Lesson 3.9

Solving Proportions

A proportion is an equation that says two ratios are equal: . The fastest way to solve? Cross-multiply.

Introduction

Now we shift from simplifying expressions to solving equations. Proportions are the simplest type of rational equation — just two fractions set equal. Cross-multiplication converts them into a polynomial equation you already know how to solve.

Past Knowledge

Solving linear and quadratic equations, domain restrictions (Lesson 3.1).

Today's Goal

Solve proportions using cross-multiplication and check for domain restrictions.

Future Success

Lesson 3.10 generalizes this to equations with more than two rational terms.

Key Concepts

Cross-Multiplication

Multiply diagonally, then set the products equal.

Why it works: You're really just multiplying both sides by (the LCD), which clears both denominators simultaneously.

Always Check!

After solving, plug your answer back in.

If it makes any original denominator equal zero, it's extraneous and must be rejected. (More on this in Lesson 3.11.)

Worked Examples

Example 1: Linear Result

Basic

Solve .

1

Cross-multiply

2

Solve

✓ (no denominators become zero)

Example 2: Variable in Denominator

Intermediate

Solve .

1

Cross-multiply

2

Distribute

3

Solve

✓ (neither nor )

Example 3: Quadratic Result

Advanced

Solve .

1

Cross-multiply

2

Rearrange and use the quadratic formula

or ✓ (neither equals 0 or −3)

Common Pitfalls

Forgetting to Check Domain

Always verify your solution doesn't make an original denominator zero. This is especially important in Examples 2 and 3.

Distribution Errors

When cross-multiplying, make sure to distribute to every term in the binomial, not just the first.

Real-Life Applications

Proportions appear everywhere: scaling recipes, converting units, similar triangles in architecture, and calculating drug dosages in medicine. Any time two ratios should be equal, cross-multiplication is the tool.

Practice Quiz

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