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A total of 100 customers purchased books at a certain bookstore last week. If these customers purchased a total of 200 books, how many of the customers purchased only 1 book each?
We need to find the exact number of customers who bought only 1 book.
Given information:
With an average of exactly 2 books per customer, any customer who buys fewer than 2 books (just 1 book) must be balanced by customers who buy more than 2 books (3 or more books). This balancing act is crucial to maintaining our average.
What makes a statement sufficient? We need to determine a single, specific number of customers who bought exactly 1 book.
Statement 1: None of the customers purchased more than 3 books.
This limits each customer to buying 1, 2, or 3 books. Let's explore what distributions are possible while maintaining our 200-book total.
Testing scenarios:
Scenario 1: All 100 customers buy exactly 2 books each
Scenario 2: Half buy 1 book, half buy 3 books
Since we found two different valid answers (0 and 50), we cannot determine a unique number.
Statement 1 alone is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choices A and D.
Now we analyze Statement 2 independently, forgetting Statement 1.
Statement 2: 20 of the customers purchased only 2 books each.
Let's work through what this tells us:
Without knowing the maximum books any customer can buy, these 80 customers could distribute their purchases in many ways:
Possibility 1: All 80 buy exactly 2 books each
Possibility 2: 40 buy 1 book, 40 buy 3 books
Possibility 3: 20 buy 1 book, 20 buy 5 books
Multiple valid answers exist, so Statement 2 alone is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choice B.
Now let's use BOTH pieces of information:
Setting up the situation:
The balancing equation:
If x customers buy 1 book each, then (80-x) customers must buy 3 books each.
Total books from the remaining 80 customers:
Solving: \(240 - 2x = 160\)
Therefore: \(2x = 80\)
So: \(x = 40\)
Unique answer: Exactly 40 customers bought 1 book.
[STOP - Sufficient!] The statements together are sufficient.
Both statements together provide exactly the information needed to determine that 40 customers bought only 1 book, but neither statement alone is sufficient.
Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."