Question

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 (arithmetic mean) of set N?

Answer 1

`(17)/(9)x`.

Explanation:

It is given that N is the set of 9 positive integers and all the values in Set N are 3x or smaller.

Median is the middle term of the data. The number of observation is 9, which is an odd number.

`Median=(n+1)/(2)\text{th term}=(9+1)/(2)\text{th term}=5\text{th term}`

It is given that the median of Set N is the integer x. It means median of the data is x. 4 terms are less than or equal to x and remaining 4 terms are more than x.

We need to find the largest possible average. So, the data values must be maximum. Let the given data set is

x, x, x, x, x, 3x, 3x, 3x, 3x

Formula for Mean:

`Mean=\frac{\text{Sum of observations}}{\text{Number of observations}}`

`Mean=(x+x+x+x+x+3x+3x+3x+3x)/(9)`

`Mean=(17x)/(9)`

Therefore, the largest possible average (arithmetic mean) of set N is
`(17)/(9)x`.