Field Trip Buses, Vans, and a Tight Budget
The eighth grade is planning a field trip for 148 students and 12 chaperones. The bus company has two vehicle types: a bus seats 48 passengers and costs $420, and a van seats 15 passengers and costs $190. Every vehicle must be filled with at least one adult, and the principal requires at least two chaperones on each bus and at least one chaperone on each van. The trip budget is $2,400. The planners want the least expensive combination that seats all 160 people. A parent suggests booking four buses because 'four buses seat 192, so there will be plenty of room.' The student council treasurer points out that the goal is to seat everyone, not to buy extra seats, and asks for a cost comparison of at least three valid combinations.
1. Find the least expensive valid combination of buses and vans that seats all 160 people. Show at least three combinations and their costs.
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Let b be buses and v be vans. The seating rule is 48b + 15v ≥ 160, and the cost is 420b + 190v. Test combinations: 3 buses and 2 vans seat 144 + 30 = 174 and cost 3(420) + 2(190) = 1,260 + 380 = $1,640. 4 buses and 0 vans seat 192 and cost $1,680. 2 buses and 5 vans seat 96 + 75 = 171 and cost 840 + 950 = $1,790. 1 bus and 8 vans seat 48 + 120 = 168 and cost 420 + 1,520 = $1,940. 0 buses and 11 vans seat 165 and cost $2,090. The least expensive valid combination is 3 buses and 2 vans at $1,640. It is valid for chaperones: 3 buses need at least 6 chaperones and 2 vans need at least 2, for a total of 8, and there are 12 chaperones available.
2. Is the parent's suggestion of four buses the cheapest? Explain using the treasurer's concern.
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No. Four buses cost 4 × $420 = $1,680 and seat 192 people. Three buses and two vans cost $1,640 and seat 174 people, which is still more than the 160 people traveling. Four buses are $40 more expensive and add 18 unnecessary seats. The treasurer's point is that the planners should compare valid combinations by total cost, not by the amount of extra room. The four-bus plan also requires at least 8 chaperones, which is possible with 12, so the issue is cost rather than supervision. Unless the school has another reason to prefer all buses, such as avoiding vans for safety or scheduling, the three-bus, two-van plan is the better budget choice.
3. The bus company adds a $50 fuel fee per bus (not per van). Does the least expensive combination change? Show calculations.
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A bus now costs $470, while a van still costs $190. Recheck the leading plans: 3 buses and 2 vans cost 3(470) + 2(190) = 1,410 + 380 = $1,790. 4 buses and 0 vans cost 4(470) = $1,880. 2 buses and 5 vans cost 2(470) + 5(190) = 940 + 950 = $1,890. The order is unchanged: 3 buses and 2 vans is still the least expensive at $1,790. It remains under the $2,400 budget. The fuel fee raises every plan that uses buses, but it raises the four-bus plan by $200 and the three-bus plan by $150, so the three-bus plan keeps its advantage. The answer should state both the new cost and the fact that the winning combination did not change.