Growth and Decay Problems for GCSE Mathematics
Introduction
Growth and decay problems are an essential part of GCSE Mathematics. These problems involve quantities that change over time, and they can be used to model a wide range of real-world situations, such as the growth of bacteria or the decay of radioactive elements.
Key Concepts and Definitions
- Exponential growth: A quantity that increases by a constant percentage over time.
- Exponential decay: A quantity that decreases by a constant percentage over time.
- Halflife: The amount of time it takes for a quantity to decrease by half.
Exponential Growth
The formula for exponential growth is:
```
y = a(1 + r)^t
```
where:
- y is the final value
- a is the initial value
- r is the growth rate (as a decimal)
- t is the time
Exponential Decay
The formula for exponential decay is:
```
y = a(1 - r)^t
```
where:
- y is the final value
- a is the initial value
- r is the decay rate (as a decimal)
- t is the time
Common Mistakes to Avoid
- Forgetting to convert percentages to decimals: Growth and decay rates are always expressed as decimals in GCSE Mathematics.
- Using the wrong formula: Make sure you use the correct formula depending on whether the quantity is growing or decaying.
- Not considering the units: Pay attention to the units of the growth or decay rate.
Worked Examples
- Example 1:
A population of bacteria doubles every hour. If there are initially 100 bacteria, how many bacteria will there be after 3 hours?
- Solution:
```
y = 100(1 + 0.5)^3 = 400
```
- Example 2:
A radioactive element has a half-life of 10 years. If there are initially 100 grams of the element, how much will be left after 20 years?
- Solution:
```
y = 100(0.5)^(20/10) = 25
```
Practice Problems
1. A population of fish grows by 10% per year. If there are initially 500 fish, how many fish will there be after 5 years?
2. A car depreciates by 20% per year. If the car is initially worth £10,000, what will it be worth after 3 years?
Conclusion
Growth and decay problems are a fundamental part of GCSE Mathematics. By understanding the key concepts and formulas, you can solve these problems with confidence. Remember to practice regularly and use the tips and resources provided to succeed in your exams.
FAQs
- Q: What is the difference between growth and decay?
A: Growth means the quantity increases over time, while decay means the quantity decreases over time.
- Q: How do I know which formula to use?
A: Use the exponential growth formula if the quantity is increasing, and the exponential decay formula if the quantity is decreasing.
- Q: What are some realworld applications of growth and decay problems?
A: Growth problems can be used to model the growth of populations, while decay problems can be used to model the decay of radioactive elements or the depreciation of assets.