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Percentages

Definition

A percentage is a way of expressing a number as a fraction of 100. The word "percent" means "per hundred." The symbol for percent is %.

For example, 25% means 25 out of 100, which can also be written as the fraction 25/100 or the decimal 0.25.

Prerequisites

To work with percentages, you should understand:

  1. Fractions and Decimals
  2. Multiplication and Division
  3. Ratios and Proportions

Learning Objectives

After mastering this topic, you should be able to:

  1. Convert between percentages, fractions, and decimals.
  2. Calculate the percentage of a given number (e.g., find 20% of 150).
  3. Find what percentage one number is of another (e.g., what percent of 50 is 10?).
  4. Calculate the original number when a percentage is known (e.g., 30 is 15% of what number?).
  5. Calculate percent increase and percent decrease.
  6. Solve real-world problems involving percentages (discounts, taxes, interest, tips).

Key Concepts & Calculations

Conversions

  • Percent to Decimal: Divide by 100 (or move the decimal point two places left). Example: 75% = 0.75
  • Decimal to Percent: Multiply by 100 (or move the decimal point two places right). Example: 0.4 = 40%
  • Percent to Fraction: Write the percent over 100 and simplify. Example: 60% = 60/100 = 3/5
  • Fraction to Percent: Convert the fraction to a decimal (divide numerator by denominator), then convert the decimal to a percent. Example: 1/4 = 0.25 = 25%

Basic Percentage Problems

  • Finding the Part: "What is P% of W?" -> Part = (P/100) × W
    • Example: What is 15% of 200? -> Part = (15/100) × 200 = 0.15 × 200 = 30.
  • Finding the Percent: "What percent of W is P?" -> Percent = (Part / Whole) × 100%
    • Example: What percent of 80 is 12? -> Percent = (12 / 80) × 100% = 0.15 × 100% = 15%.
  • Finding the Whole: "P is P% of what number?" -> Whole = Part / (P/100)
    • Example: 45 is 30% of what number? -> Whole = 45 / (30/100) = 45 / 0.30 = 150.

Percent Change

  • Percent Increase: [(New Value - Original Value) / Original Value] × 100%
  • Percent Decrease: [(Original Value - New Value) / Original Value] × 100%
  • Formula: Percent Change = (|Change| / Original Value) × 100%

Examples

Example 1: Conversion (Elementary/Middle School)

Problem: Convert 45% to a decimal and a fraction. Solution: * Decimal: 45 / 100 = 0.45 * Fraction: 45/100 = 9/20 (simplified)

Example 2: Finding Percentage of a Number (Middle School)

Problem: Calculate 60% of 80. Solution: Method 1 (Decimal): 0.60 × 80 = 48 Method 2 (Fraction): (60/100) × 80 = (3/5) × 80 = 240 / 5 = 48

Example 3: Percent Decrease (Discount) (Middle School/High School)

Problem: A shirt originally priced at $50 is on sale for 20% off. What is the sale price? Solution: 1. Calculate the discount amount: 20% of $50 = 0.20 × $50 = $10. 2. Subtract the discount from the original price: $50 - $10 = $40. Alternatively: The sale price is 100% - 20% = 80% of the original price. 0.80 × $50 = $40.

Example 4: Percent Increase (Tax) (High School)

Problem: A meal costs $75. If the sales tax is 8%, what is the total cost? Solution: 1. Calculate the tax amount: 8% of $75 = 0.08 × $75 = $6. 2. Add the tax to the original price: $75 + $6 = $81. Alternatively: The total cost is 100% + 8% = 108% of the meal cost. 1.08 × $75 = $81.

Common Misconceptions

  1. Incorrect Conversion: Errors when converting between percent, decimal, and fraction forms.
  2. Using the Wrong Base for Percent Change: Always divide the change by the original value.
  3. Adding Percentages Directly: A 10% increase followed by a 10% decrease does not result in the original value.
  4. Confusing Percent Of vs. Percent Increase/Decrease: Calculating 20% of a number is different from calculating a 20% increase.

Applications in SAT

Percentages are frequently tested on the SAT, particularly in the "Problem Solving and Data Analysis" section:

  1. Direct Calculation: Finding percent of a number, percent change.
  2. Word Problems: Discounts, taxes, tips, interest rates, markups.
  3. Data Interpretation: Analyzing data presented in tables or graphs that involve percentages.
  4. Ratios and Proportions: Problems often combine ratios and percentages.

Advanced Connections

Understanding percentages is crucial for:

  1. Statistical Analysis - Expressing proportions and interpreting statistical results.
  2. Data Interpretation - Understanding charts and graphs using percentages.
  3. Financial Mathematics: Simple and compound interest, loans, investments.
  4. Science: Expressing concentrations, error margins, and changes.

Practice Problems

  1. Basic: Convert 3/8 to a percentage.
  2. Intermediate: 70 is what percent of 200?
  3. Advanced: The price of a stock increased from $40 to $45. What was the percent increase?
  4. SAT-Level: A store buys a jacket for $60 and marks it up by 50%. If the jacket doesn't sell, the store then discounts the marked-up price by 30%. What is the final price of the jacket?