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Yesterday Bookstore B sold twice as many softcover books as hardcover books. Was Bookstore B's revenue from the sale of softcover books yesterday greater than its revenue from the sale of hardcover books yesterday?
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Let's break down what we're being asked: Yesterday, Bookstore B sold twice as many softcover books as hardcover books. Was the revenue from softcover books greater than the revenue from hardcover books?
Given Information:
What We Need to Determine:
Is softcover revenue > hardcover revenue?
Since we sold 2× more softcover books, this becomes a classic quantity versus price trade-off question. The key insight: softcover revenue will be greater if the hardcover price isn't more than 2× the softcover price.
For this yes/no question, sufficiency means we can definitively answer either "yes" or "no" to whether softcover revenue exceeded hardcover revenue.
Statement 1: The average price of hardcover books was $10 more than the average price of softcover books.
If we denote softcover price as S and hardcover price as H, then:
\(\mathrm{H = S + 10}\)
Let's test whether different prices lead to different answers:
Scenario 1: Low softcover price
Scenario 2: High softcover price
Different softcover prices lead to different answers to our question. Since we can get both "yes" and "no" answers, Statement 1 is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choices A and D.
Important: Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: The average price of all books (softcover and hardcover) sold was greater than $14.
For every hardcover book sold, 2 softcover books were sold. So in any group of 3 books, we have 1 hardcover and 2 softcover. The average price of these 3 books exceeds $14.
Let's see if this constraint alone determines our answer:
Scenario 1: Expensive hardcover, cheap softcover
Scenario 2: Moderately priced books
The average price constraint allows for different revenue outcomes. Since we can get both "yes" and "no" answers, Statement 2 is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choice B.
From both statements together:
Let's denote softcover price as P. Then:
The average price constraint becomes:
\(\mathrm{(2P + (P + 10)) ÷ 3 > 14}\)
This simplifies to:
\(\mathrm{(3P + 10) ÷ 3 > 14}\)
\(\mathrm{3P + 10 > 42}\)
\(\mathrm{P > 32/3 ≈ 10.67}\)
So softcover books must cost more than $10.67, making hardcover books cost more than $20.67.
The Critical Insight:
At these prices:
The statements together give us a definitive "yes" answer.
[STOP - Sufficient!] The combined statements are sufficient.
Both statements together are sufficient to determine that softcover revenue was greater than hardcover revenue, but neither statement alone is sufficient.
Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."