Mathematics Standard • Year 11 • Module 1 • Lesson 13
Comparing Linear Models and Break-Even Points
Practise HSC-style writing on break-even decisions — three multi-mark short answers and one extended response with marking criteria.
1. Short-answer questions
1.1 Company A charges $35 plus $10 per hour. Company B charges $65 plus $4 per hour.
(a) Write a cost equation for each company in terms of hours h.
(b) Find the break-even time. 3 marks Band 3
1.2 Using your equations from Q1.1, decide which company is cheaper for 3 hours and which is cheaper for 8 hours. Show working for each case. 3 marks Band 3-4
1.3 Explain in your own words what a break-even point means in a model-comparison question, and why it is useful to test a value either side of it. Use a worked example with numbers of your own. 3 marks Band 4
Stuck on 1.3? A break-even point is where two models cost the same. Testing either side tells you which option is cheaper in each region.2. Extended response
2.1 A small business owner is comparing two phone plans for company mobiles.
Plan X: $40 monthly access fee, plus $0.20 per call minute.
Plan Y: $70 monthly access fee, plus $0.05 per call minute.
(a) Write a cost equation for each plan in terms of m, the number of call minutes per month.
(b) Calculate the monthly cost of each plan for an employee who makes 150 minutes of calls.
(c) Calculate the monthly cost of each plan for an employee who makes 300 minutes of calls.
(d) Find the break-even number of call minutes per month and confirm both plans cost the same at this value.
(e) Write a conclusion sentence recommending which plan suits low-use employees (under the break-even) and which suits heavy-use employees (over the break-even). 7 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark
• 1 mark — both equations correct: X = 40 + 0.20m and Y = 70 + 0.05m.
Part (b) — 1 mark
• 1 mark — both 150-minute costs correct: X = $70, Y = $77.50.
Part (c) — 1 mark
• 1 mark — both 300-minute costs correct: X = $100, Y = $85.
Part (d) — 3 marks
• 1 mark — sets up 40 + 0.20m = 70 + 0.05m.
• 1 mark — correctly solves to m = 200 minutes.
• 1 mark — confirms both costs equal $80 at the break-even.
Part (e) — 1 mark
• 1 mark — clear conclusion: Plan X for low-use (under 200 min); Plan Y for heavy-use (over 200 min), with justification.
Your response:
Stuck on (d)? Subtract 0.05m and 40 from both sides: 0.15m = 30 ⇒ m = 200.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Companies A vs B (3 marks)
Sample response.
(a) A = 35 + 10h and B = 65 + 4h (where the costs are in dollars and h is the number of hours).
(b) Set equal: 35 + 10h = 65 + 4h ⇒ 6h = 30 ⇒ h = 5 hours.
Marking notes. 1 mark — equations for both companies correct. 1 mark — sets up the equation 35 + 10h = 65 + 4h. 1 mark — correctly solves to h = 5.
1.2 — Comparing at 3 h and 8 h (3 marks)
Sample response.
3 hours: A = 35 + 10(3) = $65; B = 65 + 4(3) = $77. Company A is cheaper.
8 hours: A = 35 + 10(8) = $115; B = 65 + 4(8) = $97. Company B is cheaper.
(This is consistent with the 5-hour break-even from Q1.1.)
Marking notes. 1 mark — both 3-hour costs correct with A picked. 1 mark — both 8-hour costs correct with B picked. 1 mark — clear "cheaper" sentence for each case (not just numbers).
1.3 — Meaning of break-even (3 marks)
Sample response. A break-even point is the input value at which two models produce the same output (e.g. cost). For example, if Plan A is C = 10 + 4g and Plan B is C = 25 + g, setting them equal gives 10 + 4g = 25 + g ⇒ 3g = 15 ⇒ g = 5. At 5 GB both plans cost $30. Testing g = 2 (A = $18, B = $27) shows A is cheaper below 5 GB; testing g = 10 (A = $50, B = $35) shows B is cheaper above 5 GB. Testing either side confirms the direction of the answer and prevents the common error of just stating the cut-off without naming which plan wins on each side.
Marking notes. 1 mark — defines break-even correctly. 1 mark — supplies a worked numerical example. 1 mark — explains the purpose of testing both sides (or notes the cheaper option changes across the cut-off).
2.1 — Phone plans X vs Y (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Equations.
X = 40 + 0.20m. Y = 70 + 0.05m. (Cost in dollars, m = call minutes per month.) [1 mark.]
(b) 150 minutes.
X = 40 + 0.20(150) = 40 + 30 = $70. Y = 70 + 0.05(150) = 70 + 7.50 = $77.50. [1 mark.]
(c) 300 minutes.
X = 40 + 0.20(300) = 40 + 60 = $100. Y = 70 + 0.05(300) = 70 + 15 = $85. [1 mark.]
(d) Break-even.
Set equal: 40 + 0.20m = 70 + 0.05m. [1 mark.]
Rearrange: 0.20m − 0.05m = 70 − 40 ⇒ 0.15m = 30 ⇒ m = 200 minutes. [1 mark.]
Check: X = 40 + 0.20(200) = $80; Y = 70 + 0.05(200) = $80. Both cost $80 at 200 minutes — confirmed. [1 mark.]
(e) Recommendation. Choose Plan X for low-use employees (under 200 minutes per month) because its lower $40 access fee dominates when calls are short. Choose Plan Y for heavy-use employees (over 200 minutes per month) because its lower $0.05/min rate dominates over many minutes. [1 mark — clear two-sided conclusion.]
Total: 7/7.
Band descriptors for marker.
Band 3: Writes one equation and substitutes for one value; no comparison. ≈ 2-3 marks.
Band 4: All four substitution costs correct; does not attempt the break-even equation in (d). ≈ 4-5 marks.
Band 5: Sets up and solves 40 + 0.20m = 70 + 0.05m correctly but does not confirm both costs equal $80 at m = 200; conclusion incomplete. ≈ 5-6 marks.
Band 6: All five parts complete, both ranges named in (e), check at m = 200 included. 7/7.