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Difficulty: HardPercentage Change, Markups, and Discounts

A specialized automotive repair shop orders a set of custom performance tires. The shop applies a 50%50\% markup to the wholesale cost to establish the regular customer price. During a seasonal promotion, the shop offers a 30%30\% discount off the regular customer price. If a customer pays $420.00\$420.00 for the set of tires during this promotion, what was the shop's original wholesale cost for the tires?

  1. A
    \$300.00
  2. B
    \$350.00
  3. \$400.00Answer
  4. D
    \$1,200.00

Answer

The original wholesale cost for the tires was $400.00\$400.00.
To find the original wholesale cost, we must reverse the successive percentage operations. A 50%50\% markup means the regular price is 150%150\% of the wholesale cost (a 1.501.50 multiplier). A 30%30\% discount means the customer pays 70%70\% of the regular price (a 0.700.70 multiplier). Multiplying these factors gives a net multiplier of 1.50×0.70=1.051.50 \times 0.70 = 1.05. This means the final price of $420.00\$420.00 is exactly 105%105\% of the wholesale cost. Dividing $420.00\$420.00 by 1.051.05 yields $400.00\$400.00.

Step-by-Step Solution

1
Define the variable and set up the multipliers for successive percentage changes.
Let WW be the wholesale cost. The markup multiplier is (1+0.50)=1.50(1 + 0.50) = 1.50. The discount multiplier is (1−0.30)=0.70(1 - 0.30) = 0.70.
A 50%50\% markup increases the cost to 150%150\%, and a 30%30\% discount reduces the marked-up price to 70%70\% of its value.
2
Create an equation relating the wholesale cost to the final customer price.
W×1.50×0.70=420W \times 1.50 \times 0.70 = 420
Applying the successive multipliers to the wholesale cost must equal the final price paid by the customer.
3
Simplify the equation and solve for the wholesale cost.
1.05W=420  ⟹  W=4201.05=4001.05W = 420 \implies W = \frac{420}{1.05} = 400
Multiplying 1.50×0.701.50 \times 0.70 reveals that the final price is exactly 105%105\% of the wholesale cost. Dividing isolating WW provides the initial value.

Key Concept

Solving multi-step percentage problems by tracking successive multipliers backwards from a final price.
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Topics

Percentage Change, Markups, and Discounts

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