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Set R contains five numbers that have an average value of 55 [#permalink]

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02 Oct 2010, 12:23

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Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set?

Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set?

78 77 1/5 66 1/7 55 1/7 52

{\(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_5\)} As mean of 5 numbers is 55 then the sum of these numbers is \(5*55=275\); The median of the set is equal to the mean --> \(mean=median=a_3=55\); The largest number in the set is equal to 20 more than three times the smallest number --> \(a_5=3a_1+20\).

So our set is {\(a_1\), \(a_2\), \(55\), \(a_4\), \(3a_1+20\)} and \(a_1+a_2+55+a_4+3a_1+20=275\).

The range of a set is the difference between the largest and smallest elements of a set.

\(Range=a_5-a_1=3a_1+20-a_1=2a_1+20\) --> so to maximize the range we should maximize the value of \(a_1\) and to maximize \(a_1\) we should minimize all other terms so \(a_2\) and \(a_4\).

Min possible value of \(a_2\) is \(a_1\) and min possible value of \(a_4\) is \(median=a_3=55\) --> set becomes: {\(a_1\), \(a_1\), \(55\), \(55\), \(3a_1+20\)} --> \(a_1+a_1+55+55+3a_1+20=275\) --> \(a_1=29\) --> \(Range=2a_1+20=78\)

Backsolving : Range = 2a + 20 where a = first number.

Hence the answer is even and highest options. It should be A.

That's an awesome application of number properties to solve this question is seconds. Kudos

That's not correct. Yes, the range equals to 2a+20 but without any further calculation we cannot say whether it must be even, for example if a is not an integer then 2a+20 can be odd or not an integer at all. Also the answer is not necessarily the highest option, it just happened to be so in this particular case.
_________________

Backsolving : Range = 2a + 20 where a = first number.

Hence the answer is even and highest options. It should be A.

That's an awesome application of number properties to solve this question is seconds. Kudos

That's not correct. Yes, the range equals to 2a+20 but without any further calculation we can not say whether it must be even, for example if a is not an integer then 2a+20 can be odd or not an integer at all. Also the answer is not necessarily the highest option, it just happened to be so in this particular case.

Thanks. Assuming that A is an integer is where I erred.

max range will be when 55*3 = 165 will give 110 as range.But the value isn't present. Hence go for two small numbers , 55*2 and largest number combination. thus 2x+110 + 3x+20 = 275 will give, x= 29 and 3x+20 = 97. Range = 78.
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max range will be when 55*3 = 165 will give 110 as range.But the value isn't present. Hence go for two small numbers , 55*2 and largest number combination. thus 2x+110 + 3x+20 = 275 will give, x= 29 and 3x+20 = 97. Range = 78.
_________________

Visit -- http://www.sustainable-sphere.com/ Promote Green Business,Sustainable Living and Green Earth !!

Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set?

78 77 1/5 66 1/7 55 1/7 52

{\(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_5\)} As mean of 5 numbers is 55 then the sum of these numbers is \(5*55=275\); The median of the set is equal to the mean --> \(mean=median=a_3=55\); The largest number in the set is equal to 20 more than three times the smallest number --> \(a_5=3a_1+20\).

So our set is {\(a_1\), \(a_2\), \(55\), \(a_4\), \(3a_1+20\)} and \(a_1+a_2+55+a_4+3a_1+20=275\).

The range of a set is the difference between the largest and smallest elements of a set.

\(Range=a_5-a_1=3a_1+20-a_1=2a_1+20\) --> so to maximize the range we should maximize the value of \(a_1\) and to maximize \(a_1\) we should minimize all other terms so \(a_2\) and \(a_4\).

Min possible value of \(a_2\) is \(a_1\) and min possible value of \(a_4\) is \(median=a_3=55\) --> set becomes: {\(a_1\), \(a_1\), \(55\), \(55\), \(3a_1+20\)} --> \(a_1+a_1+55+55+3a_1+20=275\) --> \(a_1=29\) --> \(Range=2a_1+20=78\)

Answer: A.

my approach was like yours, but it took me 6 min!!!

Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set?

