A value increases by 10%. What is the multiplier?
Add 10% to 1.
Multiplier = 1 + 0.10 = 1.10.
Growth and Decay
Growth and decay describe how quantities change over time using multipliers. This is a form of percentage change that is often applied repeatedly using powers in population, finance and real-world situations.
Overview
Key Definitions
Growth
An increase in value over time.
Decay
A decrease in value over time.
Multiplier
The number you multiply by to apply percentage change.
Repeated Percentage Change
The same percentage increase or decrease applied again and again.
Exponential Growth
Repeated multiplication by a growth multiplier.
Exponential Decay
Repeated multiplication by a decay multiplier.
Key Rules
Increase by \(p\%\)
Use multiplier \(1 + \frac{p}{100}\).
Decrease by \(p\%\)
Use multiplier \(1 - \frac{p}{100}\).
One change
Multiply once by the multiplier.
Repeated change
Use powers: \( (\text{multiplier})^n \).
Common Multipliers
Increase by 10%
Multiplier \(= 1.10\).
Increase by 25%
Multiplier \(= 1.25\).
Decrease by 10%
Multiplier \(= 0.90\).
Decrease by 30%
Multiplier \(= 0.70\).
How to Solve
Step 1: Recognise the type of problem
Growth and decay questions are repeated percentage change problems.
Exam thinking: First decide growth or decay, then check if it is repeated.
Growth
The value increases each time.
Decay
The value decreases each time.
Repeated change
The percentage is applied more than once.
Step 2: Use a multiplier
Instead of working out percentages each time, use a multiplier.
New value = Original value × Multiplier
Why this works: Each percentage change is applied to the new value.
Step 3: Find the multiplier
Growth
Add to 100% then convert to decimal.
Example: 6% increase → \(1.06\)
Decay
Subtract from 100% then convert to decimal.
Example: 6% decrease → \(0.94\)
Step 4: Repeated change
If the change happens multiple times, use powers.
\( \text{Final value} = \text{Original value} \times (\text{multiplier})^n \)
\(n\) is the number of times the change happens.
Exam tip: Words like 'each year' or 'each month' mean repeated change.
Step 5: Why you cannot just add percentages
Each percentage is applied to a new value, not the original.
Key idea
Repeated percentage change is multiplicative, not additive.
Step 6: Step-by-step exam method
For percentage basics, see percentages.
- Identify growth or decay.
- Convert the percentage to a multiplier.
- Check if the change is repeated.
- Use powers if needed.
- Calculate and round appropriately.
- Include correct units.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on percentage growth and decay in real-life contexts.
Edexcel
A town has a population of 24,000. The population increases by 5% each year. Calculate the population after 1 year.
Edexcel
A car worth £18,000 depreciates by 12% each year. Find its value after 1 year.
Edexcel
A salary of £28,000 increases by 3% per year. Calculate the new salary after one year.
Edexcel
A laptop costing £1,200 depreciates by 15%. Calculate its value after the depreciation.
Edexcel
The value of an investment increases from £500 to £575. Calculate the percentage increase.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, emphasising compound growth and decay using multipliers.
AQA
A population of bacteria grows by 8% per hour. If the initial population is 500, calculate the population after 2 hours.
AQA
A car worth £15,000 depreciates by 10% each year. Calculate its value after 2 years.
AQA
An investment of £800 grows by 4% per year. Find its value after 3 years.
AQA
The price of a television decreases from £650 to £520. Calculate the percentage decrease.
AQA
A house increases in value by 6% per year. If its current value is £200,000, calculate its value after 2 years.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, focusing on reasoning, compound measures, and interpreting growth and decay.
OCR
A radioactive substance decays by 5% each year. If the initial mass is 200 g, calculate the mass after 3 years.
OCR
A bank account earns 2.5% compound interest per year. If £1,000 is invested, calculate its value after 2 years.
OCR
A machine loses 20% of its value each year. If it is initially worth £5,000, calculate its value after 2 years.
OCR
The number of visitors to a website increases from 12,000 to 15,000. Calculate the percentage increase.
OCR
Explain the difference between simple growth and compound growth.
Exam Checklist
Step 1
Decide if the question is growth or decay.
Step 2
Write the percentage multiplier correctly.
