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Difficulty: HardProfit, Loss, and Markup

A commercial retailer purchased 100 identical smartwatches for a total cost of $10,000\$10,000. Before any sales were made, 10 smartwatches were damaged and could not be sold. The retailer marked up the unit cost of each remaining smartwatch by PP percent to determine its list price. During a promotional period, 50 smartwatches were sold at the full list price. The remaining 40 undamaged smartwatches were subsequently sold during a clearance sale at a 20 percent discount off the list price. If the retailer earned a total net profit of 23 percent on the initial investment of $10,000\$10,000, what is the value of PP?

  1. A
    35
  2. B
    40
  3. C
    45
  4. 50Answer
  5. E
    60

Answer

50
The unit cost of each smartwatch is $10,000/100=$100\$10,000 / 100 = \$100. A markup of P%P\% sets the list price to L=100+PL = 100 + P. The retailer sells 50 units at full list price LL and 40 units at 0.80L0.80L, earning total revenue of 50L+32L=82L=82(100+P)=8,200+82P50L + 32L = 82L = 82(100 + P) = 8,200 + 82P. To earn a 23% profit on the entire $10,000\$10,000 investment, total revenue must equal $12,300\$12,300. Solving 8,200+82P=12,3008,200 + 82P = 12,300 yields 82P=4,10082P = 4,100, so P=50P = 50.

Step-by-Step Solution

1
Calculate the unit cost and total required revenue.
Unit cost c=$10,000100=$100c = \frac{\$10,000}{100} = \$100. Required revenue R=$10,000×(1+0.23)=$12,300R = \$10,000 \times (1 + 0.23) = \$12,300.
Net profit is 23% of the total initial investment of $10,000\$10,000.
2
Express the list price LL and clearance price in terms of PP.
List price L=100×(1+P100)=100+PL = 100 \times \left(1 + \frac{P}{100}\right) = 100 + P. Clearance price =0.80×L=0.80(100+P)=80+0.8P= 0.80 \times L = 0.80(100 + P) = 80 + 0.8P.
Markup P%P\% is calculated relative to the unit cost of $100\$100, and clearance discount is 20% off list price.
3
Set up the total revenue equation based on sold units.
Total Revenue =50L+40(0.80L)=50L+32L=82L=82(100+P)=8,200+82P= 50L + 40(0.80L) = 50L + 32L = 82L = 82(100 + P) = 8,200 + 82P.
10 units are unsellable ($0 revenue), 50 units generate full list price, and 40 units generate 80% of list price.
4
Solve for PP.
8,200+82P=12,300  ⟹  82P=4,100  ⟹  P=508,200 + 82P = 12,300 \implies 82P = 4,100 \implies P = 50.
Equating total generated revenue to the required revenue of $12,300\$12,300.

Key Concept

Profit, Loss, and Markup with Damaged Inventory and Tiered Discounts
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Topics

Profit, Loss, and Markup

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