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Set R contains five numbers that have an average value of 55 [#permalink]
02 Oct 2010, 11:23

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Difficulty:

95% (hard)

Question Stats:

54% (03:24) correct
46% (02:40) wrong based on 254 sessions

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?

Re: Largest possible range in Set R [#permalink]
02 Oct 2010, 11:42

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Orange08 wrote:

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

Set R contains five numbers that have an average value of 55 [#permalink]
08 Mar 2011, 15:06

Expert's post

Yalephd wrote:

gmat1220 wrote:

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. _________________

Re: Largest possible range in Set R [#permalink]
08 Mar 2011, 15:19

Bunuel wrote:

Yalephd wrote:

gmat1220 wrote:

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.

Re: Largest possible range in Set R [#permalink]
01 May 2011, 22:15

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. _________________

Re: Largest possible range in Set R [#permalink]
01 May 2011, 22:16

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. _________________

Re: Largest possible range in Set R [#permalink]
18 Nov 2011, 03:18

Bunuel wrote:

Orange08 wrote:

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!!!

Re: Largest possible range in Set R [#permalink]
01 Nov 2012, 07:01

Bunuel wrote:

Orange08 wrote:

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 !

Re: Largest possible range in Set R [#permalink]
02 Nov 2012, 04:17

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sanjoo wrote:

Bunuel wrote:

Orange08 wrote:

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]
24 Jan 2013, 12:29

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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]
24 Jan 2013, 16:22

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.

Re: Set R contains five numbers that have an average value of 55 [#permalink]
07 Feb 2014, 14:01

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Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________

Re: Largest possible range in Set R [#permalink]
26 May 2014, 12:17

Bunuel wrote:

Orange08 wrote:

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?

gmatclubot

Re: Largest possible range in Set R
[#permalink]
26 May 2014, 12:17