OCR GCSE Maths: Growth and Decay in Real-World Contexts

OCR GCSE Maths: Growth and Decay in Real-World Contexts

Introduction

Growth and decay are fundamental concepts in mathematics, frequently encountered in real-world applications. In this tutorial, we'll explore how these concepts relate to financial calculations, including simple interest, compound interest, and depreciation. You'll also learn how to use percentage multipliers and exponential functions to solve problems accurately.

Understanding Growth and Decay

• Growth: When a quantity increases over time, we call it growth. Think of a savings account earning interest or a population expanding.
• Decay: Conversely, decay occurs when a quantity decreases over time. Examples include the depreciation of a car's value or the radioactive decay of a substance.

Percentage Multipliers

Percentage multipliers are crucial for calculating growth and decay. They represent the factor by which a quantity is multiplied after a certain period.

• Growth: A 5% increase is represented by a multiplier of 1.05 (1 + 0.05).
• Decay: A 10% decrease is represented by a multiplier of 0.9 (1 - 0.1).

Simple Interest

Simple interest is calculated only on the initial principal amount. The formula is:

Simple Interest = (Principal * Rate * Time) / 100

• Principal: The initial amount invested or borrowed.
• Rate: The annual interest rate.
• Time: The duration of the investment or loan in years.

Example:

You invest £1000 at a simple interest rate of 5% for 3 years.

• Simple Interest = (1000 * 5 * 3) / 100 = £150

Compound Interest

Compound interest is calculated on the principal amount and the accumulated interest from previous periods. This leads to exponential growth.

• Formula:

Final Amount = Principal * (1 + (Rate / 100))^Time

Example:

You invest £1000 at a compound interest rate of 5% for 3 years.

• Final Amount = 1000 * (1 + (5 / 100))^3 = £1157.63

Depreciation

Depreciation represents the decrease in value of an asset over time due to wear and tear, obsolescence, or market factors.

• Formula:

Final Value = Initial Value * (1 - (Depreciation Rate / 100))^Time

Example:

A car worth £20,000 depreciates at a rate of 10% per year. After 2 years, its value will be:

• Final Value = 20000 * (1 - (10 / 100))^2 = £16,200

Solving Problems

1. Identify the type of growth or decay: Is it simple interest, compound interest, or depreciation?
2. Determine the starting value, rate, and time: These values will be given in the problem.
3. Apply the appropriate formula: Use the formula for the relevant type of growth or decay.
4. Calculate the final value or interest: Substitute the values into the formula and solve.

Practice Problems

1. You deposit £500 into a savings account with a simple interest rate of 3% per year. How much interest will you earn after 5 years?
2. A company buys a machine for £25,000. The machine depreciates at a rate of 8% per year. What will be the machine's value after 4 years?
3. You invest £1000 at a compound interest rate of 6% per year. Calculate the value of your investment after 10 years.

Conclusion

Understanding growth and decay concepts is essential for navigating financial situations and interpreting real-world data. By mastering percentage multipliers, interest calculations, and depreciation, you'll gain valuable tools for solving a wide range of mathematical problems.