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Set R contains five numbers that have an average value of 55
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Bunuel
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Re: Largest possible range in Set R [#permalink] New post  27 May 2014, 00:16
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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: {\(a_1\), \(a_1\), \(55\), \(55\), \(3a_1+20\)} --> \(a_1+a_1+55+55+3a_1+20=275\) --> \(a_1=29\) --> \(Range=2a_1+20=78\)

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.
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russ9
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Re: Largest possible range in Set R [#permalink] New post  27 May 2014, 15:24
Bunuel wrote:
russ9 wrote:

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.


thanks for clarifying.
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Re: Set R contains five numbers that have an average value of 55 [#permalink] New post  24 Aug 2018, 02:50
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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?

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


To maximize the range, we want to make the smallest number as small as possible and the largest number as large as possible. So the five numbers would be:

x, x, 55, 55, y

The set above will minimize the smallest number (x) because the two smallest values are the same. It will maximize the largest number (y) because the next largest number (55) is the same as the median.

Since the largest number is 20 more than 3 times the smallest number, y = 20+3x. So our list of numbers can be rewritten as:

x, x, 55, 55, 20+3x

Since the average is 55, the sum of the five numbers = 5*55 = 275.
Thus, x + x + 55 + 55 + 20+3x = 275.
5x = 145
x= 29.

Thus, y = 20+3x = 20 + 3*29 = 107.

Thus, the largest possible range is y-x = 107-29 = 78.

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Re: Set R contains five numbers that have an average value of 55 [#permalink] New post  23 Feb 2019, 12:11
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.


Dear,

I have one doubt, I tried to understand the question which says that the Mean of the set is equal to the median, and this happens in case of Arithmetic progression.
When I followed that process to apply Sum = N/2(a1 + 20+3a1) and calculated, the range came 66, which was in the answers.
Could you please help me with the catch as where did I go wrong ??

Thanks in advance.
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Re: Set R contains five numbers that have an average value of 55 [#permalink] New post  28 Aug 2019, 08:17
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?

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


Given:
1. Set R contains five numbers that have an average value of 55.
2. 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

Asked: What is the largest possible range for the numbers in the set?

mean = median = 55
Since 3x + 20 increase more than x with changes in x, x has to be maximised.
Let the numbers be = {x,x,55,55,3x+20}
2x+ 55 + 55+ 3x + 20 = 55*5 = 275
5x = 275 - 130 = 145
x = 29
The numbers are = {29,29,55,55,107}
Range = 107 -29 = 78

IMO A
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Re: Set R contains five numbers that have an average value of 55 [#permalink] New post  20 Nov 2019, 06:57
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?

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


n=5, (avg,med)=55, sum=275
smallest=a, largest=tn, tn=3a+20
the median is the middle number 3, so what follows must be ≥ median;
(a,a,md,md,tn)/5=55…(a,a,55,55,tn)=275…(a,a,3a+20)=275-110…a=29
range: tn-a…3a+20-a…2a+20…2(29)+20…78

Ans (A)
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Re: Set R contains five numbers that have an average value of 55 [#permalink] New post  28 Feb 2021, 12:06
Let us say R = {a,b,c,d,e}

We know that the median should be 55 so automatically we can set c = 55.

We also know that the largest (e) is equal to 20 more than 3 times the smallest (a). Therefore we can set e = 3a+20

Now how what should be the values of b and d so that we have a maximum range? To maximize the range, we must minimize the value for both b and d. The minimum value allowable for b must be equal to a. And the minimum allowable value for d must be the median 55.

Therefore we can say that b = a, and d = 55.

Let us then solve for the value of a.

a+b+c+d+e = 55(5) = 275.

b = a
c = 55
d = 55
e = 3a + 20

a + a + 55 + 55 + 3a + 20 = 275

5a = 145

a = 29.
e = 3(29) + 20 = 107

Range = e - a = 107 - 29 = 78

The answer is A
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Re: Set R contains five numbers that have an average value of 55 [#permalink] New post  17 Mar 2021, 08:36
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.


Hello! I had one doubt. Since median = mean. we can conclude that this is an evenly spaced set. Then can we take 5 values as x, x+y, x+2y and so on. Please help.
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