78 77 1/5 66 1/7 55 1/7 52

{\(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_5\)} As mean of 5 numbers is 55 then the sum of these numbers is \(5*55=275\); The median of the set is equal to the mean --> \(mean=median=a_3=55\); The largest number in the set is equal to 20 more than three times the smallest number --> \(a_5=3a_1+20\).

So our set is {\(a_1\), \(a_2\), \(55\), \(a_4\), \(3a_1+20\)} and \(a_1+a_2+55+a_4+3a_1+20=275\).

The range of a set is the difference between the largest and smallest elements of a set.

\(Range=a_5-a_1=3a_1+20-a_1=2a_1+20\) --> so to maximize the range we should maximize the value of \(a_1\) and to maximize \(a_1\) we should minimize all other terms so \(a_2\) and \(a_4\).

Min possible value of \(a_2\) is \(a_1\) and min possible value of \(a_4\) is \(median=a_3=55\) --> set becomes: {\(a_1\), \(a_1\), \(55\), \(55\), \(3a_1+20\)} --> \(a_1+a_1+55+55+3a_1+20=275\) --> \(a_1=29\) --> \(Range=2a_1+20=78\)

Answer: A.

Bunuel sir..

few questions that cums in my mnd like ..y did bunuel take A1 is equal to A2..and y didnt he take a2=55 instead of A4=55?

i got lots of questions like this and i cant give ans correctly..

Thank u in advance bunuel..
_________________

Bole So Nehal.. Sat Siri Akal.. Waheguru ji help me to get 700+ score !

Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set?

78 77 1/5 66 1/7 55 1/7 52

{\(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_5\)} As mean of 5 numbers is 55 then the sum of these numbers is \(5*55=275\); The median of the set is equal to the mean --> \(mean=median=a_3=55\); The largest number in the set is equal to 20 more than three times the smallest number --> \(a_5=3a_1+20\).

So our set is {\(a_1\), \(a_2\), \(55\), \(a_4\), \(3a_1+20\)} and \(a_1+a_2+55+a_4+3a_1+20=275\).

The range of a set is the difference between the largest and smallest elements of a set.

\(Range=a_5-a_1=3a_1+20-a_1=2a_1+20\) --> so to maximize the range we should maximize the value of \(a_1\) and to maximize \(a_1\) we should minimize all other terms so \(a_2\) and \(a_4\).

Min possible value of \(a_2\) is \(a_1\) and min possible value of \(a_4\) is \(median=a_3=55\) --> set becomes: {\(a_1\), \(a_1\), \(55\), \(55\), \(3a_1+20\)} --> \(a_1+a_1+55+55+3a_1+20=275\) --> \(a_1=29\) --> \(Range=2a_1+20=78\)

Answer: A.

Bunuel sir..

few questions that cums in my mnd like ..y did bunuel take A1 is equal to A2..and y didnt he take a2=55 instead of A4=55?

i got lots of questions like this and i cant give ans correctly..

Thank u in advance bunuel..

After some steps we have that our set in ascending order is {\(a_1\), \(a_2\), \(55\), \(a_4\), \(3a_1+20\)} and \(Range=2a_1+20\).

We need to maximize \(Range=2a_1+20\), thus we need to maximize \(a_1\) and to maximize \(a_1\) we should minimize all other terms so \(a_2\) and \(a_4\) (remember the sum of the terms is fixed, so we cannot just make \(a_1\) as large as we want).

Now, since the set is in ascending order min possible value of \(a_2\) is \(a_1\) (it cannot be less than the first term) and min possible value of \(a_4\) is \(median=a_3=55\) (it cannot be less than the third term).

Re: Set R contains five numbers that have an average value of 55 [#permalink]

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24 Jan 2013, 13:29

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Sachin9 wrote:

Hi ,

Here's how I did..

smallest no: s largest no: 3s+20

since mean = median,

thought that numbers are in AP.

so average= (last no+first no)/2

therefore 55=(s+3s+20)/2 => s=22.5

now l=20+3s => l=87.25

range =l-s=65.. Please let me know where I am going wrong.

Sachin, you assumed that the numbers are in AP, but problem doesn't state that. This set S = {29, 29, 55, 55, 107} has the maximum range i.e. 78 and mean/median 55.

