After replacing an old member by a new member, it is 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 member and the new member?

Introduction / Context:
This is a reasoning based average age question concerning a small group of club members. We are told that after a period of 3 years and the replacement of one member by another, the average age of the group is the same as it was 3 years earlier. This means that the effect of natural ageing has been exactly balanced by the change in membership. We must translate this information into a relationship between the ages of the old and new member.

Given Data / Assumptions:

There are 5 members in the club at any time.
Consider the situation 3 years ago and the situation now after one replacement.
After 3 years and replacing one old member with a new one, the average age of the 5 members now is equal to the average age 3 years ago.
We must find the difference between the age of the replaced member and that of the new member.
All ages are measured in years.

Concept / Approach:
If no one had been replaced, the age of each of the 5 members would have increased by 3 years over 3 years, so the total age of the group would have increased by 5 * 3 = 15 years. However, we are told that after replacement, the average age has not increased at all compared to 3 years ago, which means the total age is the same as it was 3 years earlier. Therefore the replacement must have effectively decreased the total age by exactly 15 years, and this difference comes from the difference between the ages of the old and new member.

Step-by-Step Solution:
Step 1: Consider what happens in 3 years with no replacement. If all 5 original members remained, each would be 3 years older after 3 years. Total age increase would be 5 * 3 = 15 years. Step 2: Use the given condition about average age. The question states that after 3 years and replacing one member, the average age now is the same as the average age 3 years ago. That means the total age of the group now is the same as the total age 3 years ago. Step 3: Relate this to the replacement. If no one had been replaced, the total would be old total + 15. But with replacement, the total is only old total. Therefore, the replacement reduced the total age by exactly 15 years. Step 4: Interpret the reduction. This reduction in total age is exactly the difference between the age of the replaced old member and the age of the new member. Hence, difference in ages = 15 years.

Verification / Alternative Check:
To verify, we can imagine a simple example. Suppose the total age of the 5 members 3 years ago was T. If everyone stayed, the total now would be T + 15. For the average to remain the same, after replacement the total must still be T. This means the replacement change has reduced the total by 15. Hence the old member must have been 15 years older than the new member, confirming the reasoning.

Why Other Options Are Wrong:
Option 16 years would mean the total age decreased by 16 years, which would make the total age now smaller than it was 3 years ago, contradicting the balance with ageing.
Option 14 years or 13 years would not fully offset the natural 15 year total increase that would have occurred due to 5 members ageing for 3 years.

Common Pitfalls:
Some students mistakenly think the difference is 3 years or confuse the concept of average with individual changes. Others try to assign arbitrary ages instead of reasoning about total change over 3 years. Forgetting that 5 members each age by 3 years and therefore contribute a total increase of 15 years often leads to incorrect conclusions.

Final Answer:
The difference between the ages of the replaced and the new member is 15 years.