Growth and decay (AQA GCSE Maths): Revision Notes
Growth and decay
Growth and decay problems use repeated percentage changes to model real-world situations where quantities increase or decrease over time at a constant rate.
Understanding growth and decay
Growth happens when a quantity increases by the same percentage repeatedly. Examples include population growth, compound interest, and cell division.
Decay (or depreciation) occurs when a quantity decreases by the same percentage repeatedly. Examples include car depreciation, radioactive decay, and medication leaving the body.
The key to growth and decay problems is recognising that the same percentage change is applied repeatedly over multiple time periods. This creates an exponential pattern rather than a linear one.
Compound interest
Compound interest is when money in a bank account grows by earning interest on both the original amount and previously earned interest.
Step-by-step calculation method
You can calculate compound interest year by year using a table approach:
Worked Example: Compound Interest Calculation
When Saanvi invests £40,000 at 3% per annum compound interest:
- Year 1: Starting balance £40,000, interest earned £1,200, new balance £41,200
- Year 2: Starting balance £41,200, interest earned £1,236, new balance £42,436
- Year 3: Starting balance £42,436, interest earned £1,273.08, new balance £43,709.08
After 3 years, the total interest earned is £3,709.08.
Using multipliers and index notation
A much faster method uses multipliers:
- For a 3% increase, the multiplier is 1.03
- This means you keep 100% of the original plus gain 3% more
Using this method:
- Year 1: £40,000 × 1.03 = £41,200
- Year 2: £41,200 × 1.03 = £42,436
- Year 3: £42,436 × 1.03 = £43,709.08
The quickest way is using indices: £40,000 × = £43,709.08
The general growth formula
For any repeated percentage change:
Final amount = (starting amount) × (multiplier)ⁿ
Where:
- n is the number of time periods
- The multiplier depends on whether it's growth or decay
Worked example - cell growth
Let's apply the growth formula to a practical biology example.
Worked Example: Cell Growth
A petri dish contains 5000 cells at the start of an experiment. The number of cells increases by 20% each day. After 4 days:
Number of cells = cells
The multiplier is 1.2 because you keep the original 100% and add 20% more.
Repeated decrease and depreciation
When quantities decrease by the same percentage repeatedly, we use multipliers less than 1.
Worked Example: Car Depreciation
A car depreciates (loses value) by 8% each year. If it's initially worth £15,000:
After 3 years: £15,000 × = £11,680.32
The multiplier 0.92 represents keeping 92% of the value (losing 8%).
Key formulas and multipliers
Understanding how to calculate the correct multiplier is essential for all growth and decay problems. The multiplier determines whether your final answer will be larger (growth) or smaller (decay) than the starting amount.
For growth (percentage increase):
- Multiplier = 1 + (percentage ÷ 100)
- A 20% increase uses multiplier 1.20
For decay (percentage decrease):
- Multiplier = 1 - (percentage ÷ 100)
- An 8% decrease uses multiplier 0.92
General formula:
Exam guidance
When tackling growth and decay questions in exams, there are several key strategies that will help you avoid common mistakes and work efficiently.
Exam Tips:
- Per annum means "per year"
- Always identify if the problem involves growth (multiplier > 1) or decay (multiplier < 1)
- Use your calculator's power function (x^y or ^) for index calculations
- Round money answers to 2 decimal places unless told otherwise
- Check your multiplier makes sense - growth should give larger final amounts, decay should give smaller ones
Key Points to Remember:
- Growth multiplier = 1 + (percentage increase ÷ 100), always greater than 1
- Decay multiplier = 1 - (percentage decrease ÷ 100), always less than 1
- Formula: Final amount = starting amount × (multiplier)ⁿ where n = number of time periods
- Index notation makes repeated calculations much faster than working year by year
- Always double-check whether you're dealing with growth (increasing) or decay (decreasing)