Concert Ticket Revenue Plan
The student council is selling tickets for the spring concert. Floor tickets cost $45 and balcony tickets cost $30. The auditorium seats 850 people. The council must raise at least $30,000 in ticket revenue to cover costs. So far it has sold 200 floor tickets and 300 balcony tickets. A local business has pledged a $2,000 donation only if ticket revenue reaches $31,000. The council assumes every remaining seat will be sold. The president wants to know the minimum number of the remaining floor tickets that must be sold to meet the $30,000 goal, and separately what is needed to qualify for the donation. A member suggests selling all remaining seats as balcony tickets, saying that would still reach $30,000. Seats are indivisible, and ticket prices do not change.
1. Let f be the number of additional floor tickets and b the number of additional balcony tickets. Write a system for selling all remaining seats and find the minimum f needed for $30,000.
Read the worked answer
Let f and b be additional floor and balcony tickets. All remaining seats: f + b = 350. Revenue already sold: 200(45) + 300(30) = 9,000 + 9,000 = 18,000. Need 30,000 − 18,000 = 12,000 more. Substitute b = 350 − f: 45f + 30(350 − f) ≥ 12,000, which gives 45f + 10,500 − 30f ≥ 12,000, so 15f ≥ 1,500 and f ≥ 100. At least 100 of the 350 remaining seats must be floor tickets; the other 250 can be balcony. Check: 100(45) + 250(30) = 4,500 + 7,500 = 12,000.
2. Find the minimum additional floor tickets needed to reach $31,000 and qualify for the donation.
Read the worked answer
Need 31,000 − 18,000 = 13,000 more. Using b = 350 − f: 45f + 30(350 − f) ≥ 13,000, so 15f + 10,500 ≥ 13,000, 15f ≥ 2,500, and f ≥ 166.67. Since tickets are whole, f ≥ 167. Then b = 183. Check revenue: 167(45) = 7,515 and 183(30) = 5,490, total 13,005; total revenue 31,005, enough to qualify.
3. Evaluate the member's claim that selling all remaining seats as balcony still reaches $30,000.
Read the worked answer
All-balcony remaining revenue = 350(30) = 10,500. Total = 18,000 + 10,500 = 28,500, which is $1,500 short of $30,000. The member is wrong. To reach 30,000, at least 100 remaining floor tickets are needed, as found earlier; selling all balcony tickets leaves the council below its goal.