S is a set of positive integers. The average of the terms in

A - average
(1) Primes less than 11 are 2, 3, 5, and 7. Not factor of 10, we are left with 3 and 7.
If A = 3, we can have {2, 2, 3, 5} or {1, 2, 3, 4, 4, 4}.
Not sufficient.

(2) For 5 evenly spaced numbers, k - 2d, k - d, k, k + d, k + 2d the range is 4d and the average is k.
We can simply take k = 4d, and we have infinitely many sets fulfilling the condition of the form {2d, 3d, 4d, 5d, 6d}.
For example {2, 3, 4, 5, 6}, {20, 30, 40, 50, 60}...
Not sufficient.

(1) and (2) together:
We have seen above that the range can be either 3 or 7.
If the range is 3, we cannot have 5 distinct integers in the set, so only 7 is left.

We know that the sum of the 5 integers is 5 * 7 = 35, the smallest number is k and the largest number is k + 7, where k is some positive integer.
Necessarily 35 - 7 = 28 > 5k, so k must be not greater than 5.
If k = 5, the first and the last number are 5 and 12. The smallest 3 remaining integers are 6, 7, and 8 together with 5 and 12 will give a sum of 37.
If k = 4, we find sets which fulfill the condition: {4, 5, 7, 8, 11} and also (4, 5, 6, 9, 11}.
Obviously, not sufficient.

Answer E.

CORRECTION:

What a miss! We have to find the sum of the numbers and not the set of numbers. Although there is more than one possibility, the total sum is clearly 7*5 = 35.
Answer C.

And IanStewart is right about (1), in a set of numbers, each element is different.

@EvaJager - in (2) why are you considering "5 evenly spaced numbers"? This is not mentioned anywhere in the question.

It is the easiest to express the average when the numbers are evenly spaced. The goal is to find as easily as possible one/more examples which fulfill the condition.
And it is nowhere stated that the numbers cannot be evenly spaced. Also, in (1) it was not mentioned that some of the numbers cannot be equal.

^^^ what's this that you have done , can you please explain thinking why did you calculate the above, I would like to mention that I understand the problem sufficiently clear.



Thanks..

I edited my first post, see the CORRECTION attached :

What a miss! We have to find the sum of the numbers and not the set of numbers. Although there is more than one possibility, the total sum is clearly 7*5 = 35.
Answer C.

And IanStewart is right about (1), in a set of numbers, each element is different.

I was trying to find sets which fulfill the given conditions, which was in fact unnecessary. We were only asked what is the sum of the numbers.

EvaJager.... could you please explain the entire solution.....please do it independently and not as an extension of some other post...

EvaJager.... could you please explain the entire solution.....please do it independently and not as an extension of some other post...[/quote]

This is my original post:

s-is-a-set-of-positive-integers-the-average-of-the-terms-in-139444.html#p1124575

Which part didn't you understand?

As I mentioned before, I misunderstood the question and I was trying to find concrete sets of numbers which fulfill the conditions.

Please, refer to IanStewart's post for a correct solution:

s-is-a-set-of-positive-integers-the-average-of-the-terms-in-139444.html#p1124943

(A+B+C+D+E)/5=E-A
A+B+C+D+E= 5(E-A)

Range can only by a 3 or a 7: 5*3=15

Therefore, the sum of A+B+C+D+E =15
this can only happen if A+B+C+D+E = 1+2+3+4+5

n(n+1)/2=(5*6)/2=15

but it cannot be 15 because 5-1 is 4 which is not a prime number. Therefore, the range must be 7 and the sum equal to 35.

From statement 1 and statement 2, we have range = 7 ( cannot be 2,5 or 3), hence average is also 7, hence sum = 5*7 = 35

I think the answer should still be E.

Statement two says that set S is composed of 5 different integers- this IMO doesn't mean that S is composed of exactly 5 integers.

Two contrasting examples:

S= {3,6,7,9,10) - range = 7 and mean = 7. The Sum is 35
S= {3,6,7,7,9,10)- range = 7 and mean = 7. The Sum is 42

Experts- please help.

(2) says that S is composed of 5 different integers.

Nowhere stated that the numbers cannot be evenly spaced DOESN'T MEAN that the numbers are evenly spaced. Do you have another explanation?

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

Hi Bunuel - Can you pls help me in this, its really confusing.

IMO,
From Statements (1) and (2) together, we know that the range of the terms in S must be 7. This means that the average of the terms in S is also 7. It may be concluded form this that the sum of the terms in S is equal to the average (7) multiplied by the number of terms (5) = 7 × 5 = 35. However, while Statement (2) says that S is composed of 5 different integers, this does not mean that S is composed of exactly 5 integers since each integer may occur in S more than once. So I think the correct answer is E: Statements (1) and (2) TOGETHER are NOT sufficient.
Can you pls elaborate a bit by taking the above mentioned example

Hi Bunel / Experts

Can you please explain this in a new thread maybe.

Thanks

Why no set with range 3 cannot have 5 distinct integers?

Posted from my mobile device

Average = Range
For sum of integers, we need the number of terms and average or range.

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

There are 4 prime numbers less than 11: 2, 3, 5, 7
2 and 5 are factors of 10 so range will be either 3 or 7. Not sufficient.

(2) S is composed of 5 different integers.

We know the number of integers but not the average or range. Not sufficient.

Using both, we know that we have 5 distinct integers. Their range cannot be 3. Take for example, 1, 2,3 ,4 ,5 (the range will be at least 4)
Hence range is 7. This means the average is also 7.
Then sum of integers = 5 * 7 = 35
Sufficient.

Answer (C)

Now try this question on mean, median and range: https://anaprep.com/sets-statistics-mea ... -question/