Data Sufficiency

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.

Target question: Are all 15 numbers equal?

Statement 1: The sum of all the numbers in the list is 60.
There are several possible scenarios that satisfy this statement. Here are two.
Case a: numbers are: {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, in which case all of the numbers are equal
Case b: numbers are: {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 7}, in which case all of the numbers are not equal
Statement 1 is NOT SUFFICIENT

Statement 2: The sum of any 3 numbers in the list is 12.
This is a very powerful statement, because it tells us that all of the numbers in the set are equal.
Let's let a,b,c and d be four of the 15 numbers in the set.
We know that a + b + c = 12
Notice that if I replace ANY of these three values (a,b or c) with d, the sum must still be 12.
This tells us that a, b and c must all equal d.
I can use a similar approach to show that e, f and g must also equal d.
In fact, I can show that ALL of the numbers in the set must equal d, which means all of the numbers in the set must be equal.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent

dyang.sw wrote: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.

We need to determine whether all of the numbers in the certain list of 15 numbers are equal.

Statement One Alone:

The sum of all the numbers in the list is 60.

Using the information in statement one, we see that all the numbers could be equal or could not be equal. In one scenario, all the numbers could equal 4, and in another scenario 13 of 15 the numbers could equal 4 and the last two numbers could be 6 and 2. In either case, the sum would be 60. Thus, statement one alone is not sufficient to answer the question.
Statement Two Alone:

The sum of any 3 numbers in the list is 12.

Let's create variables for all 15 numbers in our list.

A, B, C, D, E, F, G, H, I, J, K, L, M, N, O

Using the information in statement two, we see that, regardless of which 3 numbers we select, they MUST sum to 12.

For example, A + B + C = 12 or A + B + N = 12 or A + C + N = 12.

We see that the only way this is possible is if all the numbers are the same value. Statement two alone is sufficient to answer the question.

Answer: B