To compare a fixed-fee plan with a pay-per-use plan, write each cost as an expression, set them equal to find the break-even point, then state which plan is cheaper on each side.
This lesson is part of choosing a model from imperfect information. It follows deciding which quantities are necessary.
How do I find the break-even point?
Follow these four steps. The last one is a sentence, not a number.
- Let one letter stand for the quantity and write each cost as an expression.
- Set the two costs equal and solve.
- Test one quantity below and one above the answer.
- Write the decision in a sentence.
Worked example: two printing shops
Shop A charges a RM15 setup fee plus RM0.10 per page. Shop B charges RM0.25 per page with no setup fee. Let p be the number of pages.
Costs. Shop A: 15 + 0.10p. Shop B: 0.25p.
Equal. 15 + 0.10p = 0.25p, so 15 = 0.15p, and p = 100.
Test below. At p = 50, Shop A costs 15 + 5 = RM20 and Shop B costs RM12.50. Shop B is cheaper.
Test above. At p = 200, Shop A costs 15 + 20 = RM35 and Shop B costs RM50. Shop A is cheaper.
Decision. For fewer than 100 pages, Shop B is cheaper. For more than 100 pages, Shop A is cheaper. At exactly 100 pages both cost RM25.
The two tests confirm the direction. Without them, the decision sentence is a guess.
The mistake that costs marks
The common slip is to stop at p = 100 and write “the break-even point is 100 pages”. It answers a different question from the one asked.
| Step | Wrong | Right |
|---|---|---|
| Solve | p = 100 | p = 100 |
| Test a side | not done | p = 50 and p = 200 |
| Conclusion | “100 pages” | “B is cheaper below 100 pages, A above 100” |
The question asked which plan to choose, so the sentence about each side is the answer.
Check yourself
Plan X charges RM20 every month plus RM2 for each film. Plan Y charges RM8 per film with no monthly fee. Let f be the number of films. Find the break-even point and decide which plan is cheaper.
Answer
Costs: Plan X is 20 + 2f and Plan Y is 8f. Setting them equal gives 20 = 6f, so f = 3⅓.
Films come in whole numbers, so test 3 and 4. At f = 3, X costs RM26 and Y costs RM24, so Y is cheaper. At f = 4, X costs RM28 and Y costs RM32, so X is cheaper.
Plan Y is cheaper for 3 films or fewer. Plan X is cheaper for 4 films or more.
What to study next
Read testing whether a model still works outside the supplied data range next. Then use the integrated practice set to mix the skills.
The algebra step repair trainer helps if solving for the break-even quantity is the slow step. For a teacher to check your decision sentences, see online one-to-one Mathematics tuition.