Hiring a DJ: Two Payment Plans
A student committee is hiring a DJ for a school dance. Plan A charges a flat $180 plus $4 per attendee over 100. Plan B charges $60 plus $7 per attendee, with no free allowance. The committee expects between 120 and 260 attendees but is unsure. Ticket price is $6 per person, and the venue costs $250 regardless of attendance. The committee wants to know the attendance at which the two DJ plans cost the same, and which plan is cheaper at their most likely attendance of 200.
1. Find the attendance at which Plan A and Plan B cost the same, and interpret it.
Read the worked answer
Let n = number of attendees.
Plan A cost: for n > 100, A = 180 + 4(n − 100) = 180 + 4n − 400 = 4n − 220. Plan B cost: B = 60 + 7n.
Set equal: 4n − 220 = 60 + 7n −280 = 3n n = −93.33...
This is negative, so within the relevant range (n > 100) the plans never cost the same. For n > 100, B − A = 60 + 7n − (4n − 220) = 3n + 280, which is always positive.
Interpretation: Plan A is always cheaper once the free allowance applies, and the gap widens as attendance grows. There is no break-even attendance in the realistic range.
2. At 200 attendees, determine the cheaper DJ plan and whether the dance makes a profit overall.
Read the worked answer
At n = 200: Plan A = 4(200) − 220 = 800 − 220 = $580. Plan B = 60 + 7(200) = 60 + 1,400 = $1,460.
Plan A is cheaper by $880.
Revenue = 6 × 200 = $1,200. Costs with Plan A = 580 + 250 (venue) = $830. Profit = 1,200 − 830 = $370.
With Plan B, costs would be 1,460 + 250 = $1,710, giving a loss of $510.
So at 200 attendees the committee should choose Plan A, which yields a $370 profit; Plan B would make the event lose money.