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Question of the week - 32 (The arithmetic mean of a sequence of 15 ..)

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Question of the week - 32 (The arithmetic mean of a sequence of 15 ..)  [#permalink]

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New post 18 Jan 2019, 00:05
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The arithmetic mean of a sequence of 15 integers is 78. The largest value of the set is five times the smallest value of the set. If the mean of the set is equal to the median, then what is the maximum possible value for the range of the set?

    A. 52
    B. 104
    C. 112
    D. 208
    E. 260

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Re: Question of the week - 32 (The arithmetic mean of a sequence of 15 ..)  [#permalink]

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New post 18 Jan 2019, 01:53
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If we want maximum range, we want 5x to be as large as possible and x to be as small as possible
x - - - - - - 78 - - - - - - 5x

This is only possible if all the other numbers are 78
13*78 = 1014

Total sum of 15 numbers = 1170
Subtracting sum of 13 no.s from sum of 15 no.s = 1170-1014
= 156

which is equal to x+5x = 6x=156
x=26

Range = 5x-x=4x=4*26=104
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Re: Question of the week - 32 (The arithmetic mean of a sequence of 15 ..)  [#permalink]

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New post 18 Jan 2019, 04:08
My approach for this problem was

Sum of nos. 78*15=1170
Sum of nos of a sequence= n/2*(a+an)
Where n total nos of digit, a = first digit and an = last digit
Therefore, 15/2*(a+an)= 1170

a+an=156
an = 5a (given in question)
6a=156
a=26, an=130
Range =130-26, 104 (B)

Feel free for correction ?

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Question of the week - 32 (The arithmetic mean of a sequence of 15 ..)  [#permalink]

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New post Updated on: 21 Jan 2019, 06:34
EgmatQuantExpert wrote:
The arithmetic mean of a sequence of 15 integers is 78. The largest value of the set is five times the smallest value of the set. If the mean of the set is equal to the median, then what is the maximum possible value for the range of the set?

    A. 52
    B. 104
    C. 112
    D. 208
    E. 260

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sequence
XXXXXXX 787878787878 5X = 78* 15
12X+ 78*7 = 1170
X= 52
5X= 260
RANGE = 5X-X = 260-52 = 208
IMO D
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Originally posted by Archit3110 on 18 Jan 2019, 07:36.
Last edited by Archit3110 on 21 Jan 2019, 06:34, edited 1 time in total.
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Question of the week - 32 (The arithmetic mean of a sequence of 15 ..)  [#permalink]

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New post Updated on: 18 Jan 2019, 11:45
We have to minimise X and maximize 5x.

So
X X X X X X X 78 78 78 78 78 78 78 5X

Is the correct sequence.


Hence
7*(78-X)=5X-78
8*78=12X

Hence X=52
4X=208

Option D is correct.

Originally posted by nitesh50 on 18 Jan 2019, 07:42.
Last edited by nitesh50 on 18 Jan 2019, 11:45, edited 2 times in total.
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Question of the week - 32 (The arithmetic mean of a sequence of 15 ..)  [#permalink]

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New post 18 Jan 2019, 09:30
EgmatQuantExpert wrote:
The arithmetic mean of a sequence of 15 integers is 78. The largest value of the set is five times the smallest value of the set. If the mean of the set is equal to the median, then what is the maximum possible value for the range of the set?

    A. 52
    B. 104
    C. 112
    D. 208
    E. 260


(x+5x)/2=78 mean/median
x=26
5x=130
130-26=104
B
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Re: Question of the week - 32 (The arithmetic mean of a sequence of 15 ..)  [#permalink]

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New post 18 Jan 2019, 11:00
1
We have to maximise the value of 5x and minimise the value of x such that the sum of the sequence remains 15*78=1170.

Hence, the sequence can be where mean is equal to median

x,x,x,x,x,x,x, 78, 78,78,78,78,78,78, 5x

The sum of the above sequence is 7x+ 78*7 + 5x = 15*78-----> 12x= 8*78-----> x=52

Hence the maximum range of the value of the set is 5x-x= 4x= 4* 52= 208

Hence answer is D.,
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Question of the week - 32 (The arithmetic mean of a sequence of 15 ..)  [#permalink]

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New post 19 Jan 2019, 00:57
My approach was:
l=5a

Median=(a+l)/2=Mean=78

-->>:::Range=l-a=104

I think I missed the point of maximizing the Range but accidentally did so...!!?
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Re: Question of the week - 32 (The arithmetic mean of a sequence of 15 ..)  [#permalink]

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New post 22 Jan 2019, 23:49

Solution


Given:
    • The arithmetic mean of a sequence of 15 integers is 78
    • The largest value of the set is five times the smallest value of the set
    • The mean of the set is equal to the median

To find:
    • The maximum possible value for the range of the set

Approach and Working:
    • We know that the range of a set = the largest value – the smallest value

So, for the range to be maximum, we need to maximise the value of largest element of the set and minimise the value of smallest element of the set.
    • We know that the mean = the median = the 8th element of the sequence = 78
    • So, to minimise the first element of the sequence, all the elements before the median element must be same. Let us assume it be “a”
    • And, to maximise the last element of the sequence, all the elements after the median except the last element must be minimum = the median = 78
    • The last element of the sequence will be = 5 * a

We are also given that the mean of the set = median = 78
    • Implies, sum of all 15 elements in the sequence = 78 * 15
    • Thus, a + a + a + a + a + a + a + 78 + 78 + 78 + 78 + 78 + 78 + 78 + 5a = 78 * 15
      o 12a = 78 * 15 – 78 * 7
      o \(a = 78 * \frac{8}{12} = 52\)

    • Therefore, range of the set = 5a – a = 4a = 4 * 52 = 208

Hence the correct answer is Option D.

Answer: D

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Re: Question of the week - 32 (The arithmetic mean of a sequence of 15 ..)  [#permalink]

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New post 25 Jan 2019, 11:41
My question is why is this being assumed
"Thus, a + a + a + a + a + a + a + 78 + 78 + 78 + 78 + 78 + 78 + 78 + 5a = 78 * 15"
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Re: Question of the week - 32 (The arithmetic mean of a sequence of 15 ..)   [#permalink] 25 Jan 2019, 11:41
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