After replacing an old member of a club by a new member, it is found that the average age of the five club members is the same as it was 3 years ago. What is the difference in years between the age of the old member who left and the new member who joined?

Introduction / Context:
This question combines changes in average age over time with the effect of replacing one member of a group. It is a classic type of age problem where time progression and replacement both influence the average.

Given Data / Assumptions:
- Original number of club members = 5.
- Consider the average age 3 years ago to be A years.
- After 3 years, each original member is 3 years older.
- One old member is replaced by a new member.
- The average age after replacement is equal to the average age 3 years ago, that is A years.

Concept / Approach:
We express the total age 3 years ago, the total age just before replacement, and the total age after replacement. Then we equate the average after replacement to the old average from 3 years ago to find the difference between the age of the member who left and the age of the new member.

Step-by-Step Solution:
Step 1: Let the average age 3 years ago be A.Step 2: Total age of the 5 members 3 years ago = 5A.Step 3: After 3 years, each gains 3 years, so total age becomes 5A + 5 * 3 = 5A + 15.Step 4: Suppose the age of the member who leaves now is x years.Step 5: The new member who joins has age y years.Step 6: Total age after replacement = (5A + 15 - x) + y.Step 7: The new average is equal to the old average A, so (5A + 15 - x + y) / 5 = A.Step 8: Multiply both sides by 5: 5A + 15 - x + y = 5A.Step 9: Simplify: 15 - x + y = 0, so y - x = -15 and therefore x - y = 15.Step 10: Thus the old member is 15 years older than the new member, so the difference in their ages is 15 years.

Verification / Alternative check:
If the member who leaves is 15 years older than the new member, the total age of the group decreases by 15 due to replacement. Over the 3 years, the total age had increased by 15 (since 5 members each aged 3 years). The increase of 15 from aging and the decrease of 15 from replacement cancel, keeping the total age equal to what it was 3 years ago, which makes the average the same as 3 years ago.

Why Other Options Are Wrong:
- 16, 14 and 13 years would not exactly balance the change in total age due to 3 years of aging of 5 people, which is 15 years. Only a 15 year difference correctly offsets the 15 year increase.

Common Pitfalls:
- Misinterpreting the statement and equating the average after replacement to the average just before replacement, instead of to the average 3 years ago.
- Forgetting that all 5 original members age by 3 years, which adds 15 to the total.

Final Answer:
The difference between the old and new member's ages is 15 years.