A data set consists of 6 different whole numbers. The difference between the largest value and the smallest value is 21 and the median is 40. What is the largest possible mean of this data set?

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

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step-by-step explanation: