Mathematics Standard • Year 11 • Module 1 • Lesson 13
Comparing Linear Models and Break-Even Points
Apply break-even analysis to real Australian decisions — phone plans, hire vs. buy, two savings plans, gym memberships and competing tradies.
Problem 1 — Choosing a phone plan
Plan A costs $20 per month plus $5 per gigabyte of data. Plan B costs $50 per month plus $2 per gigabyte. Let g be the number of gigabytes used.
Set up: What are we solving for?
(i) Write a cost equation for each plan. 1 mark
(ii) Find the break-even number of gigabytes. 2 marks
(iii) Calculate the cost of each plan at the break-even and confirm both costs are equal. 1 mark
(iv) Recommend a plan for a user who normally uses 6 GB per month and another for a user who uses 16 GB per month. Justify each choice in one sentence. 2 marks
Stuck? Revisit lesson § Worked Example 2 — Decide which option is cheaper.Problem 2 — Hire vs. buy a carpet cleaner
A carpet cleaner can be hired for $60 per day, or bought for $480 plus $20 per day of cleaning fluid. Let d be the number of days the cleaner is used.
Set up: What are we solving for?
(i) Write the cost equation for hiring (H) and the cost equation for buying (B), in terms of d. 2 marks
(ii) Find the number of days at which hiring and buying cost the same. 2 marks
(iii) A cleaning business expects to use the cleaner 20 days a year. Should they hire or buy? Justify with a cost calculation. 2 marks
Stuck on (ii)? Set 60d = 480 + 20d and solve.Problem 3 — Sam vs. Alex savings race
Sam starts with $200 in his account and saves $30 per week. Alex starts with $80 and saves $50 per week. Let w be the number of weeks.
Set up: What are we solving for?
(i) Write each person's savings equation. 1 mark
(ii) Find the number of weeks at which Sam and Alex have the same total savings. 2 marks
(iii) What is the amount each person has at that point? 1 mark
(iv) Explain in one sentence who has the higher savings 10 weeks after the start, and why. 1 mark
Stuck? Revisit lesson § Worked Example 3 — Compare savings plans.Problem 4 — Comparing gym memberships
Gym A charges a $80 joining fee plus $25 per month. Gym B charges a $20 joining fee plus $40 per month. Let m be the number of months of membership.
Set up: What are we solving for?
(i) Write the total cost equation for each gym in terms of m. 1 mark
(ii) Find the number of months at which the total costs are equal. 2 marks
(iii) Recommend a gym for someone who plans to train for 3 months only. Justify your recommendation. 2 marks
(iv) Recommend a gym for someone who plans to train for a full year (12 months). Justify with a calculation. 2 marks
Stuck? The gym with the lower joining fee wins for short memberships; the gym with the lower monthly rate wins for long memberships.Problem 5 — Competing tradies
A homeowner gets two quotes for fitting kitchen tiles. Tradie A charges $150 callout plus $60 per hour. Tradie B charges $90 callout plus $75 per hour. Let h be the number of hours on site.
Set up: What are we solving for?
(i) Write each tradie's total-cost equation. 1 mark
(ii) Find the break-even number of hours. 2 marks
(iii) Calculate the cost both tradies charge at the break-even and confirm they match. 1 mark
(iv) The job is estimated at 6 hours. Which tradie should the homeowner choose? Justify your answer with a cost comparison. 2 marks
Stuck on (ii)? Solve 150 + 60h = 90 + 75h.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Phone plans
Set up. Build A = 20 + 5g and B = 50 + 2g, set them equal, and decide which wins on each side.
(i) A = 20 + 5g; B = 50 + 2g.
(ii) 20 + 5g = 50 + 2g ⇒ 3g = 30 ⇒ g = 10 GB.
(iii) At g = 10: A = 20 + 5(10) = $70; B = 50 + 2(10) = $70. Both $70, confirmed.
(iv) 6 GB user: choose Plan A (A = $50 vs B = $62 — A's lower starting cost dominates for low usage). 16 GB user: choose Plan B (A = $100 vs B = $82 — B's lower per-GB rate dominates for high usage).
Problem 2 — Hire vs. buy carpet cleaner
Set up. Hire: only a daily rate. Buy: a fixed purchase plus ongoing fluid cost per day.
(i) H = 60d; B = 480 + 20d.
(ii) 60d = 480 + 20d ⇒ 40d = 480 ⇒ d = 12 days.
(iii) At d = 20: H = 60(20) = $1200; B = 480 + 20(20) = $880. Buy ($880 is $320 cheaper than hiring for 20 days, because 20 > the 12-day break-even).
Problem 3 — Sam vs. Alex savings
Set up. Sam starts higher but saves less per week; Alex starts lower but saves faster.
(i) Sam: S = 200 + 30w; Alex: A = 80 + 50w.
(ii) 200 + 30w = 80 + 50w ⇒ 120 = 20w ⇒ w = 6 weeks.
(iii) At w = 6: Sam = 200 + 30(6) = $380; Alex = 80 + 50(6) = $380. Both have $380.
(iv) At w = 10: Sam = 200 + 30(10) = $500; Alex = 80 + 50(10) = $580. Alex has more because his higher weekly saving rate has overtaken Sam's larger starting amount (we're past the 6-week break-even).
Problem 4 — Gym memberships
Set up. Gym A: higher joining fee, lower monthly. Gym B: lower joining fee, higher monthly.
(i) A = 80 + 25m; B = 20 + 40m.
(ii) 80 + 25m = 20 + 40m ⇒ 60 = 15m ⇒ m = 4 months.
(iii) At m = 3: A = 80 + 25(3) = $155; B = 20 + 40(3) = $140. Gym B is cheaper for 3 months (below the 4-month break-even, the lower joining fee wins).
(iv) At m = 12: A = 80 + 25(12) = $380; B = 20 + 40(12) = $500. Gym A is cheaper for 12 months ($120 saving — the lower monthly rate dominates over a long membership).
Problem 5 — Competing tradies
Set up. Tradie A: higher callout, lower hourly. Tradie B: lower callout, higher hourly.
(i) A = 150 + 60h; B = 90 + 75h.
(ii) 150 + 60h = 90 + 75h ⇒ 60 = 15h ⇒ h = 4 hours.
(iii) At h = 4: A = 150 + 60(4) = $390; B = 90 + 75(4) = $390. Both $390, confirmed.
(iv) At h = 6: A = 150 + 60(6) = $510; B = 90 + 75(6) = $540. Choose Tradie A — it saves $30 for the 6-hour job, because 6 > 4 hours and A's lower hourly rate dominates.