eskay1981 wrote:
Carcass wrote:
If the average (arithmetic mean) of five distinct positive integers is 10, what is the difference between the largest possible value of the greatest integer and the least possible value of the greatest of the five integers?
Please clarify the question, "what is the difference between the
largest possible value of the greatest integer and the least possible value of the greatest of the five integers"
OE
Quote:
Given:
Average (arithmetic mean) of 5 distinct positive integers is 10.
We have to find out the difference between the largest possible value of the greatest integer and the least possible value of the greatest of the five integers?
It is to be noted that we have to twice find out the value of the greatest integer (once the largest possible value and once the least possible value) keeping in mind that all the 5 integers are distinct i.e. they are unique (none is repeated).
Let the five numbers be v, w, x, y and z such that v>w>x>y>z; it is given that none of them are equal.
Let us first find the largest possible value of the largest integer among the five, i.e. v.
We must aim to keep the integers (w, y, x and z) as small as possible such that they are distinct.
Thus, z=1, (smallest possible positive integer), x=2, y=3 and w=4.
Since the average of the five integers =10, the sum of the integers =5×10=50.
⇒v=(Sum of the five numbers)−(Sum of the four numbers)
⇒v=50−(1+2+3+4)=40
Thus, the largest possible value of the greatest of the five numbers =40.
Let us now find the least possible value of the greatest among the five, i.e. v.
We must aim to keep all the five integers as close to each other as possible.
Let us assume that the middle-most integer x=10, because the average is 10.
Let's make y less than x by the minimum possible positive value, i.e., 1, thus y=10−1=9; similarly, z=9−1=8.
Likewise, w=10+1=11 and v=11+1=12.
Thus, we get the values:
v=12, w=11, x=10, y=9, and z=8.
Thus, the least possible value of the greatest of the five numbers =12.
Thus, the difference between the largest possible value of the greatest integer and the least possible value of the greatest of the five integers =40−12=28.