The average (arithmetic mean) of six numbers is 18 and the median of

Question

The average (arithmetic mean) of six numbers is 18 and the median of the six numbers is 16. What is the minimum possible value for the greatest number in the list?

A) 19

B) 20

C) 21

D) 22

E) 23

sumitkrocks · Answer

First four numbers in the list of ascending numbers will be equal to median and the last two will be equal in order to make a set to get the minimum value of highest number

the minimum value of highest number will be =  \frac{(18*6 - 4*16) }{2}   = 22
(D) is correct IMO

GMATinsight · Answer

- - -  16  - - -

Average = 18
Sum = 18*6 = 108

To minimize the greatest value all other values must be maximized

The maximum values before Median may be EQUAL to Median
i.e. 16. 16. 16. _. _. x

The value immediately after the median shoul dbe equal to 16 so that average two middle terms is 16
ie. i.e. 16. 16. 16. 16. _. x

The maximum values after the median must be less than or equal to the highest value of set i.e. x

ie. i.e. 16. 16. 16. 16. x. x

Sum = 2x+64 = 108
i.e. x = 22

Answer: Option D

sanjitscorps18 · Answer

To ensure we have the minimum value of the greatest number, we need to maximize the numbers around the median.

Hence, we choose 16 as the first 4 numbers to maintain 16 as median. In order to maintain mean 18, the maximum value of the greatest number would be 22

This is achieved by equating both the remaining numbers.

Final number = (108 - 64)/2 = 22

16
16
16
16
22
22

Posted from my mobile device

Nefertiti9 · Answer

why cant we take the first five items here as 16 ?

rahulm3564 · Answer

My Approach:

Numbers are a>b>c>d>e>f

a+b+c+d+e+f = 108
=> a+b+e+f = 70

now, maximum values of a,b,e would be (f-1).

Can someone please tell me where am I going wrong? Why is this approach not working?

Kinshook · Answer

Given: The average (arithmetic mean) of six numbers is 18 and the median of the six numbers is 16. 
Asked: What is the minimum possible value for the greatest number in the list?

Total of 6 numbers = 6*18 = 108
Median of 6 numbers = 16

To minimise greatest number in the list, we have to maximise other numbers.
Let the first 4 numbers be equal to median and last 2 numbers be x.

16*4 + 2x = 6*18 = 108
2x = 108 - 64 = 44
x = 22

IMO D

760Abhi · Answer

Because we have to minimize last term (6th term). You can only take first 4 terms as 16, not 5th term.

Vignesh2340 · Answer

Since they are asking for the greatest number, they are asking for that one number which is greater than the other 5 numbers in the list right? then we can reduce the 5th digit to 21, hence answer will be 23 right? kindly help me understand the question.

SKGM · Answer

If you reduce the 5th digit to 21, then the answer is 23. But the question asks for the minimum possible value for the greatest number in the list. So you have to consider the 5th digit as 22 thus getting the 6th digit as 22. In this case 22 < 23. 
Rest the problem has been nicely solved by  GMATinsight  and  sanjitscorps18

ScottTargetTestPrep · Answer

Since the average of the six numbers is 18, the sum of the six numbers is 6 * 18 = 108. Since there are six numbers, the median is the average of the numbers in the third and fourth positions. Thus, the sum of the third and fourth largest numbers is 2 * 16 = 32. It follows that the sum of the largest, second largest, smallest, and second smallest numbers is 108 - 32 = 76.

To find the minimum value of the largest number in the list, we need to maximize the remaining numbers. Since the median is 16, the maximum possible value of the smallest and second smallest numbers is 16 (which means the third and fourth numbers are also both 16). Thus, the sum of the largest and second largest numbers is 76 - 32 = 44. The smallest value of the largest number is obtained when we assume the largest and second largest numbers are equal, thus the answer is 44/2 = 22.

Answer: D

shahsahil2101 · Answer

Answer: D 
The question says, the average is 18 for six numbers, hence the summation of all the numbers will be 108.

For the median, that is 16, as the number of terms are even (6), the average of the third and fourth number should be 16, so to minimize the maximum number, and there is no condition given that the numbers need to be unique, we can take 3rd and 4th term to be 16. 
With the same logic, 1st and 2nd term can be 16.

Now the addition of the first four terms is 64 and the total is 108, subtracting both, we get 44. Now the last two terms can be equal to minimize the maximum term.
Evaluating this, we get 44/2=22 as the final answer

The average (arithmetic mean) of six numbers is 18 and the median of

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The average (arithmetic mean) of six numbers is 18 and the median of

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