Lesson 7.1.2

Properties of Proportions

Proportions can be rewritten in several equivalent forms. Mastering these properties lets you rearrange proportions to match the information given in a problem.

Introduction

When you know , you can rewrite it in many equivalent ways. These aren't new equations — they're the same relationship viewed from different angles. These properties are especially useful in similarity proofs.

Past Knowledge

Ratios & cross-multiplying (7.1.1). Equivalent fractions.

Today's Goal

Use reciprocal, exchange, add-the-denominator, and subtract properties.

Future Success

Similarity proofs (7.2), Triangle Proportionality Theorem (7.2.4).

Key Concepts

If , then all of the following are also true:

1. Reciprocal Property

Flip both fractions.

2. Exchange (Means/Extremes) Property

Swap the means (inner terms) or the extremes (outer terms).

3. Add-the-Denominator Property

Add the denominator to the numerator on both sides. Extremely useful in proofs!

4. Subtract-the-Denominator Property

Subtract the denominator from the numerator on both sides.

Worked Examples

Basic

Rewriting a Proportion

If , write three other true proportions using properties.

Reciprocal:

Exchange:

Add-denominator: →

All produce the same solution:

Intermediate

Using Add-the-Denominator

If and , show that .

Apply add-the-denominator:

Since and :

✓

Advanced

Geometric Application

In a figure, and . Find and .

From the ratio: for some

→ →

Common Pitfalls

Flipping Only One Side

The reciprocal property requires flipping both fractions: . Writing is wrong.

Adding to the Wrong Part

In the add-the-denominator property, you add the denominator to the numerator: , NOT .

Real-Life Applications

Financial Ratios

Investors compare price-to-earnings ratios across companies. Proportion properties let analysts quickly rearrange and compare these ratios in different formats.

Architectural Blueprints

Blueprints use fixed proportions (e.g., 1/4 inch = 1 foot). The exchange and reciprocal properties let architects switch between “blueprint-to-real” and “real-to-blueprint” conversions.