Interpret algebraic structure in a growth model
Explain variables, factors and a context's units · Recognise linear dependence on one variable with others fixed · Distinguish principal from growth factor and increase
Section Paper 4 · Historical Extended · calculator
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Terms, factors and coefficients
- Compound-interest percentages
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Explain variables, factors and a context's units
- Recognise linear dependence on one variable with others fixed
- Distinguish principal from growth factor and increase
Read a model in pieces
In B = P(1+r)ⁿ, P is the original amount, r is the decimal rate per period and n is the number of periods. The factors are P and (1+r)ⁿ; the latter is the multiplier. The increase is B−P, not the multiplier alone.
What is held fixed?
For fixed r and n, the multiplier does not depend on P, so doubling P doubles B. This is linear dependence on P; it does not make B linear in n or r. A rate quoted per month needs n in months. Keep model assumptions such as no fees, withdrawals or rate changes explicit.
Sources for this lesson
Pause and explain
With r and n fixed, which factor of P(1+r)ⁿ is independent of P?
Worked example
Interpret B = 200(1.1)² and compare it with a starting amount of 400 at the same rate and duration.
Show the worked solution
- The multiplier is 1.21, giving B = 242 and increase 42.
- With starting amount 400, B = 484; both final amount and increase double.
Answer Multiplier 1.21; totals 242 and 484.
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.
Interpret 100(1.05)² and calculate its final amount and increase.
Give me a hint
Separate the starting amount and multiplier.
Compare my reasoning
- The model starts at 100, growing 5% for each of two periods.
- Final amount 110.25; increase 10.25.
Final 110.25; increase 10.25.
Look for these in your work
- Name each factor
- Subtract the principal for the increase
For B = P(1.2)³, compare B at P = 50 and P = 100.
Give me a hint
The multiplier is fixed.
Compare my reasoning
- 1.2³ = 1.728.
- Totals are 86.4 and 172.8, the second twice the first.
86.4 and 172.8; doubling P doubles B.
Look for these in your work
- Keep rate and periods fixed
- Explain dependence on P
A monthly growth model is B = P(1.02)ⁿ. For one year with 12 identical months, which n is appropriate? Does doubling n double B?
Give me a hint
The exponent counts periods, not calendar labels.
Compare my reasoning
- Use n = 12 for the stated monthly model.
- B(2n)/B(n) = (1.02)ⁿ for P ≠ 0; it is not generally 2.
Use n = 12. Doubling n does not generally double B; this dependence is exponential.
Look for these in your work
- Match the rate's period
- Distinguish linear from exponential dependence
Common mistakes
- A decimal rate 0.1 is not a multiplier 0.1.
- Linearity depends on which variable is changed.
What can you explain now?
With r and n fixed, which factor of P(1+r)ⁿ is independent of P?
Compare with the explanation
(1+r)ⁿ.
It depends on the rate and periods, held fixed in this comparison.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
With r and n fixed, which factor of P(1+r)ⁿ is independent of P?
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.