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Set S contains seven distinct integers

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Set S contains seven distinct integers [#permalink] New post 18 Nov 2009, 20:35
00:00
A
B
C
D
E

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Question Stats:

70% (01:24) correct 30% (01:47) wrong based on 12 sessions
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 [#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 [#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 [#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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highest possible average [#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: highest possible average [#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: highest possible average [#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: highest possible average [#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: highest possible average [#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: highest possible average [#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: highest possible average   [#permalink] 30 May 2010, 12:09
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