Known values, in ascending order:AAPL wrote:Manhattan Prep
If x and y are unknown positive integers, is the mean of the set {6, 7, 1, 5, x, y} greater than the median of the set?
1) x + y = 7.
2) x - y = 3.
1, 5, 6, 7
Statement 1: x+y = 7
Mean = sum/quantity = (1+5+6+7+x+y)/6 = (1+5+6+7+7)/6 = 26/6 = 13/3 = 4.33.
Text EXTREMES.
Case 1: x and y are far from each other
If x=1 and y=6, we get:
1, 1. 5, 6, 6, 7
Median = (5+6)/2 = 11/2 = 5.5
Case 2: x and y are near each other
If x=3 and y=4, we get:
1, 3, 4, 5, 6, 7
Median = (4+5)/2 = 9/2 = 4.5
In both cases, the mean is LESS than the median, so the answer to the question stem is NO.
SUFFICIENT.
Statement 2: x-y = 3
Again, test EXTREMES.
Case 1: x and y are as small as possible.
If x=4 and y=1, we get:
1, 1, 4, 5, 6, 7
Mean = (1+1+4+5+6+7)/6 = 24/6 = 4
Median = (4+5)/2 = 9/2 = 4.5.
In this case, the mean is LESS than the median, so the answer to the question stem is NO.
Case 2: x and y are large
If x=100 and y=97, we get:
1, 5, 6, 7, 97, 100
Mean = (1+5+6+7+97+100)/6 = 216/6 = 108/3 = 36.
Median = (6+7)/2 = 13/2 = 6.5.
In this case, the mean is GREATER than the median, so the answer to the question stem is YES.
Since the answer is NO in Case 1 but YES in Case 2, INSUFFICIENT.
The correct answer is A.
