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

Introduction / Context:
This is a classic average-age puzzle. There are five members in a club. After 3 years, one original member leaves and a new one joins. Surprisingly, the average age of the five members now is the same as it was 3 years ago. You must determine the difference between the ages of the old member who left and the new member who joined, using the idea of total age and average.

Given Data / Assumptions:

• Number of members = 5.
• Three years pass between the two comparisons of average age.
• After 3 years, one old member leaves and a new member joins.
• The average age of the 5 members now equals the average age of the original 5 members 3 years ago.
• We need the difference between the age of the replaced member and the new member.

Concept / Approach:
Let the average age of the 5 members 3 years ago be A. Then the total age 3 years ago is 5A. After 3 years, if nobody left, each of the 5 members would be 3 years older, so the total age would increase by 5 * 3 = 15 years. However, one member leaves and a new member joins, and the resulting average returns to A. By carefully tracking the total ages, we can find the difference between the ages of the leaving and entering members.

Step-by-Step Solution:
Step 1: Let the average age 3 years ago be A years for the 5 members. Step 2: Total age 3 years ago = 5A. Step 3: After 3 years, if no one left, each person would gain 3 years, so total increase = 5 * 3 = 15 years. Thus, total age just before replacement = 5A + 15. Step 4: Let the age 3 years ago of the member who will later leave be a years. Then just before leaving, his age is a + 3. Step 5: After replacement, total age = (5A + 15) - (a + 3) + N, where N is the age of the new member at that time. Step 6: The new average age is given to be the same as 3 years ago, i.e. A. So, [(5A + 15) - (a + 3) + N] / 5 = A. Step 7: Multiply both sides by 5: (5A + 15) - (a + 3) + N = 5A. Simplify left side: 5A + 15 - a - 3 + N = 5A → 5A + 12 - a + N = 5A. Step 8: Cancel 5A on both sides to get 12 - a + N = 0 → N - a = -12. Thus, a - N = 12. The replaced member (old one) is 12 years older than the new member.
Verification / Alternative check:
The key result is that (old age now) - (new member age) = 12 years. This difference ensures that the extra 15 years gained by aging of the original group is offset in such a way that the total age comes back to 5A.
Why Other Options Are Wrong:
Options B (13), C (14), and D (15) do not follow from the balance of total ages when the averages are kept the same. Option E (9) also cannot satisfy the equation derived from the average condition.
Common Pitfalls:
Students sometimes incorrectly assume that the difference is 3 * number of members (i.e., 15) without doing the algebra. Forgetting that the average is compared across a 3-year gap is another common source of confusion.
Final Answer:
The difference between the ages of the replaced member and the new member is 12 years.