The Inter-School Science Fair Trip
Northgate Secondary is taking 174 students and 12 chaperones to a regional science fair. Every vehicle hired must carry at least one chaperone, and no chaperone can supervise two vehicles on the same trip. Two companies quote for a single out-and-back trip. Company A has 32-seat buses at $220 each. Company B has 14-seat vans at $105 each. A third company can supply at most one 50-seat coach for $600, which may be combined with either A or B vehicles. The school wants the lowest total hire cost while moving all 186 people in one departure. The deputy principal suggests hiring six buses because 'that gives 192 seats and keeps everyone together.' A parent suggests 13 vans because 'vans are cheaper per vehicle.'
1. Show why 13 vans is not a valid plan and why six buses is valid but not necessarily cheapest.
Read the worked answer
13 vans carry 13 × 14 = 182 seats, which is fewer than 186, so the plan fails on capacity. It would also need 13 chaperones, but only 12 are available. Six buses carry 6 × 32 = 192 seats and need only 6 chaperones, so that plan is valid. Its cost is 6 × $220 = $1,320. But valid does not mean cheapest; a mix of buses and vans may cost less.
2. Find the least-cost combination using Companies A and B only. Show your trials or a table.
Read the worked answer
Let b be buses and v be vans. Constraints: 32b + 14v ≥ 186 and b + v ≤ 12. Check v from 0 upward, taking the smallest b that meets capacity:
v=0, b=6: $1,320 v=1, b=6: $1,425 v=2, b=5: 5×220 + 2×105 = $1,310 v=3, b=5: $1,415 v=5, b=4: $1,405 v=7, b=3: $1,395 v=9, b=2: $1,385 v=11, b=1: $1,375 v=12, b=1: $1,480
The minimum is 5 buses and 2 vans: 160 + 28 = 188 seats, 7 vehicles and 7 chaperones, cost $1,310. This beats six buses by $10.
3. The coach option is added. Does the least-cost plan change? Show the total cost of the best plan that uses the coach and compare.
Read the worked answer
With the coach, total vehicles must still be at most 12 because each needs a chaperone, so b + v ≤ 11. Capacity: 50 + 32b + 14v ≥ 186, i.e. 32b + 14v ≥ 136. Test v:
v=0, b=5: $600 + 5×220 = $1,700 v=1, b=4: $600 + 880 + 105 = $1,585 v=3, b=3: $600 + 660 + 315 = $1,575 v=6, b=2: $600 + 440 + 630 = $1,670 v=8, b=1: $600 + 220 + 840 = $1,660 v=11, b=1: $600 + 220 + 1,155 = $1,975
The best coach plan is 3 buses + 3 vans + coach at $1,575. That is $265 more than the $1,310 plan without the coach, so the recommendation does not change.