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Difficulty: Very hardProfit, Loss, and Markup

A manufacturer produces custom electric scooters at a manufacturing cost of 250perunit.Toestablishthelistprice,themanufacturermarksupthecostpriceby60250 per unit. To establish the list price, the manufacturer marks up the cost price by 60%. During a promotional campaign, 70% of the manufactured scooters are sold at a 15% discount off the list price, and the remaining 30% are sold at a clearance discount of 35% off the list price. If the manufacturer incurs a fixed promotional overhead expense of 14,400, and the overall net profit across all scooters sold (after deducting the fixed overhead expense) is equal to 12% of the total manufacturing cost of all the scooters, how many scooters were manufactured and sold in total?

  1. A
    240
  2. 400Answer
  3. C
    720
  4. D
    800
  5. E
    3,600

Answer

400 scooters
To find the total number of scooters NN, first determine the list price: 250×1.60=$400250 \times 1.60 = \$400. Next, calculate the selling prices for the two tiers: 400×0.85=$340400 \times 0.85 = \$340 for 70% of units, and 400×0.65=$260400 \times 0.65 = \$260 for 30% of units. The weighted average revenue per scooter is 0.70(340)+0.30(260)=$3160.70(340) + 0.30(260) = \$316. The profit before fixed overhead per scooter is 316−250=$66316 - 250 = \$66, yielding a total net profit of 66N−14,40066N - 14,400. Setting this equal to the target net profit of 12% of total manufacturing costs (0.12×250N=30N0.12 \times 250N = 30N) gives 66N−14,400=30N66N - 14,400 = 30N, which simplifies to 36N=14,40036N = 14,400, or N=400N = 400.

Step-by-Step Solution

1
Calculate the list price of a single scooter.
List price = 250×(1+0.60)=$400250 \times (1 + 0.60) = \$400.
The list price is established by marking up the cost price ($250) by 60%.
2
Calculate the discounted selling prices for both sales categories.
Promotional price (70% of inventory) = 400×(1−0.15)=$340400 \times (1 - 0.15) = \$340. Clearance price (30% of inventory) = 400×(1−0.35)=$260400 \times (1 - 0.35) = \$260.
Discounts are applied directly to the list price.
3
Determine the weighted average selling price per unit and profit per unit before overhead.
Weighted average selling price = 0.70(340)+0.30(260)=238+78=$3160.70(340) + 0.30(260) = 238 + 78 = \$316. Gross profit per unit = 316−250=$66316 - 250 = \$66.
Combining the two sales segments gives the effective revenue generated per unit sold.
4
Set up and solve the net profit equation for total units NN.
Total net profit = 66N−14,40066N - 14,400. Target profit = 0.12×250N=30N0.12 \times 250N = 30N. Equating gives 66N−14,400=30N  ⟹  36N=14,400  ⟹  N=40066N - 14,400 = 30N \implies 36N = 14,400 \implies N = 400.
Net profit after deducting fixed overhead must equal 12% of total manufacturing cost.

Key Concept

Weighted average selling price, markup vs. discount percentage bases, and multi-step net profit equation modeling.

Alternative Method

Work per unit on a percentage basis: List price is 160% of cost. Revenue from 70% of inventory is 0.70×1.60×0.85=0.9520.70 \times 1.60 \times 0.85 = 0.952 of cost. Revenue from 30% of inventory is 0.30×1.60×0.65=0.3120.30 \times 1.60 \times 0.65 = 0.312 of cost. Total revenue is (0.952+0.312)=1.264(0.952 + 0.312) = 1.264 of cost. Gross profit before overhead is 1.264−1.000=0.2641.264 - 1.000 = 0.264 of cost. Net profit is 0.264×Total Cost−14,4000.264 \times \text{Total Cost} - 14,400. Setting this equal to 0.120×Total Cost0.120 \times \text{Total Cost} gives 0.144×Total Cost=14,4000.144 \times \text{Total Cost} = 14,400, so Total Cost=$100,000\text{Total Cost} = \$100,000. Since total cost is 250×N=100,000250 \times N = 100,000, we get N=400N = 400.
Estimated Time:2m 0s
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Topics

Profit, Loss, and Markup

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