The Tutoring Center's Two Constraints
The Riverside Learning Center has a $480 materials-and-staffing budget for one Saturday session. Each enrolled student receives a materials kit costing $9. Tutors are paid $18 per hour, and the center requires that every student receive at least 45 minutes of tutoring. Enrollment was capped at 24 students after last spring's waiting list, and the budget cannot be raised this term. Tutor hours must be purchased in half-hour blocks.
The director has scheduled 24 students and 14.5 tutor-hours, and she claims the plan satisfies both the budget and the 45-minute guarantee.
1. Is the director's claim correct? Show the arithmetic for both constraints.
Read the worked answer
Budget:
- Materials: 24 × $9 = $216
- Tutors: 14.5 × $18 = $261
- Total: $216 + $261 = $477, which is under $480. ✔
Guarantee:
- Total tutoring minutes: 14.5 × 60 = 870
- Per student: 870 ÷ 24 = 36.25 minutes
- Required: 45 minutes. ✘
Verdict: the claim is half right. The plan fits the budget with $3 to spare, but it falls 8.75 minutes short per student. A plan can pass one constraint and fail the other; the director checked only the one with an obvious dollar sign.
2. Ignoring the $480 cap for a moment, what is the minimum total budget that would satisfy both the 45-minute guarantee and the 24-student enrollment?
Read the worked answer
Tutor hours needed: 24 × 45 minutes = 1,080 minutes = 18 hours. (18 hours is already a whole number of hours, so half-hour blocks are not an issue here.)
Costs:
- Tutors: 18 × $18 = $324
- Materials: 24 × $9 = $216
- Total: $540
So the plan is $60 short of what the guarantee requires at full enrollment. This reframes the director's problem: it is not a scheduling problem she can fix by rearranging 14.5 hours, it is a $60 funding gap - or an enrollment decision.
3. The director cannot exceed $480 but may reduce enrollment. What is the largest number of students she can serve while keeping 45 minutes per student, given that hours are bought in half-hour blocks?
Read the worked answer
Set up the cost per student. Each student needs 0.75 tutor-hours and one kit:
- 0.75 × $18 = $13.50 tutoring + $9 materials = $22.50 per student
- $480 ÷ $22.50 = 21.33, so at most 21 students by pure division.
Check 21, because the half-hour block rule can bite:
- Hours: 21 × 0.75 = 15.75 → must round up to 16 hours
- Cost: (16 × $18) + (21 × $9) = $288 + $189 = $477 ✔ under $480
- Per student: 960 ÷ 21 = 45.7 minutes ✔
Check 22 to confirm the ceiling:
- Hours: 22 × 0.75 = 16.5, which is allowed → $297 + $198 = $495 ✘ over budget
Answer: 21 students. The rounding step matters - 15.75 hours rounds up, and a student who rounds down to 15.5 hours would wrongly conclude the center has $4 more room than it does. It does not; at 21 students the margin is $3 and no 22nd student fits.