Answer:
We have a data set with 6 whole numbers:
{A, B. C, D, E, F}
The median, in this case, will be the mean of the two middle numbers, C and D.
Median = (C + D)/2 = 40
We also know that the difference between the largest and the smallest number is 21, where the largest number is F and the smallest number is A, then:
F - A =21.
And the mean will be:
Now we want to take the largest possible numbers such that we end having the largest mean of the whole set, then we can start by taking:
(C + D)/2 = 40
Here we can just take:
C = 39
D = 41
Now our set is:
{A, B, 39, 41, E, F}
Now, A and B should be the largest possible whole numbers (and they must be smaller than 39
The two options are:
B = 38
A = 37
Now our set is:
{37, 38, 39, 41, E, F}
Now, remember that:
F - A = 21
F - 37 = 21
F = 21 + 37 = 58
Then our set will be:
{37, 38, 39, 41, E, 58}
And for E, we should pick the largest number that we could.
In this case, the only restrictions we have for E are that it must be larger than 41, and smaller than 58.
The largest number that meets those conditions is the number 57.
Then our set will be:
{37, 38, 39, 41, 57, 58}
And the mean of this set is:
Mean = (37 + 38 + 39 + 41 + 57 + 58)/6 = 45