Step 3
Use powers if the change happens repeatedly.
Step 4
Check whether your final answer is sensible.
Most common exam mistakes
Wrong multiplier
Using \(0.06\) instead of \(1.06\) for 6% growth.
Wrong decay multiplier
Using \(0.12\) instead of \(0.88\) for 12% decay.
No powers
Forgetting that repeated change needs \((\text{multiplier})^n\).
Too much rounding
Rounding too early can change the final answer.
Common Mistakes
These are common mistakes students make when working with growth and decay in GCSE Maths.
Using the percentage instead of the multiplier
Incorrect
A student uses \(0.15\) instead of \(1.15\) for 15% growth.
Correct
For growth, add 1 to the percentage as a decimal. For example, 15% growth means multiplying by \(1.15\), not \(0.15\).
Using the wrong decay multiplier
Incorrect
A student uses \(0.20\) instead of \(0.80\) for 20% decay.
Correct
For decay, subtract the percentage from 1. For example, 20% decay means multiplying by \(0.80\), not \(0.20\).
Adding instead of multiplying
Incorrect
A student adds the percentage repeatedly instead of applying a multiplier.
Correct
Growth and decay are multiplicative processes. You must multiply by the multiplier each time, not add the percentage.
Forgetting to use powers
Incorrect
A student applies the multiplier once instead of repeatedly.
Correct
For repeated change, use powers. For example, \(\text{value} = \text{initial} \times (\text{multiplier})^n\).
Rounding too early
Incorrect
A student rounds values during intermediate steps.
Correct
Avoid rounding until the final answer, as early rounding can lead to inaccurate results.
Try It Yourself
Practise solving problems involving exponential growth and decay.
Foundation
Foundation Practice
Use multipliers to calculate growth and decay.
A price of £50 increases by 20%. Find the new price.
Use multiplier 1.20.
50 × 1.20 = £60.
A value decreases by 15%. What is the multiplier?
Subtract 15% from 1.
Multiplier = 1 − 0.15 = 0.85.
A car worth £2000 decreases by 10%. Find its value after the decrease.
Use multiplier 0.9.
2000 × 0.9 = £1800.
A population increases by 5% each year. Which multiplier is used?
Increase means above 1.
5% increase → multiplier = 1.05.
A value of 100 increases by 10% twice. Find the final value.
Apply multiplier twice.
100 × 1.1 × 1.1 = 121.
A student adds 10% twice instead of multiplying. What is wrong?
Repeated change is not linear.
Growth is multiplicative, not additive.
A value of 500 decreases by 20%. Find the new value.
Use multiplier 0.8.
500 × 0.8 = 400.
Which multiplier represents a 25% decrease?
1 − 0.25.
Multiplier = 0.75.
A value of 80 increases by 25%. Find the new value.
Use multiplier 1.25.
80 × 1.25 = 100.
Higher
Higher Practice
Solve exponential growth and decay problems over multiple periods.
A value grows by 5% each year. Starting at 100, what is the value after 3 years?
Use 100 × (1.05)^3.
100 × 1.05³ = 115.76.
A population of 200 grows by 10% each year. Find the population after 2 years.
Use multiplier squared.
200 × 1.1² = 242.
A car loses 20% of its value each year. Starting at £10,000, what is its value after 2 years?
Use multiplier 0.8².
10000 × 0.8² = £6400.
A value decreases by 10% each year. Starting at 500, find the value after 3 years.
Use 0.9³.
500 × 0.9³ = 364.5.
Which formula represents exponential growth?
Look for powers.
Exponential growth uses powers: y = a(1 + r)^t.
A bacteria culture doubles every hour. Starting at 50, how many after 4 hours?
Use powers of 2.
50 × 2⁴ = 800.
A student calculates 100 × 1.1 × 2 for 2 years of growth. What is wrong?
Repeated growth uses powers.
Correct method is 100 × 1.1².
A value grows by 20% each year. Starting at 150, find the value after 2 years.
Use 1.2².
150 × 1.2² = 216.
A value halves every year. Which multiplier is used?
Halving means divide by 2.
Multiplier = 0.5.
A value of 100 halves each year. Find the value after 3 years.
Use 0.5³.
100 × 0.5³ = 12.5.
Games
Practise this topic with interactive games.
Games coming soon.