x, 3, 1, 12, 8
If x is an integer, is the median of the 5 numbers greater than the average (arithmetic mean) of the 5 numbers?

(1) x>6
(2) x is greater than the median of the 5 numbers

(1)
x must be greater than 1 or 3. if it is 8 or less, then it will be the median.
therefore:
if x = 7, then the list is 1, 3, 7, 8, 12, and median = 7. average = 31/5 = 6.2; answer to question is YES
if x = 8, then the list is 1, 3, 8, 8, 12, and median = 8. average = 32/5 = 6.4; answer to question is YES
if x = REALLY BIG, then the list is 1, 3, 8, 12, x, and median = 8. average is REALLY BIG, so the answer to the question is NO.
insufficient

(2)
this means that x is at least 9, from the observations above.
the list is either 1, 3, 8, x, 12, or 1, 3, 8, 12, x, depending on the size of x.
regardless of which one is the proper ordering, the expressions for the median and the mean are the same:
median = 8
mean (average) = (1 + 3 + 8 + 12 + x)/5 = (24 + x)/5
since x is at least 9, the mean is at least (24 + 9)/5 = 33/5 = 6.6.
this is inconclusive, because the mean could be 6.6 (if x = 9) or could be in the millions (if x is huge).
insufficient

(together)
this doesn't help, because statement (2) by itself already implies that x > 6
so, still insufficient

answer = e

For statement (1): I don't understand when you say:

"x must be greater than 1 or 3. if it is 8 or less, then it will be the median."

I understand the test cases but how did you reach the above conclusion?

For (2)I don't get it when you say:

"this means that x is at least 9, from the observations above.
the list is either 1, 3, 8, x, 12, or 1, 3, 8, 12, x, depending on the size of x."
regardless of which one is the proper ordering, the expressions for the median and the mean are the same:

Why can't be x be between 1 and 3?

Can you please explain?

if x is between 6 and 8, then the ordered list will be
1, 3, x, 8, 12
thereby making x the median.




if x is between 1 and 3, then the ordered list is
1, x, 3, 8, 12
and so x is smaller than the median, which is 3.

It takes me 1 hour to find out the best and simplest method for this.

IF THE DISTANCE BETWEEN MEDIAN AND THE NUMBERS TO THE RIGHT IS FURTHER THAN THE DISTANCE BETWEEN MEDIAN AND THE NUMBER TO THE LEFT, MEDIAN <AVERAGE

example

1,3,5,7, 9

mean=average,

1,3, 4, 7 , 9
the distance between median and the number to the right is 7-4=3 , 9-4=5, the distance is 3+5=8
the distance between median and the numbers to the left is
4-1=3, 4-3=1 , distance =4
this case, median <average because median is closer the the left numbers than the right numbers.

i hope that you didn't actually sit there for an hour working on a single gmat problem.

even if you eventually *did* find this approach -- you are building very, very bad test-taking habits. the most important skill on this entire test is the ability to QUIT after a couple of minutes if you are stuck.

also -- there is no such thing as the "best" method!
any method that WORKS is just as good as any other method that WORKS!

you should not care at all about what is "best" or "simplest". you should only be concerned with finding as many ways as possible to solve the problems.