Ticket Sales with a Ratio Constraint
A school club is selling tickets to a show. Student tickets cost $4 and adult tickets cost $7. The auditorium has 250 seats, and the club wants to raise at least $1,200. So far it has sold 90 student tickets and 40 adult tickets. Two rules apply to the rest of the sales: the club must sell at least 20 more adult tickets, and overall the number of student tickets sold must be at least twice the number of adult tickets sold. The club can sell at most 120 more tickets in total.
1. Could the club meet the money goal by selling all 120 remaining seats as adult tickets? Explain using the rules.
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The money goal would be met, but the ratio rule would be violated. Current revenue is 90($4) + 40($7) = $360 + $280 = $640. Selling 120 more adult tickets adds 120($7) = $840, for a total of $1,480, which is above $1,200. But the totals would then be 90 student tickets and 160 adult tickets. The rule requires student tickets to be at least twice adult tickets: 90 is not at least 2(160) = 320. So this plan fails the ratio rule.
2. Find one combination of additional student and adult tickets that satisfies every rule, and check each rule.
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One working combination is 90 more student tickets and 30 more adult tickets. That uses all 120 remaining seats. Revenue added: 90($4) + 30($7) = $360 + $210 = $570, so total revenue is $640 + $570 = $1,210, which is at least $1,200. Total student tickets: 90 + 90 = 180. Total adult tickets: 40 + 30 = 70. The ratio rule needs 180 ≥ 2(70) = 140, which holds. The club adds 30 adult tickets, so the “at least 20 more adult” rule also holds.
3. Explain how the ratio rule limits how adult-heavy the remaining sales can be.
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Let a be the number of additional adult tickets and s the number of additional student tickets, with s + a ≤ 120. To raise as much money as possible the club would fill the seats, so take s = 120 − a. Then total student tickets are 90 + (120 − a) = 210 − a, and total adult tickets are 40 + a. The ratio rule requires 210 − a ≥ 2(40 + a), which simplifies to 210 − a ≥ 80 + 2a, then 130 ≥ 3a, so a ≤ 43.33. At most 43 additional adult tickets are allowed. Selling 44 or more adult tickets would make the adult total grow faster than the student total can keep pace with, so the club cannot simply sell mostly adult tickets even though they bring in more money each.