Concert Ticket Mix: Feasible Plans and a Counterexample
The student council has 240 seats in the school hall for a charity concert. Adult tickets cost 1,500 AMD and student tickets 800 AMD. The principal requires at least 60 student tickets and no more than twice as many adult tickets as student tickets. The council must raise at least 250,000 AMD. A member proposes selling 100 adult and 100 student tickets. Another proposes 130 adult and 70 student tickets. The council wants to know whether either plan works and what the smallest feasible number of student tickets is if adult tickets are pushed to the maximum allowed.
1. Write the constraints as inequalities using a for adult and s for student tickets. Test both proposals against every constraint and the revenue target.
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Let a = adult tickets, s = student tickets. Constraints: a+s ≤ 240; s ≥ 60; a ≤ 2s; 1500a + 800s ≥ 250000; a and s are non-negative integers.
- 100 adult, 100 student: capacity 200 ≤ 240; s=100 ≥ 60; a=100 ≤ 200; revenue = 150000 + 80000 = 230000 AMD < 250000. Fails the revenue target.
- 130 adult, 70 student: capacity 200 ≤ 240; s=70 ≥ 60; a=130 ≤ 140; revenue = 195000 + 56000 = 251000 AMD ≥ 250000. Meets all constraints.
2. Find the smallest s that can meet the revenue target when a is set to the largest value allowed. Show the algebra and give the corresponding a. Check capacity.
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For s ≤ 80, the capacity limit gives a ≤ 240 - s, and the mix rule gives a ≤ 2s. Since 2s ≤ 240 - s when s ≤ 80, the largest allowed a is 2s. Revenue is then 1500(2s) + 800s = 3800s. Set 3800s ≥ 250000, so s ≥ 65.79; the smallest integer is s = 66. Then a = 132. Check: 132 + 66 = 198 ≤ 240; 66 ≥ 60; 132 = 2×66. Revenue = 1500×132 + 800×66 = 198000 + 52800 = 250800 AMD. At s = 65, a = 130 and revenue is 247000, so 66 is indeed the minimum in this family.
3. A council member says, 'If we sell all 240 seats, we must reach the target.' Give a counterexample with all 240 seats sold that fails the target and still satisfies the principal's ticket-mix rules.
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Counterexample: a = 80, s = 160. All 240 seats are sold. The mix rules hold: 160 ≥ 60 and 80 ≤ 2×160 = 320. Revenue = 1500×80 + 800×160 = 120000 + 128000 = 248000 AMD, which is below 250000. So selling out does not guarantee the target; the ticket mix matters.