A total of 100 customers purchased books at a certain bookstore last week. If these customers purchased a total of...

Understanding the Question

We need to find the exact number of customers who bought only 1 book.

Given information:

• Total customers: 100
• Total books purchased: 200
• Average books per customer: \(200 ÷ 100 = 2\) books

Key Insight

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.

Analyzing Statement 1

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

• Total books: \(100 × 2 = 200\) ✓
• Customers who bought 1 book: 0

Scenario 2: Half buy 1 book, half buy 3 books

• \(50\) customers × 1 book = 50 books
• \(50\) customers × 3 books = 150 books
• Total: \(50 + 150 = 200\) ✓
• Customers who bought 1 book: 50

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.

Analyzing Statement 2

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:

• 20 customers bought 2 books = 40 books accounted for
• Remaining: 80 customers who bought 160 books total
• These 80 customers still average 2 books each (\(160 ÷ 80 = 2\))

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

• Customers who bought 1 book: 0

Possibility 2: 40 buy 1 book, 40 buy 3 books

• \(40 × 1 + 40 × 3 = 40 + 120 = 160\) ✓
• Customers who bought 1 book: 40

Possibility 3: 20 buy 1 book, 20 buy 5 books

• \(20 × 1 + 20 × 5 = 20 + 100 = 120 ≠ 160\)
• Wait, we need 40 more books. So 20 buy 4 books:
• \(20 × 1 + 20 × 4 + 20 × 2 = 20 + 80 + 40 = 140 ≠ 160\)
• Actually: 20 buy 1 book, 60 buy \((160-20)/60 = 140/60 = 2.33\) books
• This doesn't work with whole books. Let me recalculate:
• If 20 buy 1 book (20 books), we need 140 more books from 60 customers
• \(20 × 1 + 20 × 5 = 20 + 100 = 120\), leaving 40 books for 40 customers
• So: 20 buy 1 book, 20 buy 5 books, 40 buy 1 book = 60 buy 1 book total
• Customers who bought 1 book: 60

Multiple valid answers exist, so Statement 2 alone is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Both Statements

Now let's use BOTH pieces of information:

• From Statement 1: Maximum 3 books per customer
• From Statement 2: 20 customers bought exactly 2 books

Setting up the situation:

• 20 customers bought 2 books = 40 books
• 80 remaining customers must buy 160 books total
• These 80 can only buy 1 or 3 books each (can't buy 2 since Statement 2 says only 20 bought 2, and can't buy more than 3)

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:

• Books from 1-book buyers: \(x × 1 = x\)
• Books from 3-book buyers: \((80-x) × 3 = 240 - 3x\)
• Total must equal 160: \(x + 240 - 3x = 160\)

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.

The Answer: C

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."