AAPL wrote:Manhattan Prep

Sarah is in a room with 6 other children. If the other children are 2, 4, 5, 8, 10, and 13 years old, is Sarah 7 years old?

1) The age of the fourth oldest child is equal to the average (arithmetic mean) of the seven children's ages.
2) Sarah is not the oldest child in the room.

Let S = Sara's age.
Sum of the ages = 2+4+5+8+10+13+S = 42+S.

Statement 1:
Since the average age is equal to an INTEGER -- the fourth largest age -- we get:
Sum of the ages = (number of ages)(average age) = (7)(INTEGER) = multiple of 7.

The blue expression and the red expression each represent the sum of the ages and thus are EQUAL:
42+S = (multiple of 7)
S = (multiple of 7) - 42
S = (multiple of 7) - (multiple of 7)
S = multiple of 7.

Case 1: S=7, with the result that the ages are 2, 4, 5, S=7, 8, 10, 13
Since the average age must be equal to the fourth largest -- 7 -- we get:
Sum of the ages = (number of ages)(average age) = 7*7 = 49.
Since the sum of the ages = 42+S, we get:
42+S = 59
S=7.
This works.

Case 2: S>7, with the result that the ages are 2, 4, 5, 8, 10, 13, S≥14
Since the average age must be equal to the fourth largest -- 8 -- we get:
Sum of the ages = (number of ages)(average age) = 7*8 = 56.
Since the sum of the ages = 42+S, we get:
42+S = 56
S=14.
This also works.

Thus, Statement 1 allows for two cases:
Case 1: S=7
Case 2: S=14
Since the answer to the question stem is YES in Case 1 but NO in Case 2, INSUFFICIENT.

Statement 2:
Since S cannot be the oldest, S<13.
If S=7, then the answer to the question stem is YES.
If S=1, then the answer to the question stem is NO.
INSUFFICIENT.

Statements combined:
Of the two options for S in Statement 1, only S=7 is such that S<13.
Thus, S=7, with the result that the answer to the question stem is YES.
SUFFICIENT.

The correct answer is C.