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Set S contains seven distinct integers. The median of set S is the

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Set S contains seven distinct integers. The median of set S is the [#permalink] New post 18 Nov 2009, 20:35
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Set S contains seven distinct integers. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?

A. m
B. 10m/7
C. 10m/7 – 9/7
D. 5m/7 + 3/7
E. 5m

OPEN DISCUSSION OF THIS QUESTION IS HERE: set-s-contains-seven-distinct-integers-the-median-of-set-s-101331.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 01 Dec 2014, 23:03, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: Set S contains seven distinct integers. The median of set S is the [#permalink] New post 18 Nov 2009, 20:51
kairoshan wrote:
Set S contains seven distinct integers. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?
m
10m/7
10m/7 – 9/7
5m/7 + 3/7
5m


10m/7 -9/7

lets consider m = 7 and set as [4,5,6,7,12,13,14] all distinct and will give highest possible average
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Re: Set S contains seven distinct integers. The median of set S is the [#permalink] New post 18 Nov 2009, 20:58
Expert's post
1. m is median --> x x x m x x x
2. 2m is the maximum value. x x x m x x 2m
3. because integers are distinct, we should find as large integers as we can under above restrictions:

x x (m-1) m x (2m-1) 1m
(m-3) (m-2) (m-1) m (2m-2) (2m-1) 2m
Sum = 10m-9
Average = 10/7m-9/7

So, C
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Re: Set S contains seven distinct integers. The median of set S is the [#permalink] New post 19 Nov 2009, 05:20
walker wrote:
1. m is median --> x x x m x x x
2. 2m is the maximum value. x x x m x x 2m
3. because integers are distinct, we should find as larger integers as we can under above restrictions:

x x (m-1) m x (2m-1) 1m
(m-3) (m-2) (m-1) m (2m-2) (2m-1) 2m
Sum = 10m+9
Average = 10/7m-9/7

So, C


Nice solution Walker!
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Re: Set S contains seven distinct integers. The median of set S is the [#permalink] New post 27 May 2010, 00:53
Set S contains seven distinct integers. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?

m
10m/7
10m/7 – 9/7
5m/7 + 3/7
5m
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Re: Set S contains seven distinct integers. The median of set S is the [#permalink] New post 27 May 2010, 01:21
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Answer is 10m/7 - 9/7.

Information provided in problem statement -
1. 7 distinct integers
2. Median is m
3. All values less than or equal to 2m

Based on this information, the 7 elements for highest possible average would be m-3, m-2, m-1, m, 2m-2, 2m-1, 2m.

And the average would be (10m - 9)/7 which is same as 10m/7 - 9/7.
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Re: Set S contains seven distinct integers. The median of set S is the [#permalink] New post 29 May 2010, 01:11
Set S contains seven distinct integers. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?

m
10m/7
10m/7 – 9/7
5m/7 + 3/7
5m


mean of 7 numbers = (Sum of 7 numbers)/7
To find the highest mean we need to maximise the numerator.

since m in the median and we've 7 numbers so m will take the 4th position i.e. the Set S must have(for max avg)

m, m, m, m, 2m, 2m, 2m

Max avg mean = (m + m + m + m + 2m + 2m + 2m)/7 = 10m/7
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Re: Set S contains seven distinct integers. The median of set S is the [#permalink] New post 30 May 2010, 03:34
OA is C

But why cant the set be

m,m,m,m,2m,2m,2m

it is a set and m can be the median, and average will be more than

m-3,m-2,m-1,m,2m-2,2m-1,2m

Amiman wrote:
Set S contains seven distinct integers. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?

m
10m/7
10m/7 – 9/7
5m/7 + 3/7
5m


mean of 7 numbers = (Sum of 7 numbers)/7
To find the highest mean we need to maximise the numerator.

since m in the median and we've 7 numbers so m will take the 4th position i.e. the Set S must have(for max avg)

m, m, m, m, 2m, 2m, 2m

Max avg mean = (m + m + m + m + 2m + 2m + 2m)/7 = 10m/7

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Re: Set S contains seven distinct integers. The median of set S is the [#permalink] New post 30 May 2010, 11:44
BlueRobin wrote:
OA is C

But why cant the set be

m,m,m,m,2m,2m,2m

it is a set and m can be the median, and average will be more than

m-3,m-2,m-1,m,2m-2,2m-1,2m

Amiman wrote:
[highlight]Set S contains seven distinct integers[/highlight]. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?

