Choosing a Tutoring Package Under a $400 Budget
A student has $400 for tutoring this term and will buy a whole number of hours. Plan A charges a one-time $45 registration fee plus $28 per hour. Plan B charges no fee and $34 per hour, but any purchase of 8 or more hours receives a 10% discount applied to the entire hourly charge. Hours may only be purchased in whole numbers, and the student's goal is the largest number of hours $400 will buy, not simply the cheapest plan at a fixed hour count. A friend advises: 'Plan A breaks even at 7.5 hours, so take Plan A for 8 hours or more.' The student wants that claim checked before committing.
1. For whole-number hour counts from 1 to 20, determine where each plan is cheaper. Evaluate the friend's claim about 7.5 hours.
Read the worked answer
Let h be hours. A(h) = 45 + 28h. B(h) = 34h for h ≤ 7, and B(h) = 0.9(34h) = 30.6h for h ≥ 8.
Region 1 (h ≤ 7): 45 + 28h = 34h → h = 7.5. So for h ≤ 7, B is cheaper. Region 2 (h ≥ 8): 45 + 28h = 30.6h → 45 = 2.6h → h ≈ 17.31. So B is cheaper through h = 17 and A is cheaper from h = 18.
Checkpoints:
- h = 7: A = $241, B = $238 → B
- h = 8: A = $269, B = $244.80 → B (the discount creates a jump down in B's price)
- h = 17: A = $521, B = $520.20 → B
- h = 18: A = $549, B = $550.80 → A
The friend is wrong. The 7.5-hour crossing is computed on the pre-discount rate; the 10% discount switches on discontinuously at exactly 8 hours, wiping out the crossing. The real switch point is 18 hours, not 8.
2. What is the largest whole number of hours the $400 budget buys under each plan, and which plan should the student choose?
Read the worked answer
Plan A: 45 + 28h ≤ 400 → h ≤ 12.68 → h = 12 hours, costing 45 + 336 = $381. ($13 of budget is unusable since a 13th hour would cost $409.)
Plan B: use the h ≥ 8 branch: 30.6h ≤ 400 → h ≤ 13.07 → h = 13 hours, costing 30.6(13) = $397.80. (The h ≤ 7 branch caps at 7 hours, so the discounted branch governs.)
The student should choose Plan B with 13 hours: one more hour than Plan A and still within budget. Because hours are integer-valued, a plan that is cheaper at a comparison point can still deliver fewer affordable hours - the binding question is the budget frontier, not the break-even point.
3. Suppose the budget rises to $500. Does the student's best choice change? Support the answer with the boundary hour count for each plan.
Read the worked answer
Plan A: 45 + 28h ≤ 500 → h ≤ 16.25 → 16 hours, cost 45 + 448 = $493. Plan B: 30.6h ≤ 500 → h ≤ 16.34 → 16 hours, cost 30.6(16) = $489.60.
Both plans now top out at 16 hours, so the maximum-hour criterion no longer separates them; the tiebreaker is cost, and Plan B is cheaper by $3.40. The best choice stays Plan B, but for a different reason than before.
Judgement point: a larger budget moved the decision from 'which plan reaches further' to 'which plan is cheaper at the same reach' - the same two functions, a different binding constraint.