Proportion Introduction

Algebra-1

1. Fundamental Concepts

Definition: A proportion is an equation stating that two ratios are equal.
Cross-Multiplication: The process of multiplying the numerator of one fraction by the denominator of the other in a proportion.
Equivalent Ratios: Ratios that have the same value when simplified.

2. Key Concepts

Basic Rule: $$\frac{{a}}{{b}} = \frac{{c}}{{d}} \implies ad = bc$$

Degree Preservation: The cross-multiplication rule always holds true for proportions

Application: Used to solve real-world problems involving comparisons and scaling

3. Examples

Example 1 (Basic)

Problem: Solve the proportion $$\frac{{3}}{{4}} = \frac{{x}}{{8}}$$

Step-by-Step Solution:

Cross-multiply: $$3 \cdot 8 = 4 \cdot x$$
Simplify: $$24 = 4x$$
Solve for $x$ : $$x = \frac{{24}}{{4}} = 6$$

Validation: Substitute $x=6$ → Original: $\frac{3}{4} = \frac{6}{8}$ ; Simplified: $\frac{3}{4} = \frac{3}{4}$ ✓

Example 2 (Intermediate)

Problem: Solve the proportion $$\frac{{5}}{{y}} = \frac{{10}}{{15}}$$

Step-by-Step Solution:

Cross-multiply: $$5 \cdot 15 = 10 \cdot y$$
Simplify: $$75 = 10y$$
Solve for $y$ : $$y = \frac{{75}}{{10}} = 7.5$$

Validation: Substitute $y=7.5$ → Original: $\frac{5}{7.5} = \frac{10}{15}$ ; Simplified: $\frac{2}{3} = \frac{2}{3}$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use diagrams to represent proportional relationships visually.
Error-Proofing: Always check if the cross-multiplication results in a consistent equation.
Concept Reinforcement: Practice with a variety of problems to reinforce understanding of proportions.