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The Range of Set A is R. A number having equal value to R

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The Range of Set A is R. A number having equal value to R [#permalink] New post 30 May 2011, 20:04
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The Range of Set A is R. A number having equal value to R, is added to set A. Will the range of Set A increase ?

(1) All numbers in Set A are positive.
(2) The mean of the new set is smaller than R.

Source: 800Score Tests.
[Reveal] Spoiler: OA

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Re: Range & Mean [#permalink] New post 30 May 2011, 20:10
i thought each statement is sufficient to answer the question.
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Re: Range & Mean [#permalink] New post 30 May 2011, 21:01
agdimple333 wrote:
i thought each statement is sufficient to answer the question.


Statement 1 cannot be sufficient.

Range = Highest - Lowest

Let, Set A =[2,3,4,5,6] Thus, Range = 6-2 = 4 . New Set A = [2,3,4,4,5,6]. Range doesn't increase.

But if, Set A = [8,9,10] Range = 10-8=2. New Set A= [2,8,9,10] New Range = 10-2=8. Range increases.

Thus statement 1 is insufficient.

I am little confused with statement 2.
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Re: Range & Mean [#permalink] New post 31 May 2011, 04:02
I'm getting E as answer.

Let Set A = {1,2,3}

=> Range = 2

If we add x = R = 2, then the range remain the same

Let Set A = {1,1,1}

Range = 0

If we add x = R = 0 in this set, then the range become 1-0 = 1

So (1) is Insufficient

(2)


In case 2 above, if we add x = 0, then mean is lesser and the Range increases.

Again, let A = {1,1,2} then if we add x = 1, the new mean = 5/4 < original mean = 4/3, but the range remains same.

So Insufficient

(1) + (2) includes all cases above

Answer - E
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Re: Range & Mean [#permalink] New post 31 May 2011, 08:58
I think E should be the answer.
For 'Yes' (Range will increase) try (1,1,1)
For 'No' (Range will not increase) try (1,1,2)
Above listed 2 sets give 'Yes' n 'No' answer for both the statements.
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Re: Range & Mean [#permalink] New post 31 May 2011, 11:08
akhileshgupta05 wrote:
The Range of Set A is R. A number having equal value to R, is added to set A. Will the range of Set A increase ?

(1)All numbers in Set A are positive.
(2)The mean of the new set is smaller than R.

Please explain your answer.
Source: 800Score Tests.


1.
CaseI:
A={1,1,1}
R:1-1=0
New Set: {0,1,1,1}
New Range=1-0=1
Range Increased.

CaseII:
A={5,10}
R=10-5=5
New Set={5,5,10}
New Range=10-5=5
Range Unchanged.

Not Sufficient.

2.
Case I:
A={-10,-5}
R=-5-(-10)=5
New Set={-10,-5,5}
New Mean = -3.XX; Mean<R
New Range: 5-(-10)=15
Range Increased.

Case II:
A={0,20}
R=20-0=20
New Set={0,20,20}
New Mean=40/3=13.XX. Mean<R
New Range=20-0=20.
Range Unchanged.

Not Sufficient.

Combining both;
A={1,101}
R=100
New Set={1,101,100}
New Mean=202/3=67 Approx < R
New Range=100
Range Not Changed.

A={1,2,3,4,5}
R=4
New Set={1,2,3,4,4,5}
New Mean=19/6=3.X<R
New Range=4
Range Not Changed.
****************************

Ans: "C"
**********************

This is a difficult question. Can the author please post the Official Explanation because substitution does not appear full proof. Thanks.
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Re: Range & Mean [#permalink] New post 31 May 2011, 23:02
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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)
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Re: Range & Mean [#permalink] New post 14 Jun 2011, 21:44
a A (1,1) gives (0,1,1) range increases.
(1,2) gives (1,1,2) range remains same.
Not sufficient.

b A(-2,-1) gives (-2,-1,1) range increases always for negative values.

A(1,5) gives (1,4,5) range remains same.
Not sufficient.

a+b (1,5) giving (1,4,5) range remaining same.
Hence C.
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Re: The Range of Set A is R . A number having equal value to R , [#permalink] New post 13 Oct 2012, 10:41
If set is (50, 54) then the range is 4. When 4 is added to the set, the mean decreases but range increases.

Consider set (3,7) range is 4. When 4 is added to the set the mean decreases but range remains the same.


So both statements together not sufficient. E?

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Re: The Range of Set A is R. A number having equal value to R [#permalink] New post 13 Oct 2012, 11:52
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akhileshgupta05 wrote:
The Range of Set A is R. A number having equal value to R, is added to set A. Will the range of Set A increase ?

(1) All numbers in Set A are positive.
(2) The mean of the new set is smaller than R.

Source: 800Score Tests.


I think we all agree that neither statement is sufficient alone.

Now, let's consider the statements together.

Since {Range}={Largest}-{Smallest}, then the range will increase if R is either the smallest or the largest element of the new set.

Now, can R be the smallest element of the new set? If it is, then according to the second statement, we would have that the mean of the new set is smaller than the smallest element of the set, which is not possible: the mean of a set cannot be smaller than its smallest element.

Next, can R be the largest element of the new set? We are given that for set A, the range, R={Largest}-{Smallest}. Since from the first statement we know that all numbers in Set A are positive, then R must be less than the largest element of set A, hence R cannot be the largest element of the new set.

So, we have that R is neither the smallest nor the largest element of the set. Therefore, the range of the new set will not be greater than the range of set A. Sufficient.

Answer: C.

Hope it's clear.
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Re: The Range of Set A is R. A number having equal value to R [#permalink] New post 13 Oct 2012, 12:11
akhileshgupta05 wrote:
The Range of Set A is R. A number having equal value to R, is added to set A. Will the range of Set A increase ?

(1) All numbers in Set A are positive.
(2) The mean of the new set is smaller than R.

Source: 800Score Tests.


Because we are given a set of numbers, we can assume that all numbers are distinct, and that we have at least two numbers.

Let's denote by M the largest value of the set, and by m its smallest value. Then the range is R = M - m.
The given question can be reformulated as "Is R < m or R>M?" If m\leq{R}\leq{M}, the range of the set A will remain the same after adding R.

(1) If R=M-m<m, then \,\,M<2m.
If R=M-m>{m}, then \,\,M>{2m}.
All the numbers in the set A being positive, both inequalities are possible. In addition, R=M-m<M.
So, R can be smaller or greater than the smallest element in the set, but for sure, it is smaller than the largest element.
Not sufficient.

(2) Let's denote by A_n the average of the set A before we add R to it.
Then \frac{A_n\cdot{n}+R}{n+1}<R, from which A_n\cdot{n}+R<nR+R, or A_n<R.
Since the average A_n\geq{m} we can deduce that R>m.
But if m is negative, R=M-m>M. If m is positive, then necessarily R=M-m<M.
Not sufficient.

(1) and (2) together: from the above analysis, we can see that if m>0, then R is between m and M, and the range will remain the same.
Sufficient.

Answer C.
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Re: The Range of Set A is R. A number having equal value to R [#permalink] New post 30 Dec 2013, 02:58
is zero a positive number ? can we consider zero in set A having all positive number ?
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Re: The Range of Set A is R. A number having equal value to R [#permalink] New post 30 Dec 2013, 03:08
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Re: The Range of Set A is R. A number having equal value to R   [#permalink] 30 Dec 2013, 03:08
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