Choosing a Packaging Option
A bakery is choosing bags for a new granola line. Option A uses stock bags with a single-color printed label: a one-time plate fee of $120 plus $0.85 per bag. Option B uses fully custom-printed bags: a one-time setup of $260 plus $0.60 per bag. The owner has $700 available for packaging this quarter and expects to sell no more than 500 bags before the next order. Unused bags can be stored and used later, but the owner does not want to run out mid-season, because a second order placed late would cost more per bag than the prices quoted here.
1. At what quantity do the two options cost the same? Use that number to advise the owner given the 500-bag sales expectation.
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Set the total costs equal, with n = number of bags:
120 + 0.85n = 260 + 0.60n 0.25n = 140 n = 560 bags
Check: Option A at 560 = 120 + 476 = $596. Option B at 560 = 260 + 336 = $596.
Advice: the expectation is at most 500 bags, which is below the 560-bag break-even. In that range Option A is cheaper, because its higher per-bag price has less distance to run than B's extra $140 in setup. At 500 bags: A = 120 + 425 = $545; B = 260 + 300 = $560. A saves $15. The margin is thin, though -- an extra 60 bags of sales would erase it -- so the recommendation should note that if demand looks likely to exceed 560, the answer flips.
2. Using the full $700 budget, what is the largest whole number of bags the owner can buy under each option?
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Option A: 120 + 0.85n <= 700 -> 0.85n <= 580 -> n <= 682.35... -> 682 bags. Check: 682 x 0.85 = 579.70; + 120 = $699.70. One more bag would be $700.55, over budget.
Option B: 260 + 0.60n <= 700 -> 0.60n <= 440 -> n <= 733.33... -> 733 bags. Check: 733 x 0.60 = 439.80; + 260 = $699.80. One more bag would be $700.40, over budget.
Both answers must be rounded down, not to the nearest integer -- a fractional bag cannot be ordered, and rounding up breaks the budget constraint. Note that B's larger ceiling only becomes relevant past 560 bags; below that, spending the same $700 buys more bags under A? No -- under A, $682 bags cost $699.70, while 682 bags under B would cost 260 + 409.20 = $669.20. The ceiling is not the cost.
3. The owner says, "Option B is cheaper per bag, so it must be the cheaper choice." Identify the error in that reasoning and give a recommendation that weighs cost against the risk of running out.
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The error: the per-bag price is only the variable cost. It ignores the fixed setup cost, which is $140 higher for B. Total cost = fixed + (rate x quantity), so a lower rate does not guarantee a lower total. The lower rate wins only past the break-even quantity of 560 bags.
Recommendation with the risk stated:
- Expected sales <= 500, which is below 560, so Option A is cheaper at the anticipated volume, and it leaves a lower fixed commitment if the granola line underperforms.
- The owner's worry about running out cuts the other way: $700 buys 733 bags under B versus 682 under A. But bags can be stored, and a mid-season reorder is possible -- at a higher per-bag cost, which is exactly the cost B was supposed to avoid.
- Decision rule: choose B only if the owner has real evidence that sales will exceed 560. At 500 bags B costs $15 more; at 700 bags B saves $35 (A = $715, B = $680). The dollar stakes are small either way, so the tie-breaker should be confidence in the sales forecast, not the headline per-bag price.