Choosing a hall for the school concert
A youth group wants to hire a hall for a concert to raise money for the school library. Hall A charges a fixed P1 200 plus P15 for every ticket sold. Hall B charges a fixed P2 000 plus P8 for every ticket sold. Every ticket is sold at P50 and the venue holds at most 150 people, so no more than 150 tickets can be sold. The hire charge must be paid out of the ticket money, and the group wants to raise at least P3 000 for the library after paying the hall.
1. For what numbers of tickets sold is Hall A the cheaper hall? Set up and solve an inequality, and state the answer in the context of the problem.
Read the worked answer
Let n be the number of tickets sold.
Cost of Hall A = 1 200 + 15n Cost of Hall B = 2 000 + 8n
Hall A is cheaper when
1 200 + 15n < 2 000 + 8n 15n − 8n < 2 000 − 1 200 7n < 800 n < 114.28...
Since n must be a whole number, Hall A is cheaper for n ≤ 114 tickets.
Check the boundary: at n = 114, Hall A costs 1 200 + 1 710 = P2 910 and Hall B costs 2 000 + 912 = P2 912, so A is cheaper by P2. At n = 115, A costs P2 925 and B costs P2 920, so B is now cheaper. The answer is consistent.
2. Write an expression for the profit (money raised for the library) with each hall, and decide which hall gives the larger profit if all 150 tickets are sold.
Read the worked answer
Money taken = 50n.
Hall A: profit = 50n − (1 200 + 15n) = 35n − 1 200
At n = 150: 35 × 150 − 1 200 = 5 250 − 1 200 = P4 050
Hall B: profit = 50n − (2 000 + 8n) = 42n − 2 000
At n = 150: 42 × 150 − 2 000 = 6 300 − 2 000 = P4 300
Hall B gives P250 more when the hall is full. This shows why the answer to the first task must be read carefully: Hall A is cheaper only while ticket sales are low. Hall A keeps more of each ticket (P35 against P42 is not the comparison - rather, Hall A's lower fixed charge helps only up to 114 tickets), but Hall B's much lower charge per ticket takes over once sales are high. Since the group expects a full hall, Hall B is the better choice.
3. The group decides on Hall A because it is cheaper to risk. Using Hall A, find the least number of tickets that must be sold to raise at least P3 000. Is this possible within the capacity of the hall?
Read the worked answer
Profit with Hall A = 35n − 1 200.
35n − 1 200 ≥ 3 000 35n ≥ 4 200 n ≥ 120
So at least 120 tickets must be sold. Check: 35 × 120 − 1 200 = 4 200 − 1 200 = P3 000 exactly; 119 tickets would give 35 × 119 − 1 200 = P2 965, which is short.
120 is less than the capacity of 150, so the target is possible, but it leaves a margin of only 30 unsold tickets. If 31 or more tickets go unsold the group misses the target. For comparison, Hall B needs 42n − 2 000 ≥ 3 000, i.e. n ≥ 119.05, so 120 tickets as well - the two halls need the same sales at this target, which is another reason the group could safely choose Hall B.