Where two plans cross
Comparing two linear models when neither is cheaper everywhere.
Scenario
Two phone plans charge a fixed monthly amount plus a rate for each minute of calls. Both are available on the same one-month contract, so a customer can move between them freely.
Plan A: £12 a month plus 8p a minute. Plan B: £20 a month plus 4p a minute.
Let m be the number of minutes used in a month.
Answers in order, explanations withheld, and a report on where the chain broke.
- 16.1Linear functionsFoundation
Which pair of expressions gives the monthly cost in pounds of each plan?
- AA: 12 + 0.08m and B: 20 + 0.04m
- BA: 12 + 8m and B: 20 + 4m
- CA: 12m + 0.08 and B: 20m + 0.04
- DA: 0.08m - 12 and B: 0.04m - 20
Show the answer and reasoningHide the reasoning
A: 12 + 0.08m and B: 20 + 0.04m
The monthly charge is a constant and the per-minute rate multiplies the minutes. Pence must be converted to pounds first, or the second choice would have 100 minutes on plan A costing £812.
- 16.2Systems of equationsStandard
Builds on: The two expressions from question 1. With the pence-and-pounds error carried in, this equation has no sensible solution at all.
At how many minutes do the two plans cost the same?
- A200 minutes
- B100 minutes
- C400 minutes
- D80 minutes
Show the answer and reasoningHide the reasoning
200 minutes
Setting the expressions equal: 12 + 0.08m = 20 + 0.04m, so 0.04m = 8 and m = 200. Both plans then cost £28. The £8 difference in the standing charge is closed at 4p a minute, which takes 200 minutes.
- 16.3Interpreting linear modelsChallenge
Builds on: Question 2 fixed the crossover at 200 minutes, which is what tells you B is the cheaper side of it before any arithmetic is done.
A customer uses about 300 minutes a month. Which plan is cheaper, and by how much?
- APlan B, by £4
- BPlan A, by £4
- CPlan B, by £8
- DThey cost the same at any usage above 200 minutes.
Show the answer and reasoningHide the reasoning
Plan B, by £4
At 300 minutes plan A costs 12 + 24 = £36 and plan B costs 20 + 12 = £32, so B is £4 cheaper. Above the crossover the plan with the lower per-minute rate wins, however much higher its standing charge is.
- 16.4Interpreting linear modelsChallenge
Builds on: Questions 2 and 3 together. One gives the crossover and the other gives the direction above it, and the pair is what rules out both "always" answers.
Which statement about the two plans is true?
- ANeither plan is cheaper at every level of use; which is cheaper depends on the minutes.
- BPlan A is cheaper at every level of use.
- CPlan B is cheaper at every level of use.
- DPlan A is cheaper only above 200 minutes.
Show the answer and reasoningHide the reasoning
Neither plan is cheaper at every level of use; which is cheaper depends on the minutes.
Below 200 minutes A is cheaper — at 100 minutes it is £20 against B's £24 — and above 200 B is. The last choice states the true fact backwards, which is the easiest mistake to make once the crossover has been found.