Tutoring Cost: Piecewise Pricing and the Wrong Break-Even
A parent is choosing between two tutoring centers for a student who needs 8 hours of algebra tutoring before a retake. Center A charges a $40 registration fee plus $35 per hour for the first 5 hours and $30 per hour for every hour beyond 5. Center B charges a $60 registration fee plus a flat $28 per hour. The parent's budget is $300, and both centers bill only for whole hours. A friend advises, 'Choose Center A: its registration fee is lower and its hourly rate drops after 5 hours.' The parent wants to know which center fits the budget, whether the friend's reasoning holds for other numbers of hours, and how the average cost per hour compares for a long-term arrangement.
1. Write each center's total cost as a function of the whole number of hours h, and determine which center the parent should choose for 8 hours under the $300 budget.
Read the worked answer
Center A (piecewise). For 0 ≤ h ≤ 5: A(h) = 40 + 35h. For h > 5: A(h) = 40 + 175 + 30(h − 5) = 65 + 30h. Check continuity at h = 5: 40 + 175 = 215 and 65 + 150 = 215.
Center B. B(h) = 60 + 28h for all h ≥ 0.
At h = 8.
- A(8) = 65 + 30(8) = $305.
- B(8) = 60 + 28(8) = $284.
Center A exceeds the $300 budget by $5; Center B costs $284 and is the only compliant choice. The lower registration fee of Center A does not compensate for its higher marginal hourly cost at this volume.
2. Find every whole-hour amount for which Center A is genuinely cheaper than Center B, and use the result to explain why the friend's advice fails at 8 hours.
Read the worked answer
Hours 0–5. 40 + 35h < 60 + 28h → 7h < 20 → h < 2.857. Whole hours: h = 1 and h = 2.
- h = 1: A = $75, B = $88.
- h = 2: A = $110, B = $116.
Hours 6 and up. 65 + 30h < 60 + 28h → 2h < −5, which has no nonnegative solution.
So Center A is cheaper only for 1 or 2 hours. At h = 8 the $20 registration advantage is overwhelmed: the tier break saves $5 per hour on the 3 hours past 5 (a $15 saving), but Center A's post-discount rate of $30 still exceeds Center B's flat $28 on all 8 hours. The friend focused on the fee structure and ignored the marginal rates.
3. Interpret the average cost per hour for each center at 8 hours and in the long run, and state what this implies for a student needing many hours.
Read the worked answer
At 8 hours. A: 305 ÷ 8 = $38.13 per hour. B: 284 ÷ 8 = $35.50 per hour.
Long run. A(h)/h = 65/h + 30 → 30 as h → ∞; B(h)/h = 60/h + 28 → 28. The fixed fees become negligible, so the limiting average cost equals the marginal rate.
Interpretation. Because 28 < 30, Center B is cheaper per hour for any sufficiently long engagement, despite the higher registration fee; the crossover is already past. Center A's effective hourly rate at 8 hours, (305 − 40) ÷ 8 = $33.13, still exceeds B's $28. The friend's rule of thumb is valid only for very short commitments.