Data Sufficiency

Hi rishianand7,

This DS question involves a Number Property rule that's based on consecutive positive integers and averages.

As a quick example, if we have an ODD number of consecutive, positive integers 1, 2, 3, 4, 5, the average = 3 (the middle term).

Here are the other rules worth knowing:
1) Even if you remove the middle term, the average of the remaining numbers will STILL = 3.
2) If you remove one of the smaller numbers, then the average will get BIGGER.
3) If you remove one of the bigger numbers, then the average will get SMALLER.

If we have an EVEN number of consecutive, positive integers 1, 2, 3, 4, the average = 2.5 (the average of the middle 2 terms)

Now, on to the DS question:

We're told that we have N consecutive integers, starting with the number 1. The questions asks what is N?

Fact 1 tells us that if you remove one of the numbers, then the average of the remaining numbers = 15

So, we could have 29 terms (1 - 29). Removing the middle term (15) will leave us with an average = 15
But, we could also have 30 terms (1 - 30). Removing the biggest term (30) will leave us with an average = 15
Fact 1 has 2 different answers, so it is INSUFFICIENT

Fact 2 tells us that if you remove one of the numbers, then the average of the remaining numbers = 16

We can use the same concept that we used in Fact 1 above!!!

We could have 31 terms (1-31). Removing the middle term (16) will leave us with an average = 16
But we could have 32 terms (1-32). Removing the biggest term (32) will leave us with an average = 16
Fact 2 has 2 different answers, so it is INSUFFICIENT

Combining facts, the only way to satisfy both Facts is with 30 terms. Here's how:
From Fact 1, we know that by removing the biggest term (30) will leave us with an average = 15
From Fact 2, if we remove the SMALLEST term (1) will leave us with an average = 16

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

this problem is pretty cool. what's the source?

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.