IGCSE Exponential Growth and Decay: Complete Guide for Cambridge IGCSE Mathematics

IGCSE exponential growth and decay are important topics in Cambridge IGCSE Mathematics that appear in both Paper 2 and Paper 4. Mastering exponential functions, growth and decay formulas, and solving exponential equations is essential for understanding real-world phenomena like population growth, radioactive decay, and compound interest.

This comprehensive IGCSE exponential growth and decay guide covers everything you need to know, including exponential growth formulas, exponential decay formulas, solving exponential equations, half-life calculations, step-by-step worked examples, common exam questions, and expert tips from Tutopiya’s IGCSE maths tutors. We’ll also show you how to avoid the most common mistakes that cost students valuable marks.

🎯 What you’ll learn: By the end of this guide, you’ll know how to model exponential growth and decay, solve exponential equations, calculate half-life, and apply these skills to solve problems in IGCSE exams.

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Why IGCSE Exponential Growth and Decay Matter

IGCSE exponential growth and decay are practical topics with wide applications. Here’s why they’re so important:

• High frequency topic: Exponential function questions appear regularly in IGCSE maths papers
• Real-world applications: Used in biology (population growth), physics (radioactive decay), finance (compound interest), and medicine
• Exam weight: Typically worth 6-10 marks per paper
• Foundation for advanced topics: Essential for understanding logarithms and advanced functions
• Problem-solving skills: Develops algebraic manipulation and equation-solving abilities

Key insight from examiners: Students often confuse growth and decay formulas, or make errors when solving exponential equations. This guide will help you master these systematically.

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Understanding Exponential Functions

An exponential function has the form:

General form: y = a × b^x

Where:

• a is the initial value (when x = 0)
• b is the base (growth/decay factor)
• x is the variable (often time)

Key characteristics:

• If b > 1: exponential growth (increases)
• If 0 < b < 1: exponential decay (decreases)
• The graph is always positive (if a > 0)
• The graph passes through (0, a)

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Exponential Growth

Exponential growth occurs when a quantity increases by a constant percentage over equal time periods.

Growth Formula

Formula: A = A₀ × (1 + r)^t

Where:

• A = final amount
• A₀ = initial amount
• r = growth rate (as a decimal, e.g., 5% = 0.05)
• t = time

Alternative form: A = A₀ × b^t where b = 1 + r

Examples of Exponential Growth

• Population growth
• Compound interest
• Bacterial growth
• Investment growth

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Exponential Decay

Exponential decay occurs when a quantity decreases by a constant percentage over equal time periods.

Decay Formula

Formula: A = A₀ × (1 - r)^t

Where:

• A = final amount
• A₀ = initial amount
• r = decay rate (as a decimal)
• t = time

Alternative form: A = A₀ × b^t where b = 1 - r (so 0 < b < 1)

Examples of Exponential Decay

• Radioactive decay
• Depreciation
• Cooling/heating
• Drug elimination from body

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Worked Examples: Exponential Growth

Example 1: Population Growth

A population of 5,000 grows at 3% per year. Find the population after 10 years.

Solution:

1. A₀ = 5000, r = 0.03, t = 10
2. A = 5000 × (1.03)^10
3. A = 5000 × 1.3439 = 6,719.5
4. Round to nearest whole: 6,720

Answer: 6,720 people

Example 2: Investment Growth

$10,000 is invested at 5% per annum, compounded annually. Find the value after 8 years.

Solution:

1. A₀ = 10000, r = 0.05, t = 8
2. A = 10000 × (1.05)^8
3. A = 10000 × 1.4775 = $14,775

Answer: $14,775

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Worked Examples: Exponential Decay

Example 1: Radioactive Decay

A radioactive substance has 800 grams initially and decays at 5% per hour. Find the amount after 6 hours.

Solution:

1. A₀ = 800, r = 0.05, t = 6
2. A = 800 × (1 - 0.05)^6 = 800 × (0.95)^6
3. A = 800 × 0.7351 = 588.08

Answer: 588 g (to nearest gram)

Example 2: Depreciation

A car worth $25,000 depreciates at 12% per year. Find its value after 5 years.

Solution:

1. A₀ = 25000, r = 0.12, t = 5
2. A = 25000 × (1 - 0.12)^5 = 25000 × (0.88)^5
3. A = 25000 × 0.5277 = $13,192.50

Answer: $13,193 (to nearest dollar)

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Half-Life

Half-life is the time it takes for a quantity to reduce to half its initial value (used mainly for decay).

Finding Half-Life

Method: Set A = A₀/2 and solve for t

Example: A substance decays at 8% per hour. Find the half-life.

Solution:

1. A = A₀/2, r = 0.08
2. A₀/2 = A₀ × (0.92)^t
3. 1/2 = (0.92)^t
4. Take logarithms: log(0.5) = t × log(0.92)
5. t = log(0.5) / log(0.92) = -0.3010 / -0.0362 = 8.31 hours

Answer: 8.3 hours (to 1 d.p.)

