Question

Are all of the numbers in a certain list of 15 numbers equal?
(1) The sum of all the numbers in the list is 60
(2) The sum of any 3 numbers in the list is 12.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is B Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

1. If there are 15 occurrences of the number 4 in the list, then the sum of the numbers in the list is 60 and all the numbers in the list are equal. If there are 13 occurrences of the number 4 in the list, 1 occurrence of the number 3 in the list, and 1 occurrence of the number 5 in the list, then the sum of the numbers in the list is 60 and not all the numbers in the list are equal; NOT sufficient.
2. Given that the sum of any 3 numbers in the list is 12, arrange the numbers in the list in numerical order, from least to greatest: .

• If a1 < 4, then a1 + a2 + a3 < 4 + a2 + a3. Therefore, from (2), 12 < 4 + a2 + a3, and so at least one of the values a2 and a3 must be greater than 4. Because it follows that a3 > 4. Since the numbers are arranged from least to greatest, it follows that a4 > 4 and a5 > 4. But then a3 + a4 + a5 > 4 + 4 + 4 = 12, contrary to (2), and so a1 < 4 is not true. Therefore, . Since a1 is the least of the 15 numbers, for n = 1, 2, 3, 4... 15.
  
• If a15 > 4, then a13 + a14 + a15 > a13 + a14 + 4. Therefore, from (2), 12 > a13 +a14 + 4 and so at least one of the values a13 and a14 must be less than 4. Because it follows that a13 < 4. Since the numbers are arranged from least to greatest, it follows that a11 < 4 and a12 < 4. But then a11 + a12 + a13 < 4 + 4 + 4 = 12, contrary to (2). Therefore, . Since a15 is the greatest of the 15 numbers, for n = 1,2,3,..15. It has been shown that, for , 2, 3, ..., 15, each of and is true. Therefore, an = 4 for n = 1, 2, 3, ..., 15; SUFFICIENT.
• The correct answer is B; statement 2 alone is sufficient.