Concert Ticket Pricing and Surplus Target
A college fest committee is finalising ticket pricing for a concert. The auditorium has 1,500 seats. Fixed costs are ₹3,20,000 for sound, lighting, and security. Each attendee costs ₹90 in variable expenses for printing, refreshments, and volunteer support. The title sponsor pays ₹1,50,000 regardless of attendance. The committee plans a base ticket price of ₹400. To fill early seats, it offers a 20% discount on the first 300 tickets sold; all remaining tickets are at ₹400. The committee’s target is a surplus of at least ₹1,75,000 after covering all costs, assuming all 1,500 seats are sold. A member proposes dropping the early-bird discount to protect the surplus; another argues the discount is needed to reach full attendance. You must test whether the target is met with the discount, find the break-even attendance at full price, and evaluate the proposal.
1. Calculate the surplus if all 1,500 seats are sold with the early-bird discount. Does it meet the ₹1,75,000 target?
Read the worked answer
Discounted price = 400 × 0.80 = ₹320. Early-bird revenue = 300 × 320 = ₹96,000. Remaining tickets = 1,500 − 300 = 1,200; revenue = 1,200 × 400 = ₹4,80,000. Ticket revenue = ₹96,000 + ₹4,80,000 = ₹5,76,000. Add sponsor ₹1,50,000, so total inflow = ₹7,26,000. Total cost = fixed ₹3,20,000 + variable 1,500 × 90 = ₹3,20,000 + ₹1,35,000 = ₹4,55,000. Surplus = ₹7,26,000 − ₹4,55,000 = ₹2,71,000. This exceeds the ₹1,75,000 target by ₹96,000.
2. Find the break-even attendance at full price with no early-bird discount, and the attendance needed to hit the target surplus.
Read the worked answer
Let n be attendance. Full-price inflow = 400n + 1,50,000. Cost = 3,20,000 + 90n. Break-even: 400n + 1,50,000 = 3,20,000 + 90n, so 310n = 1,70,000, giving n = 548.39. Attendance must be at least 549. For target surplus: 400n + 1,50,000 − (3,20,000 + 90n) ≥ 1,75,000. This simplifies to 310n − 1,70,000 ≥ 1,75,000, so 310n ≥ 3,45,000, giving n ≥ 1,112.9. At least 1,113 attendees are needed to hit the target at full price.
3. Should the committee drop the early-bird discount? Give a decision rule.
Read the worked answer
Dropping the discount adds 300 × ₹80 = ₹24,000 if attendance stays at 1,500, raising surplus from ₹2,71,000 to ₹2,95,000. But the discount may be what fills seats. With no discount, the committee needs at least 1,113 attendees to reach the ₹1,75,000 target; below that, dropping the discount fails the target. Decision rule: drop the discount only if the committee is confident of at least 1,113 full-price sales. If the discount is necessary to push attendance above that threshold, keep it; if attendance would stay near 1,500 either way, dropping it is financially better.