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

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

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

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

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

Sir, similar to this question: The median of 7 positive integers is 10, and the average of these 7 numbers is 9. The only mode is 2. What is the greatest possible number in the set?
A 22

B 23

C24

D 25

E. 26

I got 25 as an answer but answer key is saying it is 26. Where am I making mistakes sir? I arranged no like this, 1,2,2,10,11,12,25.