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Difficulty: Very hardProfit, Loss, and Markup

A merchant purchased a shipment of 200 identical items for a total cost of 8,000.Themerchantmarkedupthecostpriceofeachitemby8,000. The merchant marked up the cost price of each item by P percenttoestablishitslistprice.Duringthefirstmonthofsales,60percentoftheitemsweresoldatthelistprice.Duringthesecondmonth,50percentoftheremainingitemsweresoldata20percentdiscountoffthelistprice.Theremainingitemswerethensoldduringaclearancesaleata50percentdiscountoffthelistprice.Ifthemerchantmadeanoverallnetprofitof29percentonthetotalpurchasecostoftheshipment,whatisthevalueof percent to establish its list price. During the first month of sales, 60 percent of the items were sold at the list price. During the second month, 50 percent of the remaining items were sold at a 20 percent discount off the list price. The remaining items were then sold during a clearance sale at a 50 percent discount off the list price. If the merchant made an overall net profit of 29 percent on the total purchase cost of the shipment, what is the value of P$?

Answer: 50 %

Answer

The value of PP is 50.
The unit cost of each item is $8,000200=$40\frac{\$8,000}{200} = \$40. To make an overall net profit of 29%, total revenue must equal $8,000×1.29=$10,320\$8,000 \times 1.29 = \$10,320. Sales are divided into three tiers: 120 items sold at list price LL, 40 items sold at 0.80L0.80L, and 40 items sold at 0.50L0.50L. Summing the revenue yields 120L+32L+20L=172L120L + 32L + 20L = 172L. Solving 172L=10,320172L = 10,320 gives L=60L = 60. The percentage markup over cost price is 60−4040×100=50%\frac{60 - 40}{40} \times 100 = 50\%.

Step-by-Step Solution

1
Calculate the unit cost price and required total revenue.
Unit cost C=$8,000200=$40C = \frac{\$8,000}{200} = \$40. Total target revenue R=$8,000×(1+0.29)=$10,320R = \$8,000 \times (1 + 0.29) = \$10,320.
Determining the total required revenue sets the benchmark to solve for the list price.
2
Determine the quantity of items sold at each price tier.
Month 1 sales: 0.60×200=1200.60 \times 200 = 120 items at list price LL.
Month 2 sales: 0.50×(200−120)=400.50 \times (200 - 120) = 40 items at discounted price 0.80L0.80L.
Clearance sales: 200−120−40=40200 - 120 - 40 = 40 items at clearance price 0.50L0.50L.
Categorizing quantities by price tier enables building the total revenue expression.
3
Express total revenue in terms of list price LL and solve for LL.
Total Revenue = 120L+40(0.80L)+40(0.50L)=120L+32L+20L=172L120L + 40(0.80L) + 40(0.50L) = 120L + 32L + 20L = 172L.
Setting 172L=10,320  ⟹  L=10,320172=60172L = 10,320 \implies L = \frac{10,320}{172} = 60.
Summing revenue from all tiers and equating to target revenue yields the list price per item.
4
Calculate the markup percentage PP.
P=L−CC×100=60−4040×100=50%P = \frac{L - C}{C} \times 100 = \frac{60 - 40}{40} \times 100 = 50\%.
Percentage markup is measured relative to the initial cost price per unit.

Key Concept

Multi-tiered inventory pricing and percentage markup calculation
Estimated Time:2m 30s
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Topics

Profit, Loss, and Markup

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