After replacing an old member by 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. What is the difference between the ages of the replaced and the new member?

### Concept & Logic If no member was replaced, the average age of the 5 members today would have increased by 3 years compared to 3 years ago. Since the replacement keeps the average exactly the same as 3 years ago, the new member must be younger than the old member by the total value that would have accumulated over those 3 years. --- ### Step-by-Step Solution * **Given:** * Number of members = 5 * Time duration = 3 years * **Deduction:** * If the same 5 members continued, the total age of the group would increase by: $$5 \times 3 = 15\text{ years}$$ * Because the old member was replaced by a new member, the total age of the group dropped by exactly this cumulative increase of 15 years to bring the average back to its previous state. * Therefore, the difference between the age of the replaced member and the new member is exactly 15 years. --- ### Exam Strategy & Shortcut Direct Formula for Replacement: $$\text{Difference in age} = \text{Number of items} \times \text{Change in average for the group}$$ $$\text{Difference} = 5 \times 3 = 15\text{ years}$$ This problem can be solved mentally in under 5 seconds using this direct relation. --- ### Common Pitfall Do not confuse the change in average over time with a regular replacement where the average changes instantly. Here, the phrase "same as it was 3 years ago" means the total age dropped by $5 \times 3$ relative to what it should be today. --- ### Final Answer **Therefore, the correct answer is 15 years.**