Reams, Spoilage, and Whole-Unit Ordering at a Print Shop
A print shop has accepted a job for 4,500 finished sheets. Paper is sold only in reams of 500 sheets at $6.75 per ream, or in boxes of 5 reams for $31.50 per box; reams cannot be split or returned. On this printer, each ream that is loaded loses 8% of its sheets to jams and misprints, and the remaining sheets are usable. The shop has no paper in stock, will not open a ream it does not need for this job, and cannot deliver the job short. The shop's only cost for this job is paper, and no tax applies. The client has just called to say the order may grow by 100 sheets, so the owner wants the plan to be robust, not merely exact.
1. How many reams must the shop buy for 4,500 sheets, and what is the cheapest way to buy that many?
Read the worked answer
Usable sheets per ream: 500 × (1 − 0.08) = 500 × 0.92 = 460 sheets.
Reams required: 4,500 ÷ 460 = 9.78..., so 9 reams are not enough (9 × 460 = 4,140). Round up: 10 reams.
Cheapest purchase of exactly 10 reams:
- 10 single reams: 10 × $6.75 = $67.50
- 2 boxes of 5: 2 × $31.50 = $63.00
- 1 box + 5 singles: $31.50 + $33.75 = $65.25
Buy 2 boxes for $63.00. The rounding step matters: the answer is driven by usable sheets (460), not by the 500 sheets printed on the label.
2. Using the cheapest plan, what is the paper cost per usable sheet, and what percent does the shop save versus buying single reams?
Read the worked answer
All 4,500 sheets are deliverable, so the cost is spread over 4,500 usable sheets.
Box plan: $63.00 ÷ 4,500 = $0.014 per sheet, i.e. 1.4 cents. Single reams: $67.50 ÷ 4,500 = $0.015 per sheet, i.e. 1.5 cents.
Savings: (67.50 − 63.00) ÷ 67.50 = 4.50 ÷ 67.50 = 0.0667, about 6.7%.
The trap here is dividing by the 5,000 sheets purchased (10 × 500). That gives $0.0126, which understates the cost because 500 of those sheets are never deliverable.
3. The client confirms the order is now 4,600 sheets. Does the 10-ream plan still work, and what should the owner do?
Read the worked answer
10 reams produce 10 × 460 = 4,600 usable sheets - exactly the new order, with zero margin for error.
The 8% spoilage is an average, not a guarantee. If actual spoilage runs at 8.5%, usable sheets are 5,000 × 0.915 = 4,575, and the job ships short. Since the shop cannot deliver short, the estimate cannot be treated as an exact quantity.
Safe plan: 11 reams. Cheapest way: 2 boxes + 1 single ream = $63.00 + $6.75 = $69.75, yielding 5,060 usable sheets (460 spare). Buying 3 boxes ($94.50) would be waste.
The judgment: when the computed requirement lands exactly on the break-even quantity, round up one more ream rather than accept a zero-slack plan.