Exponential growth and decay models
Recognise a repeated percentage change, build its multiplier model and explain its limitations.
Extended only.
Before you begin
- Use percentage increase and decrease multipliers.
- Evaluate positive integer powers.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which factor represents 6% decline each period?
What you will learn
- Model growth and decay with a repeated multiplier.
- Calculate a final amount or recover an initial amount.
- Distinguish exponential change from a fixed additive change.
Repeat a ratio, not a fixed difference
A population model starting at 400 and growing by 5% each year produces 420 after one year and 441 after two. The added numbers differ, but each new amount is 1.05 times the previous one. Exponential change keeps a multiplicative factor constant. Linear change instead adds the same amount at each step.
A fixed rate does not mean a fixed number is added. The model 400 + 20n adds twenty every year, while 400(1.05)ⁿ recalculates five percent of the current amount. Compare the first few steps to decide which process a story describes before choosing a formula.
Match the factor to the time step
For a rate r% per period, growth uses 1 + r/100 and decay uses 1 − r/100. After n equal periods, multiply the initial amount by that factor n times: final = initial × factorⁿ. At 20% annual depreciation, a machine retains 0.8 of its value each year.
The exponent counts the same periods as the stated rate. Three years at an annual rate uses n = 3; eighteen months is not eighteen annual periods. Apply the model only over periods for which its assumptions are stated. This course's examples use discrete steps and do not require the exponential constant e.
Recover a start or find a threshold
If a model has reached 540 after two periods of 10% decline, its initial amount is 540/(0.9)², not 540 multiplied by two increases of 10%. Division undoes the complete repeated multiplier. Check by moving the recovered amount forward through the model again.
A threshold question can be explored with successive powers when it asks for the first whole period. For a $1000 item retaining 0.8 each year, the values after two and three years are $640 and $512. The first whole year with value below $600 is year three. Read whether the comparison is strict or includes equality.
A model has assumptions
A population projection assumes the same percentage rate in each period and does not account for resource limits, migration or changing conditions unless they are built into the question. A depreciation model describes a calculation rule, not a guaranteed resale price. State the final estimate with the precision and meaning the context permits.
Some final quantities are counts of people or organisms, so a model may produce a non-integer approximation even though an actual count is whole. Keep the unrounded model value during calculations and then report a sensible rounded prediction. The course's abstract models should not be mistaken for live measurements or forecasts.
Pause and explain
A quantity doubles each hour. Starting from 30, which gives its value after four hours?
Worked example
A machine worth $1500 depreciates by 20% each year. Find its modelled value after three years.
Show the worked solution
- A 20% decrease retains 80%, so the factor is 0.8.
- Three equal yearly steps give 1500 × 0.8³.
- 0.8³ = 0.512, so the modelled value is $768.
Answer $768.00
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 model starts at 500 and increases by 8% per year. Find its value after two years.
Give me a hint
Use the factor 1.08 twice.
Compare my reasoning
- The factor is 1 + 8/100 = 1.08.
- The model is 500 × 1.08².
- This equals 583.2; the second increase is based on 540, not 500.
583.2
Look for these in your work
- Used the current amount as the percentage base.
- Matched the exponent to two annual periods.
A quantity is 729 after three periods of 10% decay. Find its initial amount and check it.
Give me a hint
Divide by the entire decay factor cubed.
Compare my reasoning
- Each period retains 0.9, and 0.9³ = 0.729.
- Initial amount = 729/0.729 = 1000.
- Moving forward gives 1000 × 0.9³ = 729.
1000
Look for these in your work
- Reversed the whole repeated multiplier.
- Checked the recovered value through the forward model.
A $1000 device retains 80% of its value each year in a model. Find the first whole year when its value is below $600, and explain why this is only a model.
Give me a hint
Compare successive yearly values with the strict $600 threshold.
Compare my reasoning
- After one year the value is $800; after two it is $640.
- After three it is $512, the first value below $600.
- The fixed depreciation rate is assumed; a real sale price can depend on condition and demand.
Year 3
Look for these in your work
- Tested the adjacent periods around the threshold.
- Distinguished a modelled value from a guaranteed transaction price.
Common mistakes
- Multiplying a percentage rate by the number of periods.
- Using a monthly count of periods with an annual rate.
- Presenting a modelled projection as a guaranteed real outcome.
What can you explain now?
A quantity starts at 200 and grows by 10% per period. Find the value after three periods.
Compare with the explanation
266.2
Use 200 × 1.1³; multiplying the rate by three would describe a different process.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Write a growth model and a decay model for the same initial amount. Explain what stays constant in each and how to reverse two steps.
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources