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Question

Difficulty: MediumPercentages, Percent Change, and Interest

A bookstore received a shipment of NN identical books, each with an original full price of PP dollars. The sales over three consecutive months proceeded as follows:

- In May, the store sold 40%40\% of the total initial shipment at full price PP.
- In June, the store discounted the price of the unsold books by 25%25\% off the original full price and sold 50%50\% of the remaining unsold books.
- In July, the store discounted the June price of the remaining unsold books by an additional 20%20\% and sold all remaining books.

Which of the following statements regarding the bookstore's sales must be true? Select all such statements.

  1. The number of books sold in July accounted for exactly 30%30\% of the total initial shipment.Answer
  2. The total revenue generated from sales in June was greater than the total revenue generated from sales in July.Answer
  3. The total revenue from selling the entire shipment was equal to 80.5%80.5\% of the revenue that would have been earned if all books were sold at full price.Answer
  4. D
    The discounted price per book in July represented a 45%45\% decrease from the original full price PP.
  5. E
    The number of books sold in June was equal to the number of books sold in May.

Answer

The statements asserting that July sales comprised 30% of the initial shipment, that June revenue exceeded July revenue, and that overall revenue equaled 80.5% of full-price potential revenue are all correct.
The statement regarding July sales accounting for 30% of the initial shipment is correct because 50% of the 60% remaining after May leaves 30% unsold, all of which were sold in July. The statement comparing June and July revenue is correct because June revenue (0.225NP) is greater than July revenue (0.180NP). The statement regarding total revenue is correct because 0.400NP + 0.225NP + 0.180NP = 0.805NP, which is 80.5% of the total potential revenue.

Step-by-Step Solution

1
Calculate the quantity of books sold and remaining in each month as a fraction of total shipment N
May: Sold 0.40N0.40N, Remaining 0.60N0.60N. June: Sold 0.50×0.60N=0.30N0.50 \times 0.60N = 0.30N, Remaining 0.30N0.30N. July: Sold all remaining 0.30N0.30N.
Tracking the base shift after each monthly sale ensures accurate inventory proportions.
2
Calculate the selling price per book for each month relative to original price P
May price = 1.00P1.00P. June price = P×(1−0.25)=0.75PP \times (1 - 0.25) = 0.75P. July price = 0.75P×(1−0.20)=0.60P0.75P \times (1 - 0.20) = 0.60P.
Successive percent discounts compound on the previous month's price, not the original price.
3
Calculate revenue for each month and total revenue
May: 0.40N×1.00P=0.400NP0.40N \times 1.00P = 0.400NP. June: 0.30N×0.75P=0.225NP0.30N \times 0.75P = 0.225NP. July: 0.30N×0.60P=0.180NP0.30N \times 0.60P = 0.180NP. Total revenue = (0.400+0.225+0.180)NP=0.805NP(0.400 + 0.225 + 0.180)NP = 0.805NP.
Revenue equals quantity sold multiplied by price per unit for each individual phase.
4
Evaluate each given statement against calculated values
July quantity = 30%30\% of N (Statement 1 is True). June revenue 0.225NP>0.180NP0.225NP > 0.180NP July revenue (Statement 2 is True). Total revenue 0.805NP=80.5%0.805NP = 80.5\% of full potential 1.000NP1.000NP (Statement 3 is True). July price discount is 1−0.60=40%≠45%1 - 0.60 = 40\% \neq 45\% (Statement 4 is False). June sales (30%30\%) ≠\neq May sales (40%40\%) (Statement 5 is False).
Determines which set of statements must be selected.

Key Concept

Successive Percent Change and Base Shift in Multi-Step Scenarios
Estimated Time:1m 45s
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Topics

Percentages, Percent Change, and Interest

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