Coffee Stand: Two Vendor Plans
A student club is choosing a vendor for a one-day coffee stand. Every cup sells for $4.00, and the club expects to sell between 450 and 550 cups. Vendor A charges a flat $180 booth fee and lets the club buy supplies at $1.15 per cup, paid after the event. Vendor B charges a $60 booth fee plus 35% of gross sales, with supplies included. The club has no preference between vendors and wants the largest profit. A similar stand last year sold only 300 cups, so the club's adviser doubts the 450-550 estimate and asks for the sales volume at which the two plans stop being roughly equal.
1. Write a profit expression for each vendor in terms of c, the number of cups sold, and find the volume at which the two plans produce equal profit.
Read the worked answer
Vendor A: profit = revenue - supplies - fee
$$P_A = 4.00c - 1.15c - 180 = 2.85c - 180$$
Vendor B: the vendor takes 35% of gross sales, so the club keeps 65%:
$$P_B = 4.00c - 0.35(4.00c) - 60 = 2.60c - 60$$
Set them equal:
$$2.85c - 180 = 2.60c - 60 \Rightarrow 0.25c = 120 \Rightarrow c = 480$$
At 480 cups both plans net the same amount. Below 480 cups B is better; above 480 cups A is better. (For reference, A breaks even at 180/2.85 ≈ 63.2, so 64 cups; B breaks even at 60/2.60 ≈ 23.1, so 24 cups.)
2. Using the adviser's figure of 300 cups, compute each plan's profit and state the difference.
Read the worked answer
$$P_A = 2.85(300) - 180 = 855 - 180 = 675$$
$$P_B = 2.60(300) - 60 = 780 - 60 = 720$$
Vendor B yields $720, which is $45 more than Vendor A's $675 at 300 cups. Note the direction: the shortfall from optimistic forecasting falls hardest on the plan with the larger fixed fee.
3. The club must pick one vendor before sales are known. Using the 450-550 range, the 300-cup precedent, and the crossover point, recommend a plan and justify the choice arithmetically.
Read the worked answer
Check the edges of the estimate range:
- At 450 cups: $P_A = 2.85(450) - 180 = 1,102.50$; $P_B = 2.60(450) - 60 = 1,110$. B wins by $7.50.
- At 550 cups: $P_A = 2.85(550) - 180 = 1,387.50$; $P_B = 2.60(550) - 60 = 1,370$. A wins by $17.50.
- At 300 cups: B wins by $45.
Recommendation: Vendor B. The entire upside of Plan A inside the forecast range is about $17.50, while a repeat of last year's 300-cup turnout costs $45 under Plan A. Plan A only pays off if sales are confidently above 480 cups, and the club has a concrete reason (last year) to doubt that. The decision turns on the asymmetry of the risk, not on which plan has the higher break-even volume.