Growth and Decay [H]

This is a Higher-tier only topic in the AQA GCSE Mathematics specification. Growth and decay extends the compound interest and depreciation work into more general contexts, including population growth, radioactive decay, and exponential change. You need to be able to use and interpret exponential formulae and solve problems in context.

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Exponential Growth

Exponential growth occurs when a quantity increases by a fixed percentage in each time period. This is the same principle as compound interest, applied more broadly.

General Formula

Final Value = Initial Value x (Multiplier) to the power of n

Where:

• Multiplier = 1 + (percentage increase / 100)
• n = number of time periods

Recognising Exponential Growth

Feature	Description
Shape of graph	Starts slowly, then curves steeply upward
Rate of increase	Gets faster and faster over time
Multiplier	Greater than 1
Real-world examples	Population growth, bacterial growth, investment returns

Worked Example 1: A colony of bacteria doubles every hour. Starting with 500 bacteria, how many will there be after 8 hours?

• Multiplier = 2 (doubling)
• Number = 500 x 2 to the power of 8
• Number = 500 x 256
• Number = 128,000 bacteria

Worked Example 2: A town has a population of 25,000. The population grows at 3% per year. What will the population be after 10 years?

• Multiplier = 1.03
• Population = 25,000 x 1.03 to the power of 10
• Population = 25,000 x 1.343916...
• Population = 33,598 (to the nearest whole number)

graph TD
    A[Exponential Growth] --> B[Identify initial value]
    B --> C[Determine the percentage increase]
    C --> D[Calculate multiplier = 1 + rate/100]
    D --> E[Apply: Value = Initial x multiplier to the power of n]
    E --> F[Round appropriately for context]

Exam Tip: When a question says "the population grows by 3% each year", the multiplier is 1.03. When it says "the population doubles", the multiplier is 2. Read carefully to identify the correct multiplier.

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Exponential Decay

Exponential decay occurs when a quantity decreases by a fixed percentage in each time period. This is the same principle as depreciation, applied to contexts like radioactive decay and cooling.

General Formula

Final Value = Initial Value x (Multiplier) to the power of n

Where:

• Multiplier = 1 - (percentage decrease / 100)
• n = number of time periods

Recognising Exponential Decay

Feature	Description
Shape of graph	Starts high, then curves downward, approaching zero but never reaching it
Rate of decrease	Gets slower and slower over time
Multiplier	Between 0 and 1
Real-world examples	Radioactive decay, depreciation, cooling

Worked Example 3: A radioactive substance has a mass of 400 g. It decays at a rate of 12% per hour. What mass remains after 6 hours?

• Multiplier = 1 - 0.12 = 0.88
• Mass = 400 x 0.88 to the power of 6
• Mass = 400 x 0.46440...
• Mass = 185.76 g (to 2 d.p.)

Worked Example 4: The value of a painting decreases by 5% each year. It is currently worth 8,000 pounds. What will it be worth in 12 years?

• Multiplier = 0.95
• Value = 8,000 x 0.95 to the power of 12
• Value = 8,000 x 0.54036...
• Value = 4,322.89 pounds

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Half-Life

The half-life is the time it takes for a quantity to reduce to half its original value. This concept is commonly used in radioactive decay problems.

Worked Example 5: A substance has a half-life of 3 hours. If you start with 1,200 g, how much remains after 12 hours?

• Number of half-lives = 12 / 3 = 4
• After 1 half-life: 1,200 / 2 = 600 g
• After 2 half-lives: 600 / 2 = 300 g
• After 3 half-lives: 300 / 2 = 150 g
• After 4 half-lives: 150 / 2 = 75 g

Alternatively: 1,200 x (0.5) to the power of 4 = 1,200 x 0.0625 = 75 g

Half-lives passed	Fraction remaining	Amount (starting with 1,200 g)
0	1	1,200
1	1/2	600
2	1/4	300
3	1/8	150
4	1/16	75
5	1/32	37.5

Exam Tip: Half-life problems can be solved by repeatedly halving OR by using the formula with multiplier 0.5 raised to the power of (total time / half-life). The formula method is faster for large numbers of half-lives.

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Growth and Decay Graphs

Exponential Growth Graph

An exponential growth graph:

• Starts at the initial value on the y-axis
• Increases slowly at first, then more and more steeply
• Never goes below the x-axis
• Has no maximum — it keeps increasing

Exponential Decay Graph

An exponential decay graph:

• Starts at the initial value on the y-axis
• Decreases quickly at first, then more and more slowly
• Approaches the x-axis but never reaches zero (this is called an asymptote)
• Has no minimum above zero

graph LR
    A[Exponential Functions] --> B[Growth: multiplier > 1]
    A --> C["Decay: 0 < multiplier < 1"]
    B --> D[Curve goes up steeply]
    C --> E[Curve approaches zero]
    B --> F[Examples: population, bacteria]
    C --> G[Examples: radioactive decay, cooling]

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Finding the Number of Time Periods

Worked Example 6: A population of 10,000 fish decreases by 8% per year due to pollution. After how many years will the population first fall below 6,000?

Use trial and improvement with multiplier = 0.92: