GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 14 Oct 2019, 04:22 GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  Set R contains five numbers that have an average value of 55

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

Hide Tags

Manager  Joined: 25 Jul 2010
Posts: 93
Set R contains five numbers that have an average value of 55  [#permalink]

Show Tags

10
54 00:00

Difficulty:   75% (hard)

Question Stats: 61% (02:43) correct 39% (03:15) wrong based on 550 sessions

HideShow timer Statistics

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?

A. 78
B. 77 1/5
C. 66 1/7
D. 55 1/7
E. 52

Originally posted by Orange08 on 02 Oct 2010, 12:23.
Last edited by Bunuel on 10 Jun 2012, 01:25, edited 1 time in total.
Edited the question and added the OA
Math Expert V
Joined: 02 Sep 2009
Posts: 58319
Re: Largest possible range in Set R  [#permalink]

Show Tags

21
24
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: {$$a_1$$, $$a_1$$, $$55$$, $$55$$, $$3a_1+20$$} --> $$a_1+a_1+55+55+3a_1+20=275$$ --> $$a_1=29$$ --> $$Range=2a_1+20=78$$

_________________
Director  Status: Impossible is not a fact. It's an opinion. It's a dare. Impossible is nothing.
Affiliations: University of Chicago Booth School of Business
Joined: 03 Feb 2011
Posts: 655
Re: Largest possible range in Set R  [#permalink]

Show Tags

8
1
Backsolving : Range = 2a + 20 where a = first number.

Hence the answer is even and highest options. It should be A.
General Discussion
Manager  Joined: 01 Oct 2010
Posts: 63
Re: Largest possible range in Set R  [#permalink]

Show Tags

2
I took the set to be m, m, 55, 55, 3m+20. (second value has to be minimum possible - m, and fourth value has to be minimum possible - 55).

now average is 55 so 55 = (6m + 130)/5 which gives m = 29, and 3m+20=107

so range is largest - smallest = 107-29 = 78
_________________
Retired Moderator B
Joined: 16 Nov 2010
Posts: 1262
Location: United States (IN)
Concentration: Strategy, Technology
Re: Largest possible range in Set R  [#permalink]

Show Tags

2
Let smallest # = x, Largest = 3x + 20

So range = 2x + 20

x, x, 55, 55, 3x+20, For Max range lowest should be as low as possible and highest should be as high as possible

also, the 2nd value has to be minimized, so it is x, the fourth value also ahs to be kept at minimum, so it is 55

3x + 20 + 110 + 2x = 275

=> 5x = 275 - 130 = 145 => x = 29 , so range = 29*2 + 20 = 78

So answer is A.
_________________
Formula of Life -> Achievement/Potential = k * Happiness (where k is a constant)

GMAT Club Premium Membership - big benefits and savings
Manager  Joined: 03 Aug 2010
Posts: 87
GMAT Date: 08-08-2011
Re: Largest possible range in Set R  [#permalink]

Show Tags

1
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
Math Expert V
Joined: 02 Sep 2009
Posts: 58319
Set R contains five numbers that have an average value of 55  [#permalink]

Show Tags

2
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.
_________________
Manager  Joined: 03 Aug 2010
Posts: 87
GMAT Date: 08-08-2011
Re: Largest possible range in Set R  [#permalink]

Show Tags

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.
Director  Status: Impossible is not a fact. It's an opinion. It's a dare. Impossible is nothing.
Affiliations: University of Chicago Booth School of Business
Joined: 03 Feb 2011
Posts: 655
Re: Largest possible range in Set R  [#permalink]

Show Tags

I was back solving - to confirm the answer.

x^2 = 4
Implies x is not necessarily 2. It can be -2 Director  Status: There is always something new !!
Affiliations: PMI,QAI Global,eXampleCG
Joined: 08 May 2009
Posts: 865
Re: Largest possible range in Set R  [#permalink]

Show Tags

1
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.
Director  Status: There is always something new !!
Affiliations: PMI,QAI Global,eXampleCG
Joined: 08 May 2009
Posts: 865
Re: Largest possible range in Set R  [#permalink]

Show Tags

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.
Manager  Joined: 25 May 2011
Posts: 97
Re: Largest possible range in Set R  [#permalink]

Show Tags

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: {$$a_1$$, $$a_1$$, $$55$$, $$55$$, $$3a_1+20$$} --> $$a_1+a_1+55+55+3a_1+20=275$$ --> $$a_1=29$$ --> $$Range=2a_1+20=78$$

my approach was like yours, but it took me 6 min!!!  Senior Manager  Joined: 06 Aug 2011
Posts: 318
Re: Largest possible range in Set R  [#permalink]

Show Tags

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: {$$a_1$$, $$a_1$$, $$55$$, $$55$$, $$3a_1+20$$} --> $$a_1+a_1+55+55+3a_1+20=275$$ --> $$a_1=29$$ --> $$Range=2a_1+20=78$$

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 !
Math Expert V
Joined: 02 Sep 2009
Posts: 58319
Re: Largest possible range in Set R  [#permalink]

Show Tags

1
1
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: {$$a_1$$, $$a_1$$, $$55$$, $$55$$, $$3a_1+20$$} --> $$a_1+a_1+55+55+3a_1+20=275$$ --> $$a_1=29$$ --> $$Range=2a_1+20=78$$

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).

