Yearbook Printing Contract
The yearbook committee has quotes from two printers. Printer A charges a $1,200 setup fee plus $18.50 per book. Printer B charges a $2,400 setup fee plus $14.25 per book, and it offers a 6% discount on the per-book price if the order is at least 425 books. The committee expects to sell between 380 and 460 books, but sales are uncertain. The school has $2,500 in pre-sold advertising that can offset production costs. The committee must choose a printer and set a book price that covers costs even if only 380 books sell, while not exceeding the school’s $55 per-book cap. A parent volunteer argues that Printer B is “always cheaper” because its per-book cost is lower.
1. Ignoring the bulk discount, find the break-even order quantity between the two printers. Explain what it means and whether the parent’s “always cheaper” claim is valid.
Read the worked answer
Set the costs equal:
$$1200+18.50x=2400+14.25x.$$
Subtract $14.25x$ and $1200$:
$$4.25x=1200\quad\Rightarrow\quad x\approx 282.35.$$
For orders above about 283 books, Printer B costs less; for orders below about 282 books, Printer A costs less. So the parent’s claim is false in general: at 200 books, A costs $4,900 and B costs $5,250. Within the expected range of 380–460 books, B is cheaper even before the discount, but “always” is not justified. The break-even quantity is the point where the fixed-fee disadvantage of B is exactly offset by its lower per-book charge.
2. Write Printer B’s cost as a piecewise function of the number of books x. Compute the cost at 420 books and at 430 books, and explain the discontinuity at 425.
Read the worked answer
Let $x$ be the number of books. Printer B charges
$$C_B(x)=\begin{cases}2400+14.25x, & 0\le x<425\[4pt]2400+(14.25)(0.94)x, & x\ge 425.\end{cases}$$
At 420 books:
$$C_B(420)=2400+14.25(420)=2400+5985=$8{,}385.$$
At 430 books:
$$C_B(430)=2400+13.395(430)=2400+5759.85=$8{,}159.85.$$
The per-book price drops by 6%, from $14.25 to $13.395, once 425 books are ordered. This creates a jump downward in total cost at 425: ordering 430 books costs less than ordering 420 books. That is mathematically possible because the fixed setup fee is unchanged and the discount applies to every book, not just the extra ones.
3. Determine the minimum book price that guarantees break-even at 380 books with Printer B, and decide whether the committee can recommend B under the $55 cap.
Read the worked answer
At 380 books, the discount does not apply, so
$$C_B(380)=2400+14.25(380)=2400+5415=$7{,}815.$$
Advertising offsets $2,500, leaving $5,315 to be covered by book sales. The required price is
$$p=\frac{5315}{380}\approx $13.99.$$
A price of $14.00 per book covers the worst-case expected sales and is far below the $55 cap. Printer A at 380 books would cost $1,200+18.50(380)=$8,230, which is higher. The committee can recommend B, but should note two risks: sales below 380 would require a higher price or more advertising, and the $14.00 price yields almost no surplus at 380. A safer price, such as $20, preserves break-even and builds a buffer, while still respecting the cap.