If the average (arithmetic mean) of four different positive integers is greater than 3 and less than 4, what is the range of the four numbers?

(1) One number is greater than 7

(2) The median of the four numbers is 2.5

We know: average * # terms = sum

If we have four terms, and the average is between 3 and 4, then the sum will be between 3*4 and 4*4, or between 12 and 16. So that leaves us with possible sums of 13, 14, and 15.

1) One number is greater than 7.

Let's build some sets

Set One: 1, 2, 3, 8; (Sum is 14, so it satisfies the initial condition.) Range = 8 - 1 = 7
Set Two: 1, 2, 3, 9; (Sum is 15, so it satisfies the initial condition.) Range = 9 - 1 = 8
Different Results, so Not Sufficient

2) Median = 2.5; Always check the sets you've already used for S1 to see if they'll work again for S2. Both will, because (2+3)/2 = 2.5.

Set One: 1, 2, 3, 8; Range = 8 - 1 = 7
Set Two: 1, 2, 3, 9; Range = 9 - 1 = 8
Again, not sufficient.

No reason we can't use those same two sets when testing together as well - clearly they'll satisfy both statements.
Set One: 1, 2, 3, 8; Range = 8 - 1 = 7
Set Two: 1, 2, 3, 9; Range = 9 - 1 = 8
Together: not sufficient.

Answer is E