Comparing Flyer Printing Plans
A school fundraiser needs 600 flyers. A print shop offers two plans. Plan A charges a $15 setup fee plus $0.25 per flyer. Plan B charges a $30 setup fee plus $0.18 per flyer. One volunteer calculates: "Plan A costs 600 times $0.25, which is $150. Plan B has a $30 setup fee, so Plan A is cheaper." Another volunteer says the first calculation leaves something out and that the comparison is not complete. The group wants to know which plan costs less for exactly 600 flyers and whether the answer would change for a much smaller order.
1. Calculate the total cost of each plan for 600 flyers and identify the error in the first volunteer's reasoning.
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Plan A: $15 setup + 600 × $0.25 = $15 + $150 = $165. Plan B: $30 setup + 600 × $0.18 = $30 + $108 = $138. The first volunteer compared only the per-flyer portion of Plan A ($150) with only the setup fee of Plan B ($30), and also ignored Plan B's per-flyer charge. That is not a valid comparison because the two numbers represent different components of cost. For 600 flyers, Plan B is cheaper by $27. The reasoning error is mixing a variable cost with a fixed cost and then drawing a conclusion about total cost. A correct comparison must include setup plus per-flyer cost for both plans.
2. At what number of flyers do the two plans cost the same? Show the reasoning.
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Set the total costs equal: 15 + 0.25x = 30 + 0.18x. Subtract 0.18x from both sides: 15 + 0.07x = 30. Subtract 15: 0.07x = 15. Divide by 0.07: x = 214.285..., so the plans are equal at about 214.3 flyers. Since you cannot print a fraction of a flyer in a practical order, the break-even point is between 214 and 215 flyers: at 214 flyers Plan A is slightly cheaper, and at 215 flyers Plan B becomes cheaper. This matters because the volunteer's conclusion may be right for a very small order but wrong for 600. The judgement call is to recognize that the cheaper plan depends on quantity, not on a single per-unit rate.
3. The group says, "Plan B is cheaper, so we should always use Plan B." Evaluate that conclusion.
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The conclusion is too broad. Plan B is cheaper at 600 flyers, but it has a higher setup fee. For small orders, Plan A can be cheaper because its lower fixed cost outweighs its higher per-flyer charge. For example, at 100 flyers Plan A costs $15 + $25 = $40, while Plan B costs $30 + $18 = $48. The correct conclusion is conditional: Plan B is cheaper for orders above roughly 215 flyers, and Plan A is cheaper below that. The group should also consider non-cost factors such as quality, turnaround time, and whether the fundraiser might need fewer or more flyers than planned. A good answer distinguishes a valid calculation for one quantity from a general rule that applies to all quantities.