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The median value of a set of 9 positive integers is 6. If the only mod

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The median value of a set of 9 positive integers is 6. If the only mod  [#permalink]

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New post 20 Sep 2019, 05:02
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The median value of a set of 9 positive integers is 6. If the only mode in the set is 25 what is the least possible value of the average (arithmetic mean) of the set?

A. 8
B. 9
C. 10
D. 11
E. 12

The median value of a set of 9 positive integers is 6. If the only mode in the distrubution is 25 what is the least possible mean of the distribution.

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Re: The median value of a set of 9 positive integers is 6. If the only mod  [#permalink]

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New post 20 Sep 2019, 05:35
Bunuel wrote:
The median value of a set of 9 positive integers is 6. If the only mode in the set is 25 what is the least possible value of the average (arithmetic mean) of the set?

A. 8
B. 9
C. 10
D. 11
E. 12

The median value of a set of 9 positive integers is 6. If the only mode in the distribution is 25 what is the least possible mean of the distribution.


Since we have to minimize the average, and the mode is 25. So, 25 occurs only two times. Rest all digits will be distinct and occur only once.
The median is given as 6 so the 5th digit is 6.
In order to minimise the average the set of numbers will be
{1,2,3,4,6,7,8,25,25}.
Also, as the question statement says positive, so we cannot consider 0.

The sum of these numbers is 81 and the average becomes 9.
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Re: The median value of a set of 9 positive integers is 6. If the only mod  [#permalink]

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New post 20 Sep 2019, 06:00
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Bunuel wrote:
The median value of a set of 9 positive integers is 6. If the only mode in the set is 25 what is the least possible value of the average (arithmetic mean) of the set?

A. 8
B. 9
C. 10
D. 11
E. 12


Let's add 9 slots to represent the 9 values in ASCENDING order: __, __, __, __, __, __, __, __, __

Since the median is 6, we get: __, __, __, __, 6 , __, __, __, __

If the mode is 25 and we're trying to MINIMIZE the average of the 9 numbers, we should add TWO 25's to the list
We get: __, __, __, __, 6 , __, __, 25 , 25

At this point, let's make the remaining values as small as possible, without repeating any values (since we need the mode to be 25)
We get: 1, 2, 3, 4, 6, 7, 8, 25, 25

Average \(= \frac{1+2+3+4+6+7+8+25+25}{9}=\frac{81}{9}=9\)

Answer: B

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Re: The median value of a set of 9 positive integers is 6. If the only mod  [#permalink]

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New post 20 Sep 2019, 11:47
least possible combination of positive numbers:
Median - 6
Mode - 25

1, 2, 3, 4, 6, 7, 8, 25, 25

Avg : 81/9 = 9

Option B
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Re: The median value of a set of 9 positive integers is 6. If the only mod  [#permalink]

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New post 23 Sep 2019, 13:23

Solution


Given:
    • The median value of a set of 9 positive integers is 6
    • The only mode of the set is 25

To find:
    • The least possible value of the arithmetic mean of the set

Approach and Working Out:
    • From the given information, we can say that the 9 positive integers in increasing order can be written as: a, b, c, d, 6, e, f, 25, 25
      o Since, 25 is the only mode, it must be repeated at least once, and no other number can be repeated in the set

For the mean to be minimum, the values of {a, b, c, d, e, f} must be minimum
    • The minimum values that a, b, c, d can take are 1, 2, 3 and 4 respectively
    • The minimum values that e, f can take are 7 and 8 respectively

Therefore, least possible value of mean = \(\frac{(1 + 2 + 3 + 4 + 6 + 7 + 8 + 25 + 25)}{9} = \frac{81}{9} = 9\)

Hence, the correct answer is Option B.

Answer: B
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Re: The median value of a set of 9 positive integers is 6. If the only mod  [#permalink]

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New post 23 Sep 2019, 13:42
Great question . The mode is 25 . 25 has to occur more than once else it would be one of several modes.However we don't want 25 to occur too many times and we do not want to include any number greater than 25 since it would increase the mean. so we will be have a set looking like this for now : a,b,c,d,6,e,f,25,25. Now if 6 is the median,it is greater than exactly half the elements especially since it is the 5th number in a set of 9 numbers. Again 6 can not be repeated so we go for the least positive integer greater than 6, which will be 7.The smallest integer greater than 7 is 8. Now for the numbers beneath 6 we need 4 different integers that are lesser than 6. The only logical choices are 1,2,3 and 4. This is how our set looks : 1,2,3,4,6,7,8,25,25. The mean of the set is 9.


Answer is B
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Re: The median value of a set of 9 positive integers is 6. If the only mod  [#permalink]

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New post 29 Sep 2019, 19:42
Bunuel wrote:
The median value of a set of 9 positive integers is 6. If the only mode in the set is 25 what is the least possible value of the average (arithmetic mean) of the set?

A. 8
B. 9
C. 10
D. 11
E. 12

The median value of a set of 9 positive integers is 6. If the only mode in the distrubution is 25 what is the least possible mean of the distribution.


To minimize the average, we have to minimize the sum of the 9 numbers. To meet the requirements, 25 must be the only number that occurs two times, the number in the 5th position (the median) should be 6, the numbers in the first four positions should be as small as possible (i.e. 1, 2, 3 and 4), and the numbers in the 6th and 7th positions should also be as small as possible (i.e. 7 and 8). Wethus have the following sum:

1 + 2 + 3 + 4 + 6 + 7 + 8 + 25 + 25 = 81

So the average is 81/9 = 9.

Answer: B
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Re: The median value of a set of 9 positive integers is 6. If the only mod   [#permalink] 29 Sep 2019, 19:42
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