→ The answer is: " 68 " .
→ The largest integer in the set is: " 68 " .
Step-by-step explanation:
Given: There are 8 (eight) consecutive even integers in the data set.
Given: The "median" of the data set is: " 61 " .
We are asked to the find the "largest integer" in the data set.
→ Let us write the values within the data set in order from smallest value to largest value.
Let:
" x " ; represent the 1st (first) integer.
" (x + 2) " ; represent the 2nd (second) integer— (consecutive positive),
" (x + 4) " ; represent the 3rd (third) consecutive positive integer.
" (x + 6) " ; represent the 4th (fourth) consecutive positive integer.
" (x + 8) " ; represent the 5th (fifth) consecutive positive integer.
" (x + 10) " ; represent the 6th (sixth) consecutive positive integer.
" (x + 12) " ; represent the 7th (seventh) consecutive positive integer.
" (x + 14) " ; represent the 8th (eighth) consecutive positive integer ;
→ and the "largest integer" in the set.
→ By solving for "x" ; we can add " 14 " to our solved value for "x" to find:
"(x + 14)" ; → which is our answer.
Write the data set as follows:
→ " { x , (x + 2) , (x + 4) , (x + 6) , (x + 8) , (x + 10) , (x + 12) , (x + 14) . } " .
The "median" refers to the "value" of the "middle value" in a data set, when the data are are arranged in order from smallest to largest; {or from largest to smallest, for that matter}.
When there are an "even" number of values in the data set, there is no precise "middle value" in the data set that is the "median".
In our case, there are an "even" number of values in the data set — "eight" (8) .
In such cases, such as ours, we take the 2 (two) values in the data set that are closest to the "middle value" (when arranged in order from smallest to largest — {or, for that matter, largest to smallest} — and calculate the "mean" of those 2 (two) values. In such cases, the "mean" of those those 2 (two) values — is the "median" of the data set.
To calculate the "mean" :
We take the "{sum of the values}" ; and divide that value by the "number of values that have been added"} .
We are given that the "median" is: " 61 " .
The 2 (two) values closest to the middle in our data set {which we have arranged from smallest value to largest values are:
1) " (x + 6) " ; and: 2) " (x + 8) " .
So, to find the median, we add: " [ (x + 6) + (x + 8) ] " ;
and then divide the obtained value by " 2 " ;
→ {since there are 2 (two) values being added}.
We are given: The median is: " 61 " .
So:
→ " [ (x + 6) + (x + 8) ] / 2 = 61 " ;
→ " (x + 6 + x + 8) / 2 = 61 " ;
→ " (2x + 14) / 2 = 61 ;
↔ " 2x + 14 = (61 * 2) " ;
→ " 2x + 14 = 122 " ; Solve for "x" ;
→ Subtract " 14 " ; from each side of the equation:
→ " 2x + 7 - 14 = 122 - 14 " ;
to get: → 2x = 108 ;
→ Divide each side of the equation by "2" ;
to isolate "x" on one side of the equation; & to solve for "x" :
→ 2x / 2 = 108 / 2 ;
]to get: → x = 54 ;
Now, to solve for largest integer: "(x + 14)" :
→ x + 14 = 54 + 14 = 68 .
→ The answer is: " 68 " .
→ The largest integer in the set is: " 68 " .
Now, let us check our
→ Our solved value for "x" is "54" .
The eight (8) consecutive even integers, beginning with "54" , would end with the eighth (8th) -- final and largest integer -- being: " 68 " ??
→ { 54, 56, 58, 60, 62, 64, 66, 68 .} .
→ x: 54 ;
→ (x + 2) = 54 + 2 = 56 ;
→ (x + 4) = 54 + 4 = 58 ;
→ (x + 6) = 54 + 6 = 60 ;
→ (x + 8) = 54 + 8 = 62 ;
→ (x + 10) = 54 + 10 = 64 ;
→ (x + 12) = 54 + 12 = 66 ;
→ (x + 14) = 54 + 14 = 68 ;
→ Yes!
Hope this answer is helpful to you!
Wishing you well in your academic pursuits — and within the "" community!
