The range of set A is R. A number having a value equal to R is added

Yes it was pretty difficult. I had to guess on this question.

Official Explanation:
Disclosure: The following explanation has been typed word to word from an 800score practise test.


The range of a set is the difference between the largest and smallest elements of the set. Let's denote e and E as the smallest and largest element of the set, respectively. (if there is only one element, or if all elements are same, then e=E)

R, the range of the set = E-e

The questions asks whether the range of the Set A will increase when R becomes a member of the set. If R is less than the smallest member,e, or greater than the largest number,E, then the range will increase.

However, if E>R>e, then adding R to the set won't increase the range of the set.

Statement (1) Tells us that all the members of the set are positive, Since R= E-e and both E and e are positve then R must be less than E.

This makes it possible for R to be either between e and E, or less than e. In former case, the range of the new set won't increase but in the latter case it will. (see my example) Thus Statement 1 is insufficient.

Statement 2 tells us that the mean of the new set is smaller than R.

The mean of the original set A = Sum of elements /number of elements

Since the new set is just A plus the addition of R, there will be one addition element in the new set.
Thus mean of new set = (sum of elements of A + R) / Number of elements +1

Since R> mean of new set.
R> (sum of A + R) / (number of elements +1)

Multiplying both sides of the equation by (number of elements + 1)

(Number of elements +1) (R) > Sum of A + R, which is equal to

(Number of elements)(R) + (R) > Sum of A + R, subtracting R from both sides.

(Number of elements)(R)>Sum of A, dividing both sides by (number of elements)

R> Sum of A/number of elements, thus R> mean of A



Thus if know that the mean of the new set is smaller than R, we also know that the mean of the old set is smaller than R.

Furthermore, we know that the mean of any group of numbers is at least as largest as the smallest number so the mean of the old set is greater than of equal to e.

So, statment 2 tells us the R>mean of A>e. This is insufficient because R might be greater than E and, and increase the range, or it might be between e and E.

Combined two statements are sufficient. Statement (1) tells us that R is less than E and Statement (2) tells us that R is greater than e. This implies that adding R to the set will not increase the range of the set.

Final answer : (C)