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# What is the median of five integers 20, 22, 26, 30 and a?

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Math Expert
Joined: 02 Sep 2009
Posts: 46305
What is the median of five integers 20, 22, 26, 30 and a? [#permalink]

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13 Mar 2016, 09:24
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Question Stats:

41% (01:51) correct 59% (01:12) wrong based on 69 sessions

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What is the median of five integers 20, 22, 26, 30 and a?

(1) The mean of the 5 integers is equal to the median of the 5 integers.
(2) The range of the 5 integers is greater than a.

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Intern
Joined: 29 Mar 2015
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Re: What is the median of five integers 20, 22, 26, 30 and a? [#permalink]

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13 Mar 2016, 16:10
From the questions part, all I could conclude is that :

Irrespective of the value of 'a', the median of this list will be one value among 22, 23, 24, 25 and 26.

(1) The mean of the 5 integers is equal to the median of the 5 integers.

The mean and median of the list are : 98 +a/5. Since a is an integer, 98+a should be divisible by 5. Only 22 satisfies that condition. Hence 98+a/5 = 22. This gives us the value of a as 12.

Hence 1 is SUFFICIENT, as my list comes out to be 12, 20, 22, 26 and 30 and 22 is my median and mean.

(2) The range of the 5 integers is greater than a.

a can not be between 20 and 30 and greater than 30, because then the range will be less than a, hence it has to be less than 20, infact it has to be less than 15, irrespective of that the median will be 22.

SUFFICIENT.

As both 1) and 2) are SUFFICIENT, the answer should be D.

I am sure it will sound complicated and I might have made a mistake somewhere. Please correct me, if I am wrong !
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Akki

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Re: What is the median of five integers 20, 22, 26, 30 and a? [#permalink]

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13 Mar 2016, 16:23
1
akxshay wrote:
From the questions part, all I could conclude is that :

Irrespective of the value of 'a', the median of this list will be one value among 22, 23, 24, 25 and 26.

(1) The mean of the 5 integers is equal to the median of the 5 integers.

The mean and median of the list are : 98 +a/5. Since a is an integer, 98+a should be divisible by 5. Only 22 satisfies that condition. Hence 98+a/5 = 22. This gives us the value of a as 12.

Hence 1 is SUFFICIENT, as my list comes out to be 12, 20, 22, 26 and 30 and 22 is my median and mean.

(2) The range of the 5 integers is greater than a.

a can not be between 20 and 30 and greater than 30, because then the range will be less than a, hence it has to be less than 20, infact it has to be less than 15, irrespective of that the median will be 22.

SUFFICIENT.

As both 1) and 2) are SUFFICIENT, the answer should be D.

I am sure it will sound complicated and I might have made a mistake somewhere. Please correct me, if I am wrong !

Your analysis of statement 1 is not correct.

The median of 5 terms is NOT = average in general but only in particular cases.

What if the values are 20,a,22,26,30 in this case, median = 22 and mean = (98+5)/a but if the values are 20,22,26,30, a , the median = 26 and mean = (98+a)/5. Thus you get 2 different medians for the same statement and hence this statement is NOT sufficient.

Additionally, since a is an integer, it is NOT necessary that (98+a)/5 must be an integer. The main question stem only mentions that 'a' is an integer and nothing about the nature of the mean. You are assuming something that the question wants you to prove. This is a dangerous way of tackling DS questions.

Hope this helps.
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Joined: 29 Mar 2015
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Re: What is the median of five integers 20, 22, 26, 30 and a? [#permalink]

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13 Mar 2016, 16:34
Engr2012 wrote:
What if the values are 20,a,22,26,30 in this case, median = 22 and mean = (98+5)/a but if the values are 20,22,26,30, a , the median = 26 and mean = (98+a)/5. Thus you get 2 different medians for the same statement and hence this statement is NOT sufficient.

True ! Got my mistake.

Thanks !
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Akki

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Re: What is the median of five integers 20, 22, 26, 30 and a? [#permalink]

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15 Mar 2016, 00:24
2
Bunuel wrote:
What is the median of five integers 20, 22, 26, 30 and a?

(1) The mean of the 5 integers is equal to the median of the 5 integers.
(2) The range of the 5 integers is greater than a.

Good Question!

Let's place the numbers on the number line:

_______________________20 __ 22 ____ 26 ____ 30 ______________

and a

The median of 5 numbers will be the middle number. So median could be 22 or 26 or a, depending on the value of a.

(1) The mean of the 5 integers is equal to the median of the 5 integers.

The mean and median could be 22 such that a = 12
The mean and median could be 26 such that a = 32
So we see that already we do not have a unique median.
Not Sufficient

(2) The range of the 5 integers is greater than a.
The range of the given four integers is 30 - 20 = 10. If range is greater than a, a must be less than 10. But in that case, the range will increase to 30 - a. Hence the range will not be 10. a is either less than 20 or more than 30 (to get a greater range).
But if a > 30, Range = a - 20.
Range cannot be more than 'a' since it is 20 less than a.

So a must be less than 20 and median must be 22. This is possible as we saw in the analysis of statement 1.

Sufficient.

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Re: What is the median of five integers 20, 22, 26, 30 and a? [#permalink]

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19 Sep 2017, 23:56
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Re: What is the median of five integers 20, 22, 26, 30 and a?   [#permalink] 19 Sep 2017, 23:56
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