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Difficulty: Very hardProfit, Loss, and Markup

A wholesale distributor purchased a shipment of 50 units of Model A laptops and 30 units of Model B laptops for a total cost of 4,000.Tosetthelistpriceforeachmodel,thedistributormarkeduptheunitcostofModelAby404,000. To set the list price for each model, the distributor marked up the unit cost of Model A by 40% and the unit cost of Model B by 50%. During a clearance promotion, Model A was sold at a 10% discount off its list price, and Model B was sold at a 20% discount off its list price. If the distributor earned a total profit of 995 on the entire shipment of 80 laptops, what was the original purchase cost of one Model A laptop?

  1. A
    $25
  2. B
    $50
  3. C
    $60.50
  4. $65Answer
  5. E
    $75

Answer

The original purchase cost of one Model A laptop was $65.
The original purchase cost of one Model A laptop is found by setting up a system of two linear equations: one for total cost (50x+30y=4,00050x + 30y = 4,000, or 5x+3y=4005x + 3y = 400) and one for total profit (50(0.26x)+30(0.20y)=99550(0.26x) + 30(0.20y) = 995, or 13x+6y=99513x + 6y = 995). Solving this system yields x=65x = 65, which accurately reflects the unit cost of Model A.

Step-by-Step Solution

1
Define variables and set up the total cost equation.
Let xx be the unit cost of Model A and yy be the unit cost of Model B. The equation is 50x+30y=4,00050x + 30y = 4,000, which simplifies to 5x+3y=4005x + 3y = 400.
The distributor purchased 50 units of Model A and 30 units of Model B for a total cost of $4,000.
2
Calculate the selling price and profit per unit for each model.
For Model A: List Price=1.40x\text{List Price} = 1.40x, Selling Price=1.40x×0.90=1.26x\text{Selling Price} = 1.40x \times 0.90 = 1.26x, so Profit per unit=1.26x−x=0.26x\text{Profit per unit} = 1.26x - x = 0.26x.
For Model B: List Price=1.50y\text{List Price} = 1.50y, Selling Price=1.50y×0.80=1.20y\text{Selling Price} = 1.50y \times 0.80 = 1.20y, so Profit per unit=1.20y−y=0.20y\text{Profit per unit} = 1.20y - y = 0.20y.
Selling prices are derived by applying the discount to the list price, and unit profit is selling price minus cost price.
3
Set up the total profit equation.
Total profit is 50(0.26x)+30(0.20y)=99550(0.26x) + 30(0.20y) = 995, which simplifies to 13x+6y=99513x + 6y = 995.
Summing the total profit from all 50 units of Model A and 30 units of Model B equals the given total profit of $995.
4
Solve the system of linear equations for xx.
Multiply 5x+3y=4005x + 3y = 400 by 2 to get 10x+6y=80010x + 6y = 800. Subtract this from 13x+6y=99513x + 6y = 995 to obtain (13x−10x)+(6y−6y)=995−800  ⟹  3x=195  ⟹  x=65(13x - 10x) + (6y - 6y) = 995 - 800 \implies 3x = 195 \implies x = 65.
Eliminating yy isolates the variable xx, representing the unit purchase cost of Model A.

Key Concept

Profit, Loss, and Markup with Successive Percentages
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Topics

Profit, Loss, and Markup

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