in \(\frac{a}{b} = \frac{c}{d}\) and it solves for the fourth with step-by-step work shown.

Try it with the Calculator

Apply what you've learned with this tool.

Proportion Solver
Solve for x in a proportion (two equal ratios) like a/b = c/d.
Use calculator →
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