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Ratios and Proportions

Definition

A ratio compares two quantities. A proportion states that two ratios are equal. Example Ratio: 3:4 or 3/4 Example Proportion: 3/4 = 6/8

Prerequisites

  1. Multiplication and Division
  2. Fractions and Decimals

Learning Objectives

  1. Understand and write ratios.
  2. Understand and solve proportions.
  3. Apply ratios and proportions to solve real-world problems (e.g., scaling, unit rates).

Examples

Example 1: Writing Ratios

Problem: A class has 12 boys and 15 girls. What is the ratio of boys to girls? Solution: 12:15, which simplifies to 4:5.

Example 2: Solving Proportions

Problem: Solve for x: x/5 = 12/20 Solution: Cross-multiply: 20x = 5 * 12 -> 20x = 60 -> x = 3.

Common Misconceptions

  1. Mixing up the order of quantities in a ratio.
  2. Errors in cross-multiplication.

Applications in SAT

Ratios and proportions are fundamental in the Problem Solving and Data Analysis section, often appearing in word problems involving scaling, rates, and percentages.

Advanced Connections

  1. Percentages
  2. Similar Figures (Geometry)
  3. Statistical Analysis

Practice Problems

  1. Basic: Simplify the ratio 18:24.
  2. Intermediate: If 3 apples cost $2, how much do 12 apples cost?
  3. Advanced: The ratio of length to width of a rectangle is 5:2. If the perimeter is 70 cm, find the length and width.