The Canteen Queue Problem
A school canteen sells three lunch deals. The 'Small' deal costs $4 and includes 1 item. The 'Medium' deal costs $7 and includes 2 items. The 'Large' deal costs $9 and includes 3 items. On Monday, 60 students buy lunch. The canteen takes $380 in total and serves 130 items. Every student buys exactly one deal. The canteen manager thinks most students chose the Large deal. A student helper claims the number of Medium deals was exactly double the number of Small deals.
1. Using the totals only, work out how many students bought each deal. Is the helper's claim about Medium being double Small necessarily true?
Read the worked answer
Let S, M, L be the numbers of Small, Medium and Large deals.
Equations from the setup:
- Students: S + M + L = 60
- Money: 4S + 7M + 9L = 380
- Items: 1S + 2M + 3L = 130
Eliminate S. Subtract the student equation from the items equation: (1S + 2M + 3L) − (S + M + L) = 130 − 60, so M + 2L = 70.
Now subtract 4 times the student equation from the money equation: (4S + 7M + 9L) − 4(S + M + L) = 380 − 240, so 3M + 5L = 140.
Solve the two-variable system: From M + 2L = 70, M = 70 − 2L. Substitute: 3(70 − 2L) + 5L = 140, so 210 − 6L + 5L = 140, giving L = 70. But L cannot exceed 60 students.
This contradiction means the three totals are inconsistent: no whole-number solution exists. The helper's claim cannot be tested because the data itself is impossible. A careful answer states the data are inconsistent rather than forcing a false solution.
Check: If L = 70, then M = 70 − 140 = −70, which is impossible. So the totals do not describe a real Monday.
2. Suppose the money total was misrecorded and the true total was $360. Recalculate the deal counts and judge whether the manager's belief is supported.
Read the worked answer
Keep S + M + L = 60 and M + 2L = 70. Replace the money equation with 4S + 7M + 9L = 360.
Subtract 4 times the student equation: 3M + 5L = 360 − 240 = 120.
Now solve:
- M + 2L = 70 → M = 70 − 2L
- 3(70 − 2L) + 5L = 120 → 210 − 6L + 5L = 120 → L = 90
Again L = 90 exceeds 60, so $360 is also impossible. This shows the error is not a small typo in the money total; the item total of 130 is the likely culprit.
Judgement: The manager's belief that most students chose Large cannot be supported by impossible data. The right move is to flag the inconsistency and ask for a recount before drawing any conclusion about popularity.