[Flatbutt1]

Originally posted by T_Samner
You’re being too hard on yourself. I have quite a bit of math and I had to shake the cobwebs off before I could get into it.

--

The counting numbers are just the numbers 1,2,3,4 and so on. The OP said there are eight numbers in the data set. If we label the smallest number as X1 and then the next largest number as X2 and so on up to the largest number X8 we can then line them up in order of their increasing value such as

X1 X2 X3 X4 X5 X6 X7 X8.

The median is defined (see links on page 1) to be the average of the two center terms. In this case that would be (X4 + X5) /2. We are told that the median is 8 so that (X4 + X5) / 2 = 8 and solving for X5 we get

X5 = 16 - X4 (*)

We are also told that the range is 8 so that means that the largest value in the set, X8, is equal to X1 + 8 or equivalently

X1 = X8 - 8 (*)

We are also told that the average or mean of the eight values is 8 so we know that

(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8) / 8 = 8

or

(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8) = 64 (Equation #1)

Substituting the values we found for the variables X5 and X1 above into (Equation #1) yields

[ (X8 - 8) + X2 + X3 + X4 + (16 - X4) + X6 + X7 + X8 ] = 64 .

Simplifying that gives

X2 + X3 + X6 + X7 + 2(X8) + 8 = 64

or

X2 + X3 + X6 + X7 + 2(X8) = 56 (Equation #2)

The mode of a data set is the number that occurs most often in the set and we are told that the unique mode is 8. So we know the number 8 occurs more often in the data set than any other number so its frequency of occurrence must be at least two (or it wouldn’t occur more frequently than any other number in the set). The question was to find the largest possible number in the data set given the constraints that were mentioned in the original post so we want to maximize the value of the variable X8 in Equation #2 considering the fact that the terms on the left side of the equation must sum to 56. We do this by assuming that the variables X6 and X7, which are the next two largest values in equation 2 after the variable X8, are the mode - meaning both X6 and X7 are equal to 8. So substituting X6 = 8 and X7 = 8 into equation 2 and then simplifying gives

X2 + X3 + 2(X8) = 40 (Equation #3)

The only information or constraint that we can still apply at this point is that X8 - X2 must be less than or equal to 8 (because the range of the data set is 8) and similarly X8 - X3 must be less than or equal to 8. So Setting both X8 - X2 and X8 - X3 equal to 8 implies that X2 = X8 - 8 and X3 = X8 -8. Substituting these values for X2 and X3 into equation 3 gives

(X8 - 8 ) + (X8 - 8) + 2(X8) = 40

or

4(X8) = 56

and therefore

X8 = 14.

So we know that X8 is equal to 14 and in the calculations above we found that X1 = X8 - 8, X2 = X8 - 8 , X3 = X8 -8, X6 = 8 and X7 = 8 so we can solve for those variables and then fill in those values of the data set. We get

6, 6, 6, __, __, 8, 8, 14 .

We know that X4 and X5 must be between 6 and 8 in value and we also know (from the discussion above) that (X4 + X5) / 2 = 8 . This implies directly that both X4 and X5 must also equal 8. Filling in the blanks yields:


6, 6, 6, 8, 8, 8, 8, 14 .

This is waayyyyy to complicated 4me