T-Shirt Fundraiser Profit and Supplier Choice
The robotics club is selling T-shirts to raise money for a competition. Supplier A charges a one-time setup fee of $45 plus $7.50 for each shirt. The club plans to sell every shirt for $15. Supplier B has no setup fee but charges $11 per shirt. The club has a goal of at least $300 in profit after paying for the shirts. Profit means money collected from sales minus the total cost of the shirts. The club can order any whole number of shirts, but it does not want to order more than 120 because storage is limited. A member writes the profit expression 15n − (45 + 7.5n) and claims the club must sell 50 shirts to reach the goal. The advisor wants the expression simplified, the correct minimum number of shirts found, and a comparison of the two suppliers for a possible order of 80 shirts.
1. Simplify the profit expression for Supplier A and explain what each part represents.
Read the worked answer
15n − (45 + 7.5n) = 15n − 45 − 7.5n = 7.5n − 45. The 7.5n is the profit from selling n shirts before the setup fee: each shirt brings in $15 and costs $7.50, leaving $7.50 per shirt. The −45 is the one-time setup fee that must be paid no matter how many shirts are ordered. So profit is $7.50 per shirt minus $45.
2. Solve the inequality for at least $300 profit with Supplier A. Is the member's claim of 50 shirts correct? Explain.
Read the worked answer
Set 7.5n − 45 ≥ 300. Add 45 to both sides: 7.5n ≥ 345. Divide by 7.5: n ≥ 46. Since n must be a whole number, the club needs at least 46 shirts. Check n = 46: profit = 7.5 × 46 − 45 = 345 − 45 = $300. The member's claim of 50 shirts is not the minimum; 50 shirts would work, but the club can meet the goal with 46 shirts.
3. For an order of 80 shirts, compare total costs from Supplier A and Supplier B. At what order size does Supplier A become cheaper than Supplier B?
Read the worked answer
Supplier A for 80 shirts: 45 + 7.5 × 80 = 45 + 600 = $645. Supplier B for 80 shirts: 11 × 80 = $880. Supplier A is cheaper by $235 at 80 shirts. To find when A becomes cheaper, solve 45 + 7.5n < 11n. Subtract 7.5n: 45 < 3.5n. Divide by 3.5: n > 12.857..., so the first whole-number order size is n = 13. At 13 shirts, A costs 45 + 97.50 = $142.50 and B costs 11 × 13 = $143, so A is cheaper starting at 13 shirts.