Model exponential growth and decay with repeated multipliers
Represent repeated percentage change using a constant multiplier for growth, depreciation and other exponential contexts.
Section 1 · Number
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which multiplier represents a 12% decrease?
What you will learn
- Build an exponential model from a repeated percentage change
- Distinguish growth multipliers from decay multipliers
- Use a calculator without requiring knowledge of e
Each equal time interval uses the same multiplier
For a repeated increase of r%, multiply by 1 + r/100 each interval. For a decrease of r%, multiply by 1 − r/100. After n intervals, multiply the starting amount by that factor n times.
The multiplier is applied to the current amount, so equal percentage changes do not add equal amounts. This is the distinction between repeated exponential change and a fixed linear change.
Interpret the model and its units
A model such as P = P₀ × 0.92ⁿ describes 8% depreciation each period. P₀ is the starting amount and n counts whole periods. The syllabus does not require knowledge of e.
A calculated value may be rounded to a sensible amount, but keep the full power value until the final requested rounding. Check whether the question asks for the amount after n periods or the change from the start.
Pause and explain
A value grows by 10% for 2 years. What repeated factor models the change?
Worked example
A machine worth $2 500 depreciates by 8% each year. Find its value after 3 years.
Show the worked solution
- The yearly decay multiplier is 1 − 0.08 = 0.92.
- After 3 years the value is 2500 × 0.92³.
- Evaluate and round to the nearest cent.
Answer $1,946.72 after 3 years.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A population of 12 000 grows by 3% each year. Estimate the population after 4 years to the nearest whole person.
Give me a hint
Raise the growth multiplier 1.03 to the fourth power.
Compare my reasoning
- Model: 12 000 × 1.03⁴.
- Evaluate to get approximately 13 506.10.
- Round to the nearest whole person.
Approximately 13 506 people.
Look for these in your work
- Growth multiplier selected
- Four periods included
- Context rounded at the end
A phone loses 18% of its value each year. It is worth $900 now. Find its value after 2 years and the total value lost.
Give me a hint
Find the remaining-value multiplier and distinguish the final amount from the decrease.
Compare my reasoning
- Decay multiplier is 0.82.
- Value after 2 years = 900 × 0.82² = $605.16.
- Total lost = 900 − 605.16 = $294.84.
Final value $605.16; total lost $294.84.
Look for these in your work
- Decay factor applied twice
- Value calculated
- Loss found by subtracting from original
A medicine sample has 64 mg remaining after 3 equal half-life periods. It began at 512 mg. What fraction remains each period, and what will remain after one more period?
Give me a hint
Use the repeated multiplier between the initial and current values.
Compare my reasoning
- The overall factor is 64/512 = 1/8.
- Since 1/8 = (1/2)³, the factor each period is 1/2.
- After one more period, 64 × 1/2 = 32 mg.
The multiplier is 1/2 per period; 32 mg remains after one more period.
Look for these in your work
- Overall ratio reduced
- Per-period factor inferred
- Next interval applied once
Common mistakes
- Using 1.08 for a decrease
- Subtracting the same dollar amount each period
- Applying the multiplier only once for several periods
- Confusing final value with the amount lost
What can you explain now?
What multiplier represents growth of 5% per year?
Compare with the explanation
1.05
Keep the original 100% and add 5%.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How can the same repeated-multiplier model describe growth and decay, and how do you choose the factor?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.