Problem Solving

Nine positive integers are contained in Set N. The median of Set N is the integer x, and all the values in Set N are 3x or smaller. What is the largest possible average(airthemetic mean) of Set N
1- x
2- 17x/9
3- (17x/9)-1
4- (17x-4)/9
5 - 3x

Please provide approach to the above question

We require the largest possible average.
The median of 9 positive integers is given as x
So the first four integers have to be x and the last four integers have to be as large as possible, in this case, 3x

x x x x x 3x 3x 3x 3x . Average is 17x/9

Median = x
All values in N <= 3x
Largest possible average = Average of {x,x,x,x,x,3x,3x,3x,3x)
Average = (5x+12x)/9
=17x/9

Answer = 2

Thanks for posting solution.. but could you please detailed the below part -

Largest possible average = Average of {x,x,x,x,x,3x,3x,3x,3x)

HImanshu, the goal here to find the largest average possible. That is possible by ensuring that
the four integers above the median be as large as possible. In this case, we are told that it is 3x. Similarly, the integers below the median should be equal to the median in order to achieve the maximum average. If it were greater than the median the median would change

Or you could take numbers to help you understand better

say x=2

2 2 2 2 2 6 6 6 6 Avg = 34/9

if we let the first four integers to be any less than 2, you will see that the average will decrease

Hi sl750,
Thanks for replying. Appreciated
However,I am afraid to say again that I didn't get this..
Is this a general rule or what?

imhimanshu wrote:Nine positive integers are contained in Set N. The median of Set N is the integer x, and all the values in Set N are 3x or smaller. What is the largest possible average(airthemetic mean) of Set N
1- x
2- 17x/9
3- (17x/9)-1
4- (17x-4)/9
5 - 3x

Please provide approach to the above question

To maximize the average, we need to maximize every value in Set N.
Let x=10.
Thus, the median -- or middle value -- is 10.

The 4 integers BELOW the median cannot be LARGER than the median of 10.
Thus, the largest possible combination of values for the 4 integers below the median is 10,10,10,10.

The 4 integers ABOVE the median can be any value greater than or equal to the median of 10.
The problem states that no value in the set is greater than 3x.
Since 3x = 3*10 = 30, no value in the set can be greater than 30.
Thus, the largest possible combination of values for the 4 integers above the median is 30,30,30,30.

Thus, the largest possible combination of values for set N = {10,10,10,10,10,30,30,30,30}.
Average = sum/number = (10+10+10+10+10+30+30+30+30)/9 = 170/9. This is our target.

Now we plug x=10 into the answers to see which yields our target of 170/9.

Only answer choice B works:
17x/9 = (17*10)/9 = 170/9.

The correct answer is B.

imhimanshu,

In any number sequence, the way to find a median is:
Arrange the numbers in increasing order
The middle number in this increasing order is the median, if the number of items is odd
The average of the two middle numbers is the median, if the number of items is even

We have 3 pieces of information from the question
1. There are nine positive integers
2. Median of set N is x
3. All values in set N are 3x or smaller

From first two pieces & the question itself:
For the largest average of the sequence, all the numbers should be as large as possible
So, with 9 positive integers, there should be 4 integers before x for x to be the median. These 4 integers should be as large as possible. How do you build this?
They can be less than or equal to x itself
So N={x,x,x,x,x...}

Using the third piece of information:
N={x,x,x,x,x,3x,3x,3x,3x)

Sorry to bring up this old thread again but I am not sure about one thing.
In the question it says "...and all the values in Set N are 3x or smaller" From this I am lead to believe that all other values in the set are either 3x or that they are smaller, not less than or equal to. When they say smaller I assume they can't equal x and therefore all other numbers before x are x-1 thus giving us an answer of (17x-4)/9.
I think the information given is a bit misleading.

Hi Rich,

Thanks for the quick reply. It was my fault misreading the question. It was a long day of studying and my brain was tired
Anyways, the mistake I made was that in the sentence "...and all the values in Set N are 3x or smaller" I thought they meant that all values are either 3x or smaller than x, rather than smaller than 3x. Otherwise, the question is pretty straight forward.
Thanks again.