Solve the following question and choose the correct option for the question as given below:
The average age group of eight members is the same as it was \[3\] years ago when a young member is substituted for an old member. The incoming member is younger to the outgoing member by
A. \[11\] years
B. \[24\] years
C. \[28\] years
D. \[16\] years

Hint: We take the mean of the eight members as a variable and assume the ages of the eight members as the variables. Simplifying, we bring the common terms to one side and write it in the terms of the common values. We then take the age of the incoming member as a variable and substitute the values in the needed areas, simplifying then getting the final answer.

Complete step-by-step solution:
Let us assume that the average age of eight members be \[\overline x \]
Let the ages of the eight members be \[{a_1},{a_2},{a_3},{a_4},{a_5},{a_6},{a_7},{a_8}\]
The mean \[\overline x = \dfrac{{{a_1} + {a_2} + {a_3} + {a_4} + {a_5} + {a_6} + {a_7} + {a_8}}}{8}\]
Taking the denominator to the other side, we get;
$\Rightarrow$\[{a_1} + {a_2} + {a_3} + {a_4} + {a_5} + {a_6} + {a_7} + {a_8} = 8\overline x \]
This can also be written as;
$\Rightarrow$\[{a_2} + {a_3} + {a_4} + {a_5} + {a_6} + {a_7} + {a_8} = 8\overline x - {a_1}\]
Let the age of the old member be \[{a_1}\]
Let the age of the young member be \[{a_9}\]
According to the given condition, \[3\] years ago, the young member became a substitute for the old member.
Also, given that the average is the same even after substitution.
So, subtracting \[3\] from every age to find the age \[3\] years ago, then calculating the mean, we get;
$\Rightarrow$\[\overline x = \dfrac{{\left( {{a_9} - 3} \right) + \left( {{a_2} - 3} \right) + \left( {{a_3} - 3} \right) + \left( {{a_4} - 3} \right) + \left( {{a_5} - 3} \right) + \left( {{a_6} - 3} \right) + \left( {{a_7} - 3} \right) + \left( {{a_8} - 3} \right)}}{8}\]
Simplifying the above equation, we get;
$\Rightarrow$\[\overline x = \dfrac{{{a_9} + {a_2} + {a_3} + {a_4} + {a_5} + {a_6} + {a_7} + {a_8} - 3(8)}}{8}\]
Rearranging the terms, we get;
$\Rightarrow$\[{a_9} + {a_2} + {a_3} + {a_4} + {a_5} + {a_6} + {a_7} + {a_8} - 24 = 8\overline x \]
Substituting the above acquired value here, we get;
$\Rightarrow$\[{a_9} + 8\overline x - {a_1} - 24 = 8\overline x \]
Simplifying the terms, we get;
$\Rightarrow$\[{a_9} - {a_1} = 24\]

That implies that the incoming member is \[24\] years younger than the outgoing member.

Note: We have to remember that, in colloquial language, an average is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often, “average” refers to the arithmetic mean, the sum of the number divided by how many numbers are being averaged. In statistics, mean, median and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an average value.