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Juana mailed at least one letter on each day last week. Was the total number of letters that Juana mailed on the 7 days greater than 27?
We need to determine whether the total number of letters Juana mailed over 7 days is greater than 27.
For this yes/no question to be sufficient, we need to definitively answer either:
We cannot have any uncertainty about which answer is correct.
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).
Let's check the extreme cases to see if we get a consistent answer:
Minimum scenario: If Juana mailed 1 letter each day
Maximum scenario: If Juana mailed 7 letters each day
Since we can get both YES and NO answers depending on the specific distribution, we cannot definitively answer the question.
Statement 1 alone is NOT sufficient.
This eliminates choices A and D.
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.
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:
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.
Statement 2 alone is sufficient.
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.