Ratio and Rate - Algebra-1

1. Fundamental Concepts

Definition: A ratio is a comparison of two quantities by division, often expressed as $$\text{{a}} \cdot \text{{b}}$$ where $$\text{{a}}$$ and $$\text{{b}}$$ are the quantities being compared.
Rate: A rate is a special type of ratio that compares quantities with different units, such as speed ( $$\text{{distance}} \cdot \text{{time}}$$ ).
Unit Rate: A unit rate is a rate in which the second term is 1, such as miles per hour ( $$\text{{miles}} \cdot \text{{hour}}$$ ).

2. Key Concepts

Simplifying Ratios: $$\frac{\text{{a}}}{\text{{b}}} = \frac{\text{{c}}}{\text{{d}}} \Rightarrow \text{{ad}} = \text{{bc}}$$

Proportional Relationships: If $$\frac{\text{{a}}}{\text{{b}}} = \frac{\text{{c}}}{\text{{d}}}$$ , then $$\text{{a}} \cdot \text{{d}} = \text{{b}} \cdot \text{{c}}$$

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

3. Examples

Example 1 (Basic)

Problem: Simplify the ratio $$\frac{10}{15}$$ .

Step-by-Step Solution:

Find the greatest common divisor (GCD) of 10 and 15, which is 5.
Divide both the numerator and the denominator by the GCD: $$\frac{10 \div 5}{15 \div 5} = \frac{2}{3}$$

Validation: The simplified ratio $$\frac{2}{3}$$ is in its lowest terms.

Example 2 (Intermediate)

Problem: If the ratio of boys to girls in a classroom is $$\frac{3}{4}$$ and there are 12 boys, how many girls are there?

Step-by-Step Solution:

Set up the proportion: $$\frac{3}{4} = \frac{12}{x}$$
Cross-multiply to solve for $$x$$ : $$3x = 48$$
Solve for $$x$$ : $$x = \frac{48}{3} = 16$$

Validation: Substituting back, the ratio $$\frac{12}{16}$$ simplifies to $$\frac{3}{4}$$ , confirming the solution.

4. Problem-Solving Techniques

Visual Strategy: Use diagrams or tables to represent ratios visually.
Error-Proofing: Always check if the final answer makes logical sense in the context of the problem.
Concept Reinforcement: Practice setting up and solving proportions using real-life scenarios.