m
10m/7
10m/7 – 9/7
5m/7 + 3/7
5m


mean of 7 numbers = (Sum of 7 numbers)/7
To find the highest mean we need to maximise the numerator.

since m in the median and we've 7 numbers so m will take the 4th position i.e. the Set S must have(for max avg)

m, m, m, m, 2m, 2m, 2m

Max avg mean = (m + m + m + m + 2m + 2m + 2m)/7 = 10m/7




Note that "Set S contains seven distinct integers".
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Re: Set S contains seven distinct integers. The median of set S is the [#permalink] New post 30 May 2010, 12:09
Fistail wrote:
BlueRobin wrote:
OA is C

But why cant the set be

m,m,m,m,2m,2m,2m

it is a set and m can be the median, and average will be more than

m-3,m-2,m-1,m,2m-2,2m-1,2m

Amiman wrote:
[highlight]Set S contains seven distinct integers[/highlight]. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?

m
10m/7
10m/7 – 9/7
5m/7 + 3/7
5m


mean of 7 numbers = (Sum of 7 numbers)/7
To find the highest mean we need to maximise the numerator.

since m in the median and we've 7 numbers so m will take the 4th position i.e. the Set S must have(for max avg)

m, m, m, m, 2m, 2m, 2m

Max avg mean = (m + m + m + m + 2m + 2m + 2m)/7 = 10m/7




Note that "Set S contains seven distinct integers".



Yeah thanks for point zzzzzzzzzzzz that out, i am awake now.
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Re: Set S contains seven distinct integers. The median of set S is the [#permalink] New post 01 Dec 2014, 09:04
walker wrote:
1. m is median --> x x x m x x x
2. 2m is the maximum value. x x x m x x 2m
:-D 3. because integers are distinct, we should find as large integers as we can under above restrictions:

x x (m-1) m x (2m-1) 1m
(m-3) (m-2) (m-1) m (2m-2) (2m-1) 2m
Sum = 10m-9
Average = 10/7m-9/7

So, C



Had the integers not been distinct would it have been like this:
m m m m 2m 2m 2m :?: And the median would still have been m :?: trying to clarify an imp concept
P.S. I'm a technology handicap :cry: so please dont beat your head at my posts' misplacements and redundancies :roll:
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Set S contains seven distinct integers. The median of set S is the [#permalink] New post 01 Dec 2014, 23:07
Expert's post
Motivatedtowin wrote:
walker wrote:
1. m is median --> x x x m x x x
2. 2m is the maximum value. x x x m x x 2m
:-D 3. because integers are distinct, we should find as large integers as we can under above restrictions:

x x (m-1) m x (2m-1) 1m
(m-3) (m-2) (m-1) m (2m-2) (2m-1) 2m
Sum = 10m-9
Average = 10/7m-9/7

So, C



Had the integers not been distinct would it have been like this:
m m m m 2m 2m 2m :?: And the median would still have been m :?: trying to clarify an imp concept
P.S. I'm a technology handicap :cry: so please dont beat your head at my posts' misplacements and redundancies :roll:


Yes, that's correct.

Set S contains seven distinct integers. The median of set S is the integer m, and all values in set S are equal to or less than 2m. What is the highest possible average (arithmetic mean) of all values in set S ?

A. m
B. 10m/7
C. 10m/7 – 9/7
D. 5m/7 + 3/7
E. 5m

If a set has odd number of terms the median of a set is the middle number when arranged in ascending or descending order;
If a set has even number of terms the median of a set is the average of the two middle terms when arranged in ascending or descending order.


So median of S, which contains seven terms is 4th term when arranged in ascending order;: median=4th \ term=m.

Now, to maximize the mean we should maximize the terms. As numbers in S are distinct integers and the highest number in S could be equal to 2m, then maximum values of the terms would be: m-3, m-2, m-1, median=m, 2m-2, 2m-1, 2m.

Mean=\frac{(m-3)+(m-2)+(m-1)+m+(2m-2)+(2m-1)+2m}{7}=\frac{10m-9}{7}.

Answer: C.

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OPEN DISCUSSION OF THIS QUESTION IS HERE: set-s-contains-seven-distinct-integers-the-median-of-set-s-101331.html
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Set S contains seven distinct integers. The median of set S is the   [#permalink] 01 Dec 2014, 23:07
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