A set of five different positive integers has a mean of 19 and a median of 21. What is the greatest value of the greatest integer in the set?
If the mean of 5 numbers is 19 then the sum of the numbers is 5*19 = 95.
If the median is 21, then there are two numbers greater than 21 and two less.
so, the greatest value of the largest member of the set will be when the set contains
1 2 21 22 49
1,2,21,22,49
Yes
To find the greatest value of the greatest integer in the set, we need to consider the given conditions.
Let's call the five positive integers a, b, c, d, and e. Since the median is 21, the third-largest integer out of these five numbers should be 21.
This means that there are two numbers smaller than 21 and two numbers larger than 21. Since the mean is 19, the sum of all five integers divided by 5 equals 19.
Thus, the sum of the five integers is 19 * 5 = 95.
Now, if we want to maximize the greatest value in the set, we should minimize the remaining four numbers. The sum of the four smallest numbers should be the smallest possible while being larger than 21 (to keep the median value).
To minimize the sum of the four smallest numbers, we can assign them the smallest possible values that are greater than 21. The smallest positive integers greater than 21 are 22, 23, and so on.
So, let's assign the four smallest numbers as follows:
a = 22
b = 23
c = 24
d = 25
Now, we can find the value of the greatest integer (e).
To do this, we subtract the sum of a, b, c, and d from the total sum of 95:
e = 95 - (a + b + c + d) = 95 - (22 + 23 + 24 + 25) = 95 - 94 = 1
Therefore, the greatest value of the greatest integer in the set is 1.