IMO E

S is a set of positive integers. The average of the terms in S is equal to the range of the terms in S. What is the sum of all the integers in S?

Let b be the largest integer, and
Let a be the smallest integer, and
Let n be the number of integers in the set

Range = b - a = Sum of all integers in the set / n
The question can be rephrased to what is the value of (b - a) * n

(1) The range of S is a prime number that is less than 11 and is not a factor of 10.

b - a = 3 or 7 and the value of n is unknown. Hence, Insufficient.

(2) S is composed of 5 different integers

Insufficient data. Hence, Insufficient

From 1 and 2

b - a cannot be 3 as it has 5 different integers.
a,a+1,a+2,a+3 is the only case(I can think of) and the series can have only 4 different integers.
So, the value b-a = 7. What is the value of n ? It can be 5 or 6 or 7.. Hence Insufficient

kumadil2011 wrote:S is a set of positive integers. The average of the terms in S is equal to the range of the terms in S. What is the sum of all the integers in S?

(1) The range of S is a prime number that is less than 11 and is not a factor of 10.
(2) S is composed of 5 different integers


Tx

Avg = range
Total/count = highest-lowest term
hence total = count * (high-low)

1. only 3 and 7 fits in as the range.. But no info about the count.
2. No info about the range

Combined we have a solution. That is total = 5* 7 = 35

** Range can never be 3 since the numbers are different and total numbers is 5, unless some number is negative.

C

jm2011 wrote: Combined we have a solution. That is total = 5* 7 = 35
** Range can never be 3 since the numbers are different and total numbers is 5, unless some number is negative.
C

IMO (2) S is composed of 5 different integers - doesn't imply that set S has 5 integers.

e.g. let set S be 1,2,3,4,5,5,5,3,3,3,2,2,8 - This set is composed of 5 different integers but has 13 integers in total. I reckon it is impossible to find the value of the total number of integers in the set. Correct me if I am wrong.

kumadil2011 wrote:S is a set of positive integers. The average of the terms in S is equal to the range of the terms in S. What is the sum of all the integers in S?

(1) The range of S is a prime number that is less than 11 and is not a factor of 10.
(2) S is composed of 5 different integers

'

We are given that S is a set of positive integers and the average of the terms in S is equal to the range of the terms in S. We need to determine the sum of all the integers in S.

Statement One Alone:

The range of S is a prime number that is less than 11 and is not a factor of 10.

Using information in statement one, we know that the range of S is 3 or 7. Thus, the average of S is also 3 or 7. However, since we don't know whether it is 3 or 7, nor do we know the number of integers in S, statement one alone is not sufficient to answer the question.

Statement Two Alone:

S is composed of 5 different integers.

Since we don't know any of the 5 integers, statement two alone is not sufficient to answer the question.

Statements One and Two Together:

From statement one, we know that the range and average are either both 3 or both 7. From statement two, we know S is composed of 5 different positive integers. Thus, the sum of these 5 integers is either 15 (if the average is 3) or 35 (if the average is 7). Therefore, we have two cases to consider: range = average = 3 (case 1) and range = average = 7 (case 2).

Case 1: range = average = 3

We can let x = the smallest number, so the largest number = x + 3. We can "squeeze" 2 more integers between x and x + 3, namely x + 1 and x + 2. So, there could be only 4 different total integers. However, remember that there should be 5 different integers in S; thus, case 1 is not possible.

So, it must be case 2: range = average = 7. If that is the case, the sum of the 5 integers is 35.

For example, the 5 integers could be 4, 5, 7, 8, and 11. We see that the range is 11 - 4 = 7 and the sum is 4 + 5 + 7 + 8 + 11 = 35 with an average of 7.

Answer: C