Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed...

Understanding the Question

We need to determine whether the total number of letters Juana mailed over 7 days is greater than 27.

Given Information

• Juana mailed at least one letter on each day last week (7 days)
• This means each day had \(\geq 1\) letter

What We Need to Determine

For this yes/no question to be sufficient, we need to definitively answer either:

• YES, the total is greater than 27, or
• NO, the total is not greater than 27

We cannot have any uncertainty about which answer is correct.

Analyzing Statement 1

Statement 1: Juana mailed fewer than 8 letters on each of the 7 days.

This means each day had between 1 and 7 letters (inclusive).

Testing the Range of Possibilities

Let's check the extreme cases to see if we get a consistent answer:

Minimum scenario: If Juana mailed 1 letter each day

• Total = \(1 \times 7 = 7\) letters
• Is \(7 > 27\)? NO

Maximum scenario: If Juana mailed 7 letters each day

• Total = \(7 \times 7 = 49\) letters
• Is \(49 > 27\)? YES

Since we can get both YES and NO answers depending on the specific distribution, we cannot definitively answer the question.

Conclusion

Statement 1 alone is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

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

Statement 2: Juana mailed a different number of letters on any two of the 7 days.

This means all seven daily amounts must be distinct positive integers.

The Key Insight

If each day must have a different number of letters, and we know each day has at least 1 letter, what's the smallest possible total?

The seven distinct positive integers must be at minimum:

• Day 1: 1 letter
• Day 2: 2 letters
• Day 3: 3 letters
• Day 4: 4 letters
• Day 5: 5 letters
• Day 6: 6 letters
• Day 7: 7 letters

Minimum total = \(1 + 2 + 3 + 4 + 5 + 6 + 7 = 28\) letters

Since the minimum possible total is 28, and \(28 > 27\), the answer must always be YES.

[STOP - Sufficient!] We can definitively answer YES to the question.

Conclusion

Statement 2 alone is sufficient.

The Answer: B

Statement 2 alone gives us a definitive YES answer because requiring all values to be different creates a minimum total of 28, which exceeds 27.

Answer Choice B: Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.