here's another approach. as with other "pick integers" problems, i like to think about extreme cases.
STATEMENT 1:
* One extreme case would be if removing the biggest number in the set gives an average of 15.
If the biggest number "n" is removed, then you have 1, 2, ..., n - 1 (with no missing values). If the average of these is 15, then 15 is the middle number, so the set after "n" is removed is 1, 2, ..., 15, ..., 28, 29.
This means that the set started with the numbers 1 through 30. This is the biggest possible starting set (since we took the biggest number away from it to make the average 15): n = 30.
* The other extreme would be if removing the smallest number in the set gives an average of 15.
The smallest number is 1, so this removal would leave the set 2, 3, ..., n. If the average of these is 15, then 15 is again the middle number, meaning that this set (after removal of the "1") is 2, 3, ..., 15, ..., 27, 28.
This means that the set started with the numbers 1 through 29. This is the smallest possible starting set (since we took away the smallest number this time): n = 29.
So n could be either 29 or 30; not sufficient.
STATEMENT 2:
Basically, do the same things shown above.
* In the first case, you'll discover that the set must have started out as 1 through 32. That's the biggest possible "n", 32.
* In the second case, you'll discover that the set must have started out as 1 through 30. That's the smallest possible "n", 30.
Not sufficient.
(You can make 31 values happen, too, but why think about something else if we don't have to.)
TOGETHER:
The biggest value of "n" in statement 1 is the same as the smallest value of "n" in statement 2 -- both are 30 -- so that's the only common value . Sufficient.
Ron has been teaching various standardized tests for 20 years.
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