After replacing an old member by a new number, it was found that the average age of five members of a club is the same as it was $3$ years ago. What is the difference between the ages of the replaced and the new member?
A. $2$ years
B. $4$ years
C. $8$ years
D. $15$ years

Hint: To find the difference between the ages of the replaced and the new member, we need to first find the value of the new member. For that, we need to average the five members. The average is nothing but the ratio of the sum of ages to the total number of members whose ages are added in the numerator. If it was $3$ years ago, we need to subtract each age by $3$.

Complete step-by-step answer:
Given,
After replacing an old member with a new member, it was found that the average age of five members of a club is the same as it was $3$ years ago.
If it is written in mathematical form,
Let the new member be ${x_6}$
The five members of the age are ${x_5},{x_2},{x_3},{x_4}$and ${x_6}$
The average of the new members is $\dfrac{{{x_5} + {x_2} + {x_3} + {x_4} + {x_6}}}{5}$
The five members of the age before $3$years ago are ${x_1} - 3,{x_2} - 3,{x_3} - 3,{x_4} - 3$and ${x_5} - 3$
The average of the members of the age before $3$years ago is $\dfrac{{{x_1} - 3 + {x_2} - 3 + {x_3} - 3 + {x_4} - 3 + {x_5} - 3}}{5}$
Equating both equations we get,
$\dfrac{{{x_5} + {x_2} + {x_3} + {x_4} + {x_6}}}{5} = \dfrac{{{x_1} - 3 + {x_2} - 3 + {x_3} - 3 + {x_4} - 3 + {x_5} - 3}}{5}$
Canceling the terms on the denominator, we get
${x_5} + {x_2} + {x_3} + {x_4} + {x_6} = {x_1} - 3 + {x_2} - 3 + {x_3} - 3 + {x_4} - 3 + {x_5} - 3$
Equating the like terms in the above equation, we get
${x_5} + {x_2} + {x_3} + {x_4} + {x_6} - ({x_1} - 3) - ({x_2} - 3) - ({x_3} - 3) - ({x_4} - 3) - ({x_5} - 3) = 0$
Subtracting the like terms in the above equation, we get
${x_6} - {x_1} + 3 + 3 + 3 + 3 + 3 = 0$
Adding the terms in the above equation, we get
${x_6} - {x_1} + 15 = 0$
We need to find the difference between the ages of the replaced and the new member are
${x_1} - {x_6} = 15$
The difference between the ages of the replaced and the new member is $15$.

So, the correct answer is “Option A”.

Note: We need to find the difference between the ages of the replaced and the new member. We can not find the new member age or old member age because the given data is not enough. The average is the ratio of the sum of ages to the total number of members whose ages are added in the numerator. If it was $3$ years ago, we need to subtract each age by $3$. Do not ever forget this.