Table of Contents

Percentage increases and decreases Level 8

Introduction

Have you ever wondered how much a product costs after a sale or how much your savings grow over time? Understanding percentage increases and decreases is essential in making sense of these everyday situations. In this article, we’ll explore what percentage changes are, how to calculate them, and why they matter in real life.

Definition and Concept

A percentage increase occurs when a quantity grows compared to its original amount, while a percentage decrease happens when a quantity shrinks. Both are expressed as a fraction of 100.

Example: If a shirt originally costs $50 and is now $60, the increase is calculated as follows:
Increase = (New Price – Original Price) / Original Price × 100 = (60-50)/50 × 100 = 20%.

Relevance:

  • Mathematics: Understanding percentages is vital for algebra and statistics.
  • Real-world applications: Used in finance, shopping, and data analysis.

Historical Context or Origin​

The concept of percentages dates back to ancient civilizations, where fractions were used to represent parts of a whole. The word ‘percent’ comes from the Latin ‘per centum,’ meaning ‘by the hundred.’ Over time, this concept evolved, and today, percentages are an essential part of mathematics and finance.

Understanding the Problem

To solve problems involving percentage increases and decreases, follow these steps:
1. Identify the original amount.
2. Determine the new amount.
3. Calculate the difference (increase or decrease).
4. Divide the difference by the original amount and multiply by 100 to find the percentage.

Methods to Solve the Problem with different types of problems​

Method 1: Direct Calculation

  • Identify the original value and the new value.
  • Calculate the difference.
  • Use the formula: Percentage Change = (Difference / Original Value) × 100.

    • Example:
    Original Price: $40, New Price: $50.

  • Difference: 50 – 40 = 10.
  • Percentage Increase: (10 / 40) × 100 = 25%.

    • Method 2: Using Proportions
    Set up a proportion to find the percentage.
    Example:
    If a product’s price increases from $30 to $36, set it up as:

  • (New Value – Original Value) / Original Value = x / 100.
  • (36 – 30) / 30 = x / 100.
  • 6/30 = x/100, which simplifies to x = 20%.

    • Exceptions and Special Cases​

  • Zero Increase/Decrease: If the new amount is the same as the original, the percentage change is 0%.
  • Negative Changes: If the new amount is less than the original, it indicates a percentage decrease.

    • Step-by-Step Practice​

    Problem 1: A jacket costs $80 and is on sale for $64. What is the percentage decrease?
    Solution:

  • Difference: 80 – 64 = 16.
  • Percentage Decrease: (16 / 80) × 100 = 20%.

    • Problem 2: A stock price rises from $150 to $180. What is the percentage increase?
    Solution:

  • Difference: 180 – 150 = 30.
  • Percentage Increase: (30 / 150) × 100 = 20%.

    • Examples and Variations

    Easy Example:

    • Problem: A book originally costs $20 and is now $15. What is the percentage decrease?
    • Solution:
      • Difference: 20 – 15 = 5.
      • Percentage Decrease: (5 / 20) × 100 = 25%.

    Moderate Example:

    • Problem: A car’s value decreases from $25,000 to $20,000. What is the percentage decrease?
    • Solution:
      • Difference: 25,000 – 20,000 = 5,000.
      • Percentage Decrease: (5,000 / 25,000) × 100 = 20%.

    Advanced Example:

    • Problem: A population of a city increases from 200,000 to 250,000. What is the percentage increase?
    • Solution:
      • Difference: 250,000 – 200,000 = 50,000.
      • Percentage Increase: (50,000 / 200,000) × 100 = 25%.

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    Common Mistakes and Pitfalls

    • Confusing percentage increase with percentage decrease.
    • Forgetting to subtract the original value from the new value.
    • Not converting the final answer into a percentage (e.g., forgetting to multiply by 100).

    Tips and Tricks for Efficiency

    • Always double-check your original and new values before calculating.
    • Use a calculator for quick calculations when dealing with large numbers.
    • Visualize the problem using graphs or charts if possible.

    Real life application

    • Shopping: Understanding sales and discounts.
    • Finance: Calculating interest rates and investment growth.
    • Health: Monitoring weight changes or fitness progress.

    FAQ's

    A negative percentage indicates a decrease in value. Just follow the same formula to calculate it.
    Absolutely! Percentages are used in shopping, cooking, budgeting, and many other daily activities.
    To find a percentage of a number, multiply the number by the percentage (as a decimal). For example, to find 20% of 50, do 50 × 0.20 = 10.
    Percentage points measure the absolute difference between two percentages, while percentages measure relative changes.
    Percentages help us understand proportions and make comparisons easier in various contexts, from finances to statistics.

    Conclusion

    Understanding percentage increases and decreases is crucial for making informed decisions in everyday life. By mastering these calculations, you can navigate financial situations, analyze data, and make comparisons with confidence.

    References and Further Exploration

    • Khan Academy: Lessons on percentages and their applications.
    • Book: Math for Everyday Life by John Doe.

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