Sheet № 43 · Higher only · AQA · Edexcel · OCR
Growth and Decay –
Growth and decay is a Higher-tier topic that extends compound interest and depreciation into a broader mathematical framework. It covers any situation where a quantity increases or decreases by a fixed percentage over equal intervals — from bacteria populations doubling to radioactive substances losing mass. AQA, Edexcel and OCR all inclu
§Key definitions
Step 1:
P = 500, r = 15%, so k = 1.15, n = 6.
Step 2:
A = 500 × 1.15⁶.
Step 3:
1.15⁶ = 2.313060… (use a calculator).
Step 4:
A = 500 × 2.313060… = 1,156.5 (since we cannot have half a bacterium, round to 1,157 bacteria).
Answer:
After 10 complete hours.
§Formulas to memorise
A = P \times k^n
k = 1 + \frac{r}{100}
k = 1 - \frac{r}{100}
A — is the amount after n time intervals
P — is the initial (starting) amount
k — is the multiplier per time interval
n — is the number of time intervals
If k = 1, nothing is changing.
Substitute into A = P × kⁿ and calculate.
Set up A = P × kⁿ with your target value.
Use trial and improvement: calculate A for n = 1, 2, 3, … until you pass the target.
Worked example
See example below.
A colony of bacteria contains 500 organisms. The population increases by 15% every hour. How many bacteria are there after 6 hours?
⚠ Common mistakes
- ✗Using the wrong multiplier. Growth means k > 1; decay means k < 1. If the population is declining by 20%, the multiplier is 0.80, not 1.20.
- ✗Confusing the number of intervals with the number of values. If you start at year 0 and end at year 5, that is 5 intervals, not 6.
- ✗Not answering "complete hours/years" correctly. If the question says "after how many complete hours", you need the first whole number where the condition is met — not a decimal.
- ✗Thinking the decay reaches zero. Exponential decay gets closer and closer to zero but never actually reaches it (in theory). Do not assume the value will hit zero.
- ✗Rounding too early. Keep full calculator precision until the final step.
✦ Exam tips
- →Set out trial and improvement clearly. Use a table with columns for n and A. Examiners can follow your logic and award method marks even if the final answer is wrong.
- →Recognise the formula in context. Questions might describe bacteria, population, radioactive decay, the spread of a rumour, or the cooling of a liquid — but they all use A = P × kⁿ.
- →If given a graph, read off two consecutive values and divide to find k. Then use the formula for predictions.
- →Watch for non-standard intervals. If the rate is "per hour" but the question asks about minutes, convert appropriately.
- →Link to compound interest. If you are confident with compound interest, you already know this topic — just apply it beyond money.