IMO A

Range is either 7-5, if x<=7 or x-5, if x>7
Median is
(6+7)/2 = 6.5 if x>=7
(6+x)/2 if 6<x<7
6 if x<=6

From 2:
For median to be 6: x<=6
therefore range could be =2,<2 or >2 depending upon value of x
NOT SUFFICIENT

From 1:
median is greater than mean only if x<5
therefore range is 7-x, which is greater than 2
SUFFICIENT

IMO B -


what is the OA?

iamseer wrote:IMO A

Range is either 7-5, if x<=7 or x-5, if x>7
Median is
(6+7)/2 = 6.5 if x>=7
(6+x)/2 if 6<x<7
6 if x<=6

From 2:
For median to be 6: x<=6
therefore range could be =2,<2 or >2 depending upon value of x
NOT SUFFICIENT

From 1:
median is greater than mean only if x<5
therefore range is 7-x, which is greater than 2
SUFFICIENT

One more for A
Median can be > mean only when X<6. In that case range is NOT 2. ST 1 gives a consistent YES.Pick A

How many minutes did you guys take in solving this one. I would consider this one bit on the tougher side.

How can it be A?
if x>7, let's say, 7.5
Mean = (31+7.5)/6=38.5/6=6.4
whereas Median=6.5 which is greater than 6.4

There are six numbers is 5, 6, 6, 7, 7, x. Is the range greater than 2?
1. The median of the six numbers is greater than the mean.
2. The median is 6

Sum of the 6 numbers = x+5+6+6+7+7 = x+31.
In each case, test whether it's possible for the range of the 6 numbers to be EQUAL TO 2.

Statement 1: The median of the six numbers is greater than the mean.
Test one case that also satisfies Statement 2.
Case 1: median = 6
If x=5, then the range is equal to 2 and the set looks as follows: x=5, 5, 6, 6, 7, 7.
Average = (5+31)/6 = 6.
Not viable: the average must be LESS than the median.
To decrease the average, x must be LESS than 5.
Thus, the set must look as follows:
x<5, 5, 6, 6, 7, 7.
Since the smallest value is less than 5 and the greatest value is 7, the range of the 6 numbers is GREATER THAN 2.

Test one case that does NOT also satisfy Statement 2.
Case 2: median = 6.5
If x=7, then the range is equal to 2 and the set looks as follows: 5, 6, 6, 7, 7, x=7.
Average = (7+31)/6 = 38/6 = 6.33.
This works, since the average is less than the median.
Since the smallest value is 5 and the greatest value is 7, the range of the 6 numbers is EQUAL TO 2.

Since the range is greater than 2 in Case 1 but equal to 2 in Case 2, INSUFFICIENT.

Statement 2: The median is 6.
Case 1 also satisfies statement 2.
In Case 1, the range > 2.

Case 3:
If x=6, then the range is EQUAL TO 2 and the set looks as follows: 5, x=6, 6, 6, 7, 7.

Since the range is greater than 2 in Case 1 but equal to 2 in Case 3, INSUFFICIENT.

Statements combined:
Both statements are satisfied only by Case 1.
Thus, the range > 2.
SUFFICIENT.

The correct answer is C.