Water Bottle Order: Tiered Pricing, Case Packs, and Break-Even
The student council of Northfield High is ordering custom stainless-steel water bottles for its annual 5K fundraiser. They expect 240 registered runners and want 10% extra bottles for volunteers and late entries. Supplier A charges $45 for setup, $6.40 per bottle for the first 150 bottles, and $5.20 for each bottle beyond 150. Supplier B charges $5.75 per bottle with no setup fee, but requires orders in whole multiples of 24 and cannot split a case. The council has a firm budget of $1,800 and wants at least 264 bottles. A local shop has offered to print a logo on any bottles the council buys for an additional $0.35 per bottle, but only if the order is at least 250 bottles. The council also plans to sell leftover bottles at the race for $8 each, though past events suggest about 8% of the ordered bottles may remain unsold after the race. They need to decide which supplier to use and what price to charge to recover costs.
1. Determine the minimum number of bottles the council needs and compare the total cost from Supplier A and Supplier B, including logo printing where applicable.
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Bottles needed: 240 × 1.10 = 264, so the minimum order is 264. Supplier A: first 150 × $6.40 = $960.00; remaining 114 × $5.20 = $592.80; setup $45; logo 264 × $0.35 = $92.40. Total = $960 + $592.80 + $45 + $92.40 = $1,690.20. Supplier B: 264 is exactly 11 × 24, so no rounding up is needed. Bottles: 264 × $5.75 = $1,518.00; logo = $92.40; no setup. Total = $1,610.40. B is cheaper by $79.80, and both totals fit the $1,800 budget. The logo condition is met because 264 ≥ 250.
2. Using Supplier B, calculate the net surplus if 8% of ordered bottles remain unsold and all other bottles sell for $8. Does the council recover its cost?
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At an 8% unsold rate, 264 × 0.08 = 21.12, so about 21 bottles remain and 264 − 21 = 243 bottles are sold. Revenue = 243 × $8 = $1,944. Cost from B = $1,610.40. Net surplus = $1,944 − $1,610.40 = $333.60. Using the exact fraction, 0.92 × 264 = 242.88 bottles sold, revenue = $1,943.04 and surplus = $332.64; the difference is rounding to whole bottles. The council does recover its cost, but the margin is thin enough that the unsold-rate assumption matters. At 15% unsold, revenue is 264 × 0.85 × 8 = $1,795.20, still above cost; at 25% unsold, revenue falls to $1,584, below cost.
3. Supplier B is considering raising its per-bottle price. At 264 bottles, what is the highest price at which B is still cheaper than A once logo printing is included?
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Compare at 264 bottles with the logo on both sides. A's total is $1,690.20. B's cost at price p is 264p + $92.40. Set 264p + 92.40 < 1,690.20 → 264p < 1,597.80 → p < $6.052. So B remains cheaper up to about $6.05 per bottle. At exactly $6.05, B costs 264 × 6.05 + 92.40 = $1,689.60, just $0.60 below A. That margin is too thin to rely on without written quotes: a small change in A's setup fee or a volume discount would flip the decision.