Calculator Purchase Order: Tiered Pricing vs. Total Cost
A seventh-grade team needs calculators, and the school business office has two quotes on file. Vendor A lists $11.75 per calculator for orders of 1–49, $10.40 for 50–99, and $9.85 for 100 or more, with a flat $48 shipping charge on every order. Vendor B charges $12.20 per calculator with no minimum; shipping is free on orders over $250 and $15 otherwise. The district adds 6.5% sales tax to the merchandise subtotal only - shipping is not taxed. The team must supply 72 students, but the principal will approve up to 100 units if a larger order is genuinely cheaper. A team member's draft note reads: 'Buy 100 from Vendor A - the unit price falls from $10.40 to $9.85, so we save 5.3% and get spare calculators for free.' The business office wants a written justification before it issues the purchase order.
1. Compute the total cost to the district of (a) 72 calculators from Vendor A, (b) 100 calculators from Vendor A, and (c) 72 calculators from Vendor B. Which option is cheapest, and what happens to the draft note's claim?
Read the worked answer
Vendor A, 72 units (tier 50–99):
- Merchandise: 72 × $10.40 = $748.80
- Shipping: $48.00
- Tax: 0.065 × $748.80 = $48.67
- Total: $845.47
Vendor A, 100 units (tier 100+):
- Merchandise: 100 × $9.85 = $985.00
- Shipping: $48.00
- Tax: 0.065 × $985.00 = $64.03
- Total: $1,097.03
Vendor B, 72 units:
- Merchandise: 72 × $12.20 = $878.40 (over $250, so free shipping)
- Tax: 0.065 × $878.40 = $57.10
- Total: $935.50
Cheapest is Vendor A at 72 units, $845.47 - $90.03 less than Vendor B. The draft note fails: buying 100 costs $251.56 more, which is $8.98 for each of the 28 extra calculators, and those units may never be used. A lower unit price is not the same as a lower total cost.
2. The team member defends the claim: '5.3% is just the change from $10.40 to $9.85, so the district saves 5.3% either way.' Identify the reasoning errors and state what comparison actually decides whether a price-tier jump saves money.
Read the worked answer
Two errors compound here.
- Unit-price percentage is not total-cost percentage. Shipping is a fixed $48 no matter the quantity, and the district pays tax on merchandise, so total cost is not proportional to unit price. A 5.3% merchandise saving translates into a smaller percentage of the total.
- Quantity is ignored. Saving per unit while buying more units can still raise spending. The relevant quantity is total dollars out the door, not dollars per calculator.
The correct test is a boundary comparison: total cost at the last quantity in the lower tier versus total cost at the first quantity in the higher tier.
- 99 units: 99 × $10.40 = $1,029.60; + $48.00; + $66.92 tax = $1,144.52
- 100 units: $1,097.03
Crossing the 100-unit boundary saves $47.49, so ordering 100 instead of 99 is smart. Similarly, 49 units cost $661.17 while 50 units cost $601.80 - a $59.37 saving for one more calculator. But jumping from 72 to 100 is not a boundary crossing; it simply buys 28 unwanted units at $8.98 each.
3. Using Vendor A, what is the largest number of calculators the team can buy on a $900 purchase order? Show the inequality and verify your answer.
Read the worked answer
The quantity must land in the 50–99 tier, so the per-unit price is $10.40 and total cost is 10.40n + 48 + 0.065(10.40n) = 11.076n + 48.
Set 11.076n + 48 ≤ 900:
- 11.076n ≤ 852
- n ≤ 76.92
Since n must be a whole number, the maximum is 76 calculators.
Checks:
- n = 76: 76 × $10.40 = $790.40; tax = $51.38; plus $48.00 = $889.78, within budget.
- n = 77: $800.80 + $52.05 + $48.00 = $900.85, over budget by $0.85.
Also confirm a cheaper-tier order cannot do better: in the 1–49 tier, 11.75n + 48 + 0.065(11.75n) = 12.5138n + 48 ≤ 900 gives n ≤ 68.1, so at most 68 units - well below 76. The 76-unit answer stands, and the $10.22 remaining cannot buy anything else.