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Question: After replacing an old member by a new number, it was found that the average age of five members of ...

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 33 years ago. What is the difference between the ages of the replaced and the new member?
A. 22 years
B. 44 years
C. 88 years
D. 1515 years

Explanation

Solution

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 33 years ago, we need to subtract each age by 33.

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 33 years ago.
If it is written in mathematical form,
Let the new member be x6{x_6}
The five members of the age are x5,x2,x3,x4{x_5},{x_2},{x_3},{x_4}and x6{x_6}
The average of the new members is x5+x2+x3+x4+x65\dfrac{{{x_5} + {x_2} + {x_3} + {x_4} + {x_6}}}{5}
The five members of the age before 33years ago are x1−3,x2−3,x3−3,x4−3{x_1} - 3,{x_2} - 3,{x_3} - 3,{x_4} - 3and x5−3{x_5} - 3
The average of the members of the age before 33years ago is x1−3+x2−3+x3−3+x4−3+x5−35\dfrac{{{x_1} - 3 + {x_2} - 3 + {x_3} - 3 + {x_4} - 3 + {x_5} - 3}}{5}
Equating both equations we get,
x5+x2+x3+x4+x65=x1−3+x2−3+x3−3+x4−3+x5−35\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
x5+x2+x3+x4+x6=x1−3+x2−3+x3−3+x4−3+x5−3{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
x5+x2+x3+x4+x6−(x1−3)−(x2−3)−(x3−3)−(x4−3)−(x5−3)=0{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
x6−x1+3+3+3+3+3=0{x_6} - {x_1} + 3 + 3 + 3 + 3 + 3 = 0
Adding the terms in the above equation, we get
x6−x1+15=0{x_6} - {x_1} + 15 = 0
We need to find the difference between the ages of the replaced and the new member are
x1−x6=15{x_1} - {x_6} = 15
The difference between the ages of the replaced and the new member is 1515.

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 33 years ago, we need to subtract each age by 33. Do not ever forget this.