At a certain library, 1600 books were checked out by patrons yesterday. If each patron checked out at least 1...

Understanding the Question

We need to determine whether the number of patrons who checked out books yesterday was greater than 300.

Given Information

• Total books checked out: 1600
• Each patron checked out at least 1 book and at most 10 books
• This is a yes/no question

What We Need to Determine

For this yes/no question to be sufficient, we need to be able to definitively answer either "Yes, more than 300 patrons" or "No, 300 or fewer patrons."

Key Insight

Think of this as a "book distribution puzzle": We have 1600 books to distribute among patrons, where each patron takes 1-10 books. To minimize the number of patrons (and potentially get ≤ 300), we'd want patrons to take as many books as possible. To maximize the number of patrons (and likely get > 300), we'd want patrons to take as few books as possible.

Without any constraints:

• Minimum possible patrons: 160 (if all took 10 books each)
• Maximum possible patrons: 1600 (if all took 1 book each)

Since this range spans both above and below 300, we need additional information to answer the question.

Analyzing Statement 1

Statement 1 tells us: 80 patrons checked out 1 or 2 books each.

These 80 patrons are "light borrowers" - they're using up patron spots while taking very few books. This leaves potentially 1400+ books for other patrons.

Let's test whether we can construct scenarios on both sides of 300:

Scenario A - Trying to stay below 300 patrons:

• If the 80 patrons each took 2 books → 160 books used
• Remaining books: \(\mathrm{1600 - 160 = 1440}\) books
• If all other patrons took the maximum (10 books each), we'd need only 144 more patrons
• Total: \(\mathrm{80 + 144 = 224}\) patrons (< 300) ✓

Scenario B - Trying to exceed 300 patrons:

• If the 80 patrons each took 1 book → 80 books used
• Remaining books: \(\mathrm{1600 - 80 = 1520}\) books
• If all other patrons took the minimum (1 book each), we'd need 1520 more patrons
• Total: \(\mathrm{80 + 1520 = 1600}\) patrons (> 300) ✓

Since we can construct valid scenarios both above and below 300, Statement 1 alone is NOT sufficient.

[This eliminates choices A and D]

Analyzing Statement 2

Important: Now we forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: 150 patrons checked out 3 or 4 books each.

These 150 patrons are "moderate borrowers" - not as light as 1-2 books, but not maximizing either.

Again, let's test scenarios on both sides of 300:

Scenario A - Trying to stay below 300 patrons:

• If the 150 patrons each took 4 books → 600 books used
• Remaining books: \(\mathrm{1600 - 600 = 1000}\) books
• If all other patrons took the maximum (10 books each), we'd need only 100 more patrons
• Total: \(\mathrm{150 + 100 = 250}\) patrons (< 300) ✓

Scenario B - Trying to exceed 300 patrons:

• If the 150 patrons each took 3 books → 450 books used
• Remaining books: \(\mathrm{1600 - 450 = 1150}\) books
• If all other patrons took the minimum (1 book each), we'd need 1150 more patrons
• Total: \(\mathrm{150 + 1150 = 1300}\) patrons (> 300) ✓

We can still create scenarios on both sides of 300, so Statement 2 alone is NOT sufficient.

[This eliminates choices B and D]

Combining Statements

Now we use both pieces of information:

• 80 patrons checked out 1 or 2 books each
• 150 patrons checked out 3 or 4 books each

This gives us \(\mathrm{80 + 150 = 230}\) confirmed patrons.

Here's the critical insight: We already have 230 patrons. The question becomes: Can the remaining books be distributed among just 70 more patrons to stay at or below 300 total?

Let's find the minimum books these 230 patrons could have taken:

• \(\mathrm{80 \times 1 = 80}\) books (minimum from first group)
• \(\mathrm{150 \times 3 = 450}\) books (minimum from second group)
• Total minimum: \(\mathrm{80 + 450 = 530}\) books

This leaves at most \(\mathrm{1600 - 530 = 1070}\) books for other patrons.

The crucial question: Can we pack 1070 books into just 70 patrons?

Even at maximum efficiency (10 books per patron), \(\mathrm{70 \times 10 = 700}\) books.

But we have 1070 books remaining! We MUST have more than 70 additional patrons to account for these books.

Therefore, the total number of patrons MUST exceed \(\mathrm{230 + 70 = 300}\).

[STOP - Sufficient!] The combined statements allow us to answer with a definitive "Yes, more than 300 patrons."

[This eliminates choice E]

The Answer: C

Both statements together are sufficient to determine that more than 300 patrons checked out books, but neither statement alone is sufficient.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."