Similar questions to practice:
if-the-average-of-5-positive-integers-is-40-and-the-127038.html
the-average-arithmetic-mean-of-the-5-positive-integers-k-107059.html
a-certain-city-with-population-of-132-000-is-to-be-divided-76217.html
five-peices-of-wood-have-an-average-length-of-124-inches-and-123513.html
three-boxes-of-supplies-have-an-average-arithmetic-mean-105819.html
a-set-of-25-different-integers-has-a-median-of-50-and-a-129345.html
three-people-each-took-5-tests-if-the-ranges-of-their-score-127935.html
each-senior-in-a-college-course-wrote-a-thesis-the-lengths-126964.html
in-a-certain-set-of-five-numbers-the-median-is-128514.html
shaggy-has-to-learn-the-same-71-hiragana-characters-and-126948.html

Other min/max questions:
PS: search.php?search_id=tag&tag_id=63
DS: search.php?search_id=tag&tag_id=42

Hope it helps.
_________________
Senior Manager  Joined: 06 Aug 2011
Posts: 318
Re: Set R contains five numbers that have an average value of 55  [#permalink]

Show Tags

1
Thanks alot Bunuel..now i got that ..

i think in REAL GMAT these type of question cum frequenlty..!!.
_________________
Bole So Nehal.. Sat Siri Akal.. Waheguru ji help me to get 700+ score !
Senior Manager  Status: Gonna rock this time!!!
Joined: 22 Jul 2012
Posts: 423
Location: India
GMAT 1: 640 Q43 V34 GMAT 2: 630 Q47 V29 WE: Information Technology (Computer Software)
Re: Set R contains five numbers that have an average value of 55  [#permalink]

Show Tags

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

Who says you need a 700 ?Check this out : http://gmatclub.com/forum/who-says-you-need-a-149706.html#p1201595

My GMAT Journey : http://gmatclub.com/forum/end-of-my-gmat-journey-149328.html#p1197992
Senior Manager  Joined: 27 Jun 2012
Posts: 350
Concentration: Strategy, Finance
Schools: Haas EWMBA '17
Re: Set R contains five numbers that have an average value of 55  [#permalink]

Show Tags

1
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,
Prashant Ponde

Tough 700+ Level RCs: Passage1 | Passage2 | Passage3 | Passage4 | Passage5 | Passage6 | Passage7
VOTE GMAT Practice Tests: Vote Here
PowerScore CR Bible - Official Guide 13 Questions Set Mapped: Click here
Finance your Student loan through SoFi and get \$100 referral bonus : Click here
Senior Manager  Status: Gonna rock this time!!!
Joined: 22 Jul 2012
Posts: 423
Location: India
GMAT 1: 640 Q43 V34 GMAT 2: 630 Q47 V29 WE: Information Technology (Computer Software)
Re: Set R contains five numbers that have an average value of 55  [#permalink]

Show Tags

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.

Who says you need a 700 ?Check this out : http://gmatclub.com/forum/who-says-you-need-a-149706.html#p1201595

My GMAT Journey : http://gmatclub.com/forum/end-of-my-gmat-journey-149328.html#p1197992
Manager  Joined: 15 Aug 2013
Posts: 228
Re: Largest possible range in Set R  [#permalink]

Show Tags

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: {$$a_1$$, $$a_1$$, $$55$$, $$55$$, $$3a_1+20$$} --> $$a_1+a_1+55+55+3a_1+20=275$$ --> $$a_1=29$$ --> $$Range=2a_1+20=78$$

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?
Math Expert V
Joined: 02 Sep 2009
Posts: 58319
Re: Largest possible range in Set R  [#permalink]

Show Tags

russ9 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: {$$a_1$$, $$a_1$$, $$55$$, $$55$$, $$3a_1+20$$} --> $$a_1+a_1+55+55+3a_1+20=275$$ --> $$a_1=29$$ --> $$Range=2a_1+20=78$$

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.
_________________ Re: Largest possible range in Set R   [#permalink] 27 May 2014, 01:16

Go to page    1   2    Next  [ 24 posts ]

Display posts from previous: Sort by

Set R contains five numbers that have an average value of 55

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne  