T-Shirt Printing Quotes
The class council needs T-shirts printed for a fundraiser and has quotes from two shops. Print Shop A charges a $45 setup fee plus $6 per shirt. Print Shop B charges a $20 setup fee plus $8 per shirt, but it takes 25% off the entire order when the order is more than 100 shirts. The council expects to order between 40 and 150 shirts, in whole numbers. A member claims that Shop B is always cheaper once the discount applies and that there is an order size above 100 where the two shops cost the same. The treasurer wants a clear rule for choosing a shop based on the number of shirts ordered. Both shops confirm there are no other fees or taxes.
1. Write a piecewise function B(n) for Shop B's cost and find the order size where A(n) = B(n) for n ≤ 100. Interpret the result for whole-number orders.
Read the worked answer
A(n) = 45 + 6n. B(n) = 20 + 8n for n ≤ 100. Set equal: 45 + 6n = 20 + 8n, so 25 = 2n and n = 12.5. Since shirts are whole, no order of exactly 12.5 shirts exists. At n = 12, A = 45 + 72 = 117 and B = 20 + 96 = 116, so B is cheaper by $1. At n = 13, A = 123 and B = 124, so A becomes cheaper. The break-even is between 12 and 13 shirts.
2. Determine which shop is cheaper for 80 shirts and for 120 shirts.
Read the worked answer
For n = 80, B has no discount: A = 45 + 6(80) = 525; B = 20 + 8(80) = 660. Shop A saves $135. For n = 120, the discount applies: A = 45 + 6(120) = 765; B = 0.75(20 + 8(120)) = 0.75(980) = 735. Shop B saves $30.
3. Evaluate the member's two claims using algebra for n > 100.
Read the worked answer
For n > 100, B(n) = 0.75(20 + 8n) = 15 + 6n. A(n) = 45 + 6n. Then B(n) = A(n) − 30 for every n > 100, so Shop B is always $30 cheaper once the discount applies, but the costs are never equal above 100. The first claim is true; the second is false. The only algebraic equality is at n = 12.5, which is not a possible order.