Note that these numbers are not in AP/sequence. Hence you cannot take average of last & first to find the mean.
_________________

Re: Set R contains five numbers that have an average value of 55 [#permalink]

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24 Jan 2013, 17:22

PraPon wrote:

Sachin9 wrote:

Hi ,

Here's how I did..

smallest no: s largest no: 3s+20

since mean = median,

thought that numbers are in AP.

so average= (last no+first no)/2

therefore 55=(s+3s+20)/2 => s=22.5

now l=20+3s => l=87.25

range =l-s=65.. Please let me know where I am going wrong.

Sachin, you assumed that the numbers are in AP, but problem doesn't state that. This set S = {29, 29, 55, 55, 107} has the maximum range i.e. 78 and mean/median 55.

Note that these numbers are not in AP/sequence. Hence you cannot take average of last & first to find the mean.

Thanks mate.. I thought that the numbers would be in AP since their median and mean were same.

I now understand that if the nos are in AP , then their median and mean will be same but the vice versa is not necessarily true.
_________________

hope is a good thing, maybe the best of things. And no good thing ever dies.

Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set?

78 77 1/5 66 1/7 55 1/7 52

{\(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_5\)} As mean of 5 numbers is 55 then the sum of these numbers is \(5*55=275\); The median of the set is equal to the mean --> \(mean=median=a_3=55\); The largest number in the set is equal to 20 more than three times the smallest number --> \(a_5=3a_1+20\).

So our set is {\(a_1\), \(a_2\), \(55\), \(a_4\), \(3a_1+20\)} and \(a_1+a_2+55+a_4+3a_1+20=275\).

The range of a set is the difference between the largest and smallest elements of a set.

\(Range=a_5-a_1=3a_1+20-a_1=2a_1+20\) --> so to maximize the range we should maximize the value of \(a_1\) and to maximize \(a_1\) we should minimize all other terms so \(a_2\) and \(a_4\).

Min possible value of \(a_2\) is \(a_1\) and min possible value of \(a_4\) is \(median=a_3=55\) --> set becomes: {\(a_1\), \(a_1\), \(55\), \(55\), \(3a_1+20\)} --> \(a_1+a_1+55+55+3a_1+20=275\) --> \(a_1=29\) --> \(Range=2a_1+20=78\)

Answer: A.

Hi Bunuel,

Since the statement says that the median = mean, aren't we supposed to assume that it's an evenly spaced set? If so, wouldn't a2 and a4 be different from a1 and a3?

Set R contains five numbers that have an average value of 55. If the median of the set is equal to the mean, and the largest number in the set is equal to 20 more than three times the smallest number, what is the largest possible range for the numbers in the set?

78 77 1/5 66 1/7 55 1/7 52

{\(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_5\)} As mean of 5 numbers is 55 then the sum of these numbers is \(5*55=275\); The median of the set is equal to the mean --> \(mean=median=a_3=55\); The largest number in the set is equal to 20 more than three times the smallest number --> \(a_5=3a_1+20\).

So our set is {\(a_1\), \(a_2\), \(55\), \(a_4\), \(3a_1+20\)} and \(a_1+a_2+55+a_4+3a_1+20=275\).

The range of a set is the difference between the largest and smallest elements of a set.

\(Range=a_5-a_1=3a_1+20-a_1=2a_1+20\) --> so to maximize the range we should maximize the value of \(a_1\) and to maximize \(a_1\) we should minimize all other terms so \(a_2\) and \(a_4\).

Min possible value of \(a_2\) is \(a_1\) and min possible value of \(a_4\) is \(median=a_3=55\) --> set becomes: {\(a_1\), \(a_1\), \(55\), \(55\), \(3a_1+20\)} --> \(a_1+a_1+55+55+3a_1+20=275\) --> \(a_1=29\) --> \(Range=2a_1+20=78\)

Answer: A.

Hi Bunuel,

Since the statement says that the median = mean, aren't we supposed to assume that it's an evenly spaced set? If so, wouldn't a2 and a4 be different from a1 and a3?

For evenly spaced set mean = median, but the reverse is not necessarily true. Consider {1, 1, 2, 2, 4} --> mean = median = 2, but the set is not evenly spaced.
_________________