Growth And Decay Quizzes

Introduction

Growth and decay is a key topic in GCSE Maths that involves quantities increasing or decreasing over time. It is widely applied in finance (interest, investments), population studies, biology (bacteria growth), and physics (radioactive decay). Understanding growth and decay allows students to solve real-life problems, recognize patterns, and apply exponential reasoning. Mastery of this topic also reinforces percentages, ratios, and proportional reasoning.

Core Concepts

What is Growth?

Growth occurs when a quantity increases over time. Examples include:

• Bank savings increasing with interest.
• Population growth.
• Investment value appreciation.

Growth can be modeled using the formula:

$$ \text{Final amount} = \text{Initial amount} \times (1 + r)^t $$>

Where:

• $r$ is the growth rate per period (as a decimal).
• $t$ is the number of periods (years, months, etc.).

What is Decay?

Decay occurs when a quantity decreases over time. Examples include:

• Depreciation of car value.
• Radioactive decay.
• Population decrease in a habitat.

Decay can be modeled using the formula:

$$ \text{Final amount} = \text{Initial amount} \times (1 - r)^t $$>

Where $r$ is the decay rate per period (as a decimal).

Understanding Percentages in Growth and Decay

Growth and decay rates are often given as percentages. Convert percentages to decimals before using formulas:

$$ r\% = \frac{r}{100} $$>

Example: 5% growth → $r = 0.05$

Rules & Steps

1. Identify whether the problem involves growth or decay.
2. Convert the percentage rate to a decimal.
3. Identify the initial amount ($P_0$) and number of periods ($t$).
4. Use the appropriate formula:
   • Growth: $P = P_0 \times (1 + r)^t$
   • Decay: $P = P_0 \times (1 - r)^t$
5. Perform calculations carefully and round as required.
6. Check your answer for reasonableness.

Worked Examples

1. Example 1 (Growth): A savings account has £500, growing at 5% per year. Find the amount after 3 years.
   Calculation: $$ P = 500 × (1 + 0.05)^3 = 500 × 1.157625 = £578.81 $$
2. Example 2 (Decay): A car is worth £12,000 and depreciates at 8% per year. Find its value after 4 years.
   Calculation: $$ P = 12000 × (1 - 0.08)^4 = 12000 × 0.7164 ≈ £8596.80 $$
3. Example 3 (Higher Level Growth): A population of 1,200 bacteria increases by 20% every hour. Find the population after 5 hours.
   Calculation: $$ P = 1200 × (1 + 0.20)^5 = 1200 × 2.48832 ≈ 2986 $$
4. Example 4 (Higher Level Decay): A radioactive substance has half-life of 6 hours. Initial mass 100 g. Find remaining mass after 18 hours.
   Step 1: Decay factor per half-life = 0.5 Step 2: Number of half-lives = 18 ÷ 6 = 3 Step 3: Remaining mass: $$ P = 100 × (0.5)^3 = 100 × 0.125 = 12.5 \text{ g} $$
5. Example 5: A town’s population decreases by 2% per year. If current population is 25,000, what will it be in 10 years?
   Calculation: $$ P = 25000 × (1 - 0.02)^{10} = 25000 × 0.8171 ≈ 20,428 $$

Common Mistakes

• Confusing growth and decay formulas.
• Not converting percentages to decimals before using formulas.
• Using the wrong number of periods $t$.
• Rounding too early in multi-step calculations.
• For half-life problems, forgetting to calculate the number of half-lives first.

Applications

• Finance: Compound interest, savings, and investment growth.
• Population Studies: Predicting population growth or decline.
• Science: Radioactive decay, chemical reactions, and biological growth.
• Business: Depreciation of assets and forecasting stock levels.

Strategies & Tips

• Always identify whether it is growth or decay before choosing the formula.
• Convert all percentages to decimals for calculation.
• Keep a consistent number of periods; check whether time is in years, months, or hours.
• Use a calculator for exponentiation to maintain accuracy.
• Practice a variety of real-life contexts to strengthen understanding.

Summary

Growth and decay describe how quantities increase or decrease over time. Key formulas:

• Growth: $P = P_0 \times (1 + r)^t$
• Decay: $P = P_0 \times (1 - r)^t$

Ensure proper conversion of percentages to decimals, correct identification of periods, and careful calculation. Mastery of growth and decay enables students to solve a wide range of exam problems and apply mathematics in real-life situations. Reinforce your skills by attempting the quizzes in this subcategory and exploring more challenging scenarios!