Model related quantities and choose a meaningful domain
Model one quantity as a function of another · Interpret average cost C(n)/n · Restrict domains for counts, capacity and physical quantities
Section E3.3 · Extended · Models of related quantities and average cost
Exam scope and assessment notes
Core and Extended foundations. 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
- Function notation and graphs
- Units and ratios
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
- Model one quantity as a function of another
- Interpret average cost C(n)/n
- Restrict domains for counts, capacity and physical quantities
Define inputs, outputs and units
A model should state what each variable measures and the units. A printing cost C(n) = 30 + 2n means a fixed setup charge of 30 plus 2 per item. For 1 to 40 items, n is an integer in {1, …, 40}; the plotted feasible inputs are separate points, even if a smooth formula is drawn as a guide.
Physical measurements such as time may vary continuously within a stated interval. Algebraically possible inputs can be impossible in context: negative ticket counts, a half-person or values beyond capacity are not allowed.
Total and average cost
Average cost per item is A(n) = C(n)/n = 30/n + 2. It is defined only for n > 0 in this count model. The setup charge is spread across the batch as n increases. Total cost rises while average cost falls, so always identify which quantity a question asks for.
Use inequalities to impose a budget, then enforce integer counts and any capacity limit. A model is based on stated assumptions; a price rule may change outside its quoted range.
Sources for this lesson
Pause and explain
Which inputs are allowed for a count of 1 to 40 items?
Worked example
C(n) = 30 + 2n for integer n from 1 to 40. Find total and average costs for 10 and 20 items.
Show the worked solution
- C(10) = 50, so A(10) = 5 per item.
- C(20) = 70, so A(20) = 3.50 per item.
- Doubling the count increases the total but reduces the average.
Answer 10 items: total 50, average 5; 20 items: total 70, average 3.50.
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.
For C(n) = 24 + 3n, find the total and average cost of 6 items.
Give me a hint
Average is total divided by the count.
Compare my reasoning
- C(6) = 24 + 18 = 42.
- A(6) = 42/6 = 7.
Total 42; average 7 per item.
Look for these in your work
- Label units
- Divide the full total
A bus charge is B(p) = 80 + 5p for 1 to 30 passengers. State the domain and find B(12) and B(12)/12.
Give me a hint
Only whole passenger counts within capacity are feasible.
Compare my reasoning
- Domain: integers 1 ≤ p ≤ 30.
- B(12) = 140 and the average is 140/12 = 35/3.
Domain {1, …, 30}; total 140; average 35/3 ≈ 11.67 per passenger.
Look for these in your work
- Exclude half-passengers
- Separate exact and rounded averages
Printing costs 30 + 2n for integer n from 1 to 40. Under a budget of 85, what is the largest batch and its average cost?
Give me a hint
Solve the budget inequality, then round the count down.
Compare my reasoning
- 30 + 2n ≤ 85 gives n ≤ 27.5; capacity is 40.
- The greatest feasible integer is 27, costing 84.
- 84/27 = 28/9 ≈ 3.11 per item.
27 items; total 84; average 28/9 ≈ 3.11 per item.
Look for these in your work
- Enforce integer feasibility
- Check the budget with the actual batch
Common mistakes
- C(0) may represent a setup amount, but C(0)/0 is undefined.
- A continuous curve does not make half an item possible.
What can you explain now?
Which inputs are allowed for a count of 1 to 40 items?
Compare with the explanation
The integers 1 through 40.
Counts are discrete and the capacity is part of the model.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Which inputs are allowed for a count of 1 to 40 items?
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.