Yesterday Bookstore B sold twice as many softcover books as hardcover books. Was Bookstore B's revenue from the sale of...

markdown

Understanding the Question

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:

• Softcover books sold = 2 × Hardcover books sold
• Revenue = Number of books × Average price per book

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.

Analyzing Statement 1

Statement 1: The average price of hardcover books was $10 more than the average price of softcover books.

What Statement 1 Tells Us

If we denote softcover price as S and hardcover price as H, then:
\(\mathrm{H = S + 10}\)

Testing Different Scenarios

Let's test whether different prices lead to different answers:

Scenario 1: Low softcover price

• Softcover price: $5
• Hardcover price: $15 (which is 3× the softcover price)
• Softcover revenue = \(\mathrm{2 \times \$5 = \$10}\)
• Hardcover revenue = \(\mathrm{1 \times \$15 = \$15}\)
• Answer: NO - Hardcover revenue is greater

Scenario 2: High softcover price

• Softcover price: $20
• Hardcover price: $30 (which is only 1.5× the softcover price)
• Softcover revenue = \(\mathrm{2 \times \$20 = \$40}\)
• Hardcover revenue = \(\mathrm{1 \times \$30 = \$30}\)
• Answer: YES - Softcover revenue is greater

Conclusion

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.

Analyzing Statement 2

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.

What Statement 2 Provides

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.

Testing Different Scenarios

Let's see if this constraint alone determines our answer:

Scenario 1: Expensive hardcover, cheap softcover

• Hardcover: $30, Softcover: $10
• Average of 3 books: \(\mathrm{(1 \times \$30 + 2 \times \$10) ÷ 3 = \$50 ÷ 3 = \$16.67 > \$14}\) ✓
• Softcover revenue = \(\mathrm{2 \times \$10 = \$20}\)
• Hardcover revenue = \(\mathrm{1 \times \$30 = \$30}\)
• Answer: NO - Hardcover revenue is greater

Scenario 2: Moderately priced books

• Hardcover: $15, Softcover: $14
• Average of 3 books: \(\mathrm{(1 \times \$15 + 2 \times \$14) ÷ 3 = \$43 ÷ 3 = \$14.33 > \$14}\) ✓
• Softcover revenue = \(\mathrm{2 \times \$14 = \$28}\)
• Hardcover revenue = \(\mathrm{1 \times \$15 = \$15}\)
• Answer: YES - Softcover revenue is greater

Conclusion

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.

Combining Statements

Combined Information

From both statements together:

• Hardcover price = Softcover price + $10
• The average price of 1 hardcover + 2 softcover books > $14

Why Together They Are Sufficient

Let's denote softcover price as P. Then:

• Hardcover price = P + 10
• For 3 books total: 2 softcover + 1 hardcover

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:

• Hardcover price ÷ Softcover price < 2 (since \(\mathrm{\$20.67 ÷ \$10.67 < 2}\))
• Since we sold 2× more softcover books, and the price ratio is less than 2:1, softcover revenue must be greater

The statements together give us a definitive "yes" answer.

[STOP - Sufficient!] The combined statements are sufficient.

The Answer: C

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."