Using Half-Life

If you know the half-life, you can find the amount after any time.

Example: A substance has a half-life of 5 hours. If there are 200 grams initially, find the amount after 15 hours.

Solution:

1. Number of half-lives: 15 ÷ 5 = 3
2. After 3 half-lives: 200 × (1/2)³ = 200 × 1/8 = 25 grams

Answer: 25 grams

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Solving Exponential Equations

Method 1: Using Logarithms

Example: Solve 3^x = 81

Solution:

1. 3^x = 81 = 3⁴
2. Therefore: x = 4

Answer: x = 4

Example: Solve 2^(x+1) = 32

Solution:

1. 32 = 2⁵
2. 2^(x+1) = 2⁵
3. x + 1 = 5
4. x = 4

Answer: x = 4

Method 2: Taking Logarithms

Example: Solve 5^x = 100

Solution:

1. Take log of both sides: log(5^x) = log(100)
2. x × log(5) = log(100)
3. x = log(100) / log(5) = 2 / 0.6990 = 2.86

Answer: x = 2.86 (to 2 d.p.)

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Finding the Growth/Decay Rate

Sometimes you need to find the rate from given information.

Example: A population grows from 4,000 to 5,000 in 3 years. Find the annual growth rate.

Solution:

1. A₀ = 4000, A = 5000, t = 3
2. 5000 = 4000 × (1 + r)³
3. (1 + r)³ = 5000/4000 = 1.25
4. 1 + r = ∛1.25 = 1.0794
5. r = 0.0794 = 7.94%

Answer: 7.9% per year (to 1 d.p.)

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Step-by-Step Method for Exponential Problems

1. Identify the type - Growth or decay?
2. List given information - A₀, A, r, t (identify what’s missing)
3. Choose the formula:
   • Growth: A = A₀(1 + r)^t
   • Decay: A = A₀(1 - r)^t
4. Substitute and solve - Rearrange if necessary
5. Check your answer - Does it make sense?

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Common Examiner Traps (and How to Dodge Them)

• Confusing growth and decay - Growth: 1 + r, Decay: 1 - r
• Rate conversion errors - Always convert percentage to decimal (divide by 100)
• Time unit errors - Ensure time units match the rate (e.g., if rate is per year, time must be in years)
• Formula errors - Growth uses +, decay uses -
• Solving exponential equations - Use logarithms or recognize powers
• Half-life confusion - Half-life is for decay, not growth

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IGCSE Exponential Growth and Decay Practice Questions

Question 1: Growth

A population of 8,000 grows at 4% per year. Find the population after 6 years.

Solution: A = 8000 × (1.04)^6 = 8000 × 1.2653 = 10,122.4

Answer: 10,122 people (to nearest whole)

Question 2: Decay

A substance has 500 grams and decays at 6% per hour. Find the amount after 4 hours.

Solution: A = 500 × (0.94)^4 = 500 × 0.7807 = 390.35

Answer: 390 g (to nearest gram)

Question 3: Finding Rate

An investment grows from $5,000 to $6,500 in 4 years. Find the annual growth rate.

Solution:

1. 6500 = 5000 × (1 + r)⁴
2. (1 + r)⁴ = 1.3
3. 1 + r = 1.0678
4. r = 0.0678 = 6.78%

Answer: 6.8% per year (to 1 d.p.)

Question 4: Half-Life

A substance decays at 10% per hour. Find the half-life.

Solution:

1. A₀/2 = A₀ × (0.9)^t
2. 0.5 = (0.9)^t
3. t = log(0.5) / log(0.9) = 6.58 hours

Answer: 6.6 hours (to 1 d.p.)

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Frequently Asked Questions About IGCSE Exponential Growth and Decay

What’s the difference between growth and decay?

• Growth: Quantity increases, formula uses 1 + r (where r > 0)
• Decay: Quantity decreases, formula uses 1 - r (where 0 < r < 1)

What is the exponential growth formula?

A = A₀(1 + r)^t where A₀ is initial amount, r is growth rate, t is time.

What is the exponential decay formula?

A = A₀(1 - r)^t where A₀ is initial amount, r is decay rate, t is time.

What is half-life?

Half-life is the time for a quantity to reduce to half its initial value (used for decay).

How do I solve exponential equations?

Use logarithms or recognize when both sides can be written as powers of the same base.

How do I find the growth/decay rate?

Rearrange the formula to solve for r, then convert to percentage.

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Related IGCSE Maths Resources

Strengthen your IGCSE Mathematics preparation with these comprehensive guides:

• IGCSE Compound Interest: Complete Guide - Master compound interest calculations
• IGCSE Exponents and Surds: Complete Guide - Master index laws
• IGCSE Maths Revision Notes, Syllabus and Preparation Tips - Complete syllabus overview, topic breakdown, and revision strategies
• IGCSE Past Papers Guide - Access free IGCSE past papers and exam resources

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