Concert Fund-Raiser Break-Even Decision
The student council plans a fund-raising concert. The venue costs P18,000, the band costs P22,000, and sound and lights cost P10,000. Tickets will sell for P250 each, but the council expects to give 40 free tickets to sponsors and volunteers. The council has P15,000 in savings that it can use, but it does not want to spend more than that from savings. A local business offers to cover P8,000 of the venue cost if the council prints the business logo on 500 posters. Printing 500 posters costs P6,000. The council must decide whether to accept the offer, and it wants to know how many tickets it must sell to break even without using more than P15,000 of savings. All amounts are in pesos.
1. Without the business offer, write an equation for the number of paid tickets t needed so that the council uses no more than P15,000 of savings. Solve for the minimum whole number of tickets.
Read the worked answer
Total cost without offer = 18,000 + 22,000 + 10,000 = 50,000 pesos. Revenue from t paid tickets = 250t. Savings used = 50,000 − 250t. The condition is 50,000 − 250t ≤ 15,000. Subtract 50,000: −250t ≤ −35,000. Divide by −250 and flip the inequality: t ≥ 140. So the council must sell at least 140 paid tickets. The 40 free tickets do not add revenue, but the problem gives no seating limit, so they do not change this equation.
2. With the business offer, the council pays P6,000 for posters and receives P8,000, so net benefit is P2,000. Does accepting the offer lower or raise the break-even number of paid tickets? Show the new equation and solve.
Read the worked answer
Net effect of the offer: the council receives 8,000 but pays 6,000 for posters, so net cost falls by 2,000. New total cost = 50,000 − 2,000 = 48,000. New savings condition: 48,000 − 250t ≤ 15,000, so 250t ≥ 33,000, t ≥ 132. The offer lowers the break-even number from 140 to 132 paid tickets, a reduction of 8 tickets. It raises the council's savings cushion at any given ticket sales level.
3. The council expects to sell 150 paid tickets. Should it accept the offer? Consider both the break-even rule and a non-mathematical factor.
Read the worked answer
At 150 paid tickets, revenue is 250 × 150 = 37,500. Without the offer, savings used = 50,000 − 37,500 = 12,500, which is within the 15,000 limit, so the concert is feasible. With the offer, savings used = 48,000 − 37,500 = 10,500, also within the limit and better by 2,000. Mathematically, accept the offer. A non-mathematical factor is whether the council is willing to print the business logo on 500 posters and whether that poster quantity is useful; if the sponsor's condition conflicts with school policy or the posters would be wasted, the council could decline despite the better number.