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Math Expert V
Joined: 02 Sep 2009
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Difficulty:   45% (medium)

Question Stats: 55% (01:32) correct 45% (01:14) wrong based on 65 sessions

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If set $$S$$ consists of positive integers, is the median of set $$S$$ larger than its range?

(1) All elements of set $$S$$ are smaller than 40.

(2) All elements of set $$S$$ are larger than 20.

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Math Expert V
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Official Solution:

Statement (1) by itself is insufficient. The range can be smaller than median (as in one-element set) or bigger than median (if the elements are spread wide enough: $${1, 39}$$).

Statement (2) by itself is insufficient. The range can be smaller than median (as in one-element set) or bigger than median (if the elements are spread wide enough: $${21, 100}$$).

Statements (1) and (2) combined are sufficient. If all the elements are confined between 21 and 39, the range of the set cannot exceed $$39 - 21 = 18$$, whereas the median is between 21 and 39. Thus, the median of set $$S$$ is larger than its range.

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Bunuel wrote:
If set $$S$$ consists of positive integers, is the median of set $$S$$ larger than its range?

(1) All elements of set $$S$$ are smaller than 40.

(2) All elements of set $$S$$ are larger than 20.

I did it this way:

Let's say Median = M, Smallest element = S, and Largest element = L

Q: is M > L - S? or M + S > L?

(1) L <= 39, S & M unknown. Not Sufficient

(2) All elements > 20, so L, M & S unknown. Not sufficient

Together (1) & (2)
From (1) we know that L>=39, so let's say L is the maximum possible integer, so L = 39

From (2), we know that S could be any positive integer from 21 to 39, so let's say S is the smallest possible integer in the range, S = 21. We are left with M. Let's keep M as low as S, so M= S = 21

Q: Q: is M > L - S? or M + S > L? 21 > 39 -21? or 21 + 21 > 39? Yes. SUFFICIENT.

Hope this way of getting to the answer is ok.
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Intern  B
Joined: 29 Apr 2016
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but what if all the elements are same? I think you had to emphasize that set contains of distinct numbers.
Math Expert V
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Tuvshinbagana wrote:
but what if all the elements are same? I think you had to emphasize that set contains of distinct numbers.

It does not matter whether the elements are distinct or not. Please re-read:
If all the elements are confined between 21 and 39, the range of the set cannot exceed $$39 - 21 = 18$$, whereas the median is between 21 and 39. Thus, the median of set $$S$$ is larger than its range.
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Current Student D
Joined: 12 Aug 2015
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Bunuel wrote:
Tuvshinbagana wrote:
but what if all the elements are same? I think you had to emphasize that set contains of distinct numbers.

It does not matter whether the elements are distinct or not. Please re-read:
If all the elements are confined between 21 and 39, the range of the set cannot exceed $$39 - 21 = 18$$, whereas the median is between 21 and 39. Thus, the median of set $$S$$ is larger than its range.

Hi Bunuel.
What if the set has only one integer.

In that case => Median = Range ??
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Math Expert V
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stonecold wrote:
Bunuel wrote:
Tuvshinbagana wrote:
but what if all the elements are same? I think you had to emphasize that set contains of distinct numbers.

It does not matter whether the elements are distinct or not. Please re-read:
If all the elements are confined between 21 and 39, the range of the set cannot exceed $$39 - 21 = 18$$, whereas the median is between 21 and 39. Thus, the median of set $$S$$ is larger than its range.

Hi Bunuel.
What if the set has only one integer.

In that case => Median = Range ??

The range of a single-element set is 0.
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Intern  B
Joined: 03 Apr 2018
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I think this is a high-quality question and I agree with explanation.
Intern  B
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I think this is a high-quality question and I agree with explanation. Re M17-01   [#permalink] 13 Jul 2019, 14:43
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