Dance Ticket Pricing and Break-Even Safety
The student council is choosing a ticket price for the spring dance. The venue costs $450, the DJ $250, and decorations $80. Refreshments cost $3 per attendee. The gym can hold 220 people. Last year, 160 students attended when tickets cost $8. This year the council is considering $10, but a survey suggests attendance would be 15% lower than last year’s 160. The council must cover all costs and wants to avoid a loss. They also want a cushion: if attendance is 20% lower than expected, the dance should still not lose money. The treasurer proposes comparing profit at the two prices and checking each break-even point. A member argues that the lower price is safer because it attracts more people. The council has not decided whether to prioritize expected profit or downside protection.
1. Write a profit expression for each ticket price and find the break-even attendance. Use n for attendance.
Read the worked answer
Fixed costs: 450 + 250 + 80 = 780. Refreshments are $3 per attendee, so contribution per ticket is price − 3.
At $8: P8 = (8 − 3)n − 780 = 5n − 780. Break-even: 5n − 780 = 0 → n = 156.
At $10: P10 = (10 − 3)n − 780 = 7n − 780. Break-even: 7n − 780 = 0 → n = 111.43, so 112 whole tickets. At 111 tickets P10 = −3; at 112 tickets P10 = 4.
The $10 price has a lower break-even because each ticket contributes $7 instead of $5.
2. The survey predicts $10 attendance will be 15% below last year’s 160. Compare expected profits at $10 and at $8 with 160 attendees. Which price is expected to raise more?
Read the worked answer
Expected $10 attendance = 160(1 − 0.15) = 136.
P10 = 7(136) − 780 = 952 − 780 = 172.
At $8 with 160 attendees: P8 = 5(160) − 780 = 800 − 780 = 20.
The $10 plan is expected to earn $152 more. The lower attendance is more than offset by the higher contribution per ticket, because 136 is well above the $10 break-even of 112.
3. A member says the $8 price is safer. Test that claim using a 20% shortfall from each expected attendance and using the attendance at which $10 matches the $8 profit at 160.
Read the worked answer
20% shortfall at $10: 136(0.8) = 108.8, so about 109 attendees. P10 = 7(109) − 780 = 763 − 780 = −17, a loss.
20% shortfall at $8: 160(0.8) = 128. P8 = 5(128) − 780 = 640 − 780 = −140, a larger loss.
To match the $8 profit of $20, $10 needs 7n − 780 > 20 → 7n > 800 → n > 114.29, so 115 attendees.
Thus $10 is safer on break-even and on the shortfall test, but it is not risk-free: below 112 attendees it loses money. The member is only partly right: $8 attracts more people, but it requires 156 attendees just to break even.