The average age of a team of 12 players is 23 years. If the age of the coach is also included, the average age of the group increases by 2 years. What is the age (in years) of the coach?

Introduction / Context:
This problem checks understanding of averages when a new member is added to a group. It is a standard aptitude question where the average age changes when another person joins, and we are asked to find that person's age from the change in the overall average.

Given Data / Assumptions:
- Number of players in the team = 12. - Average age of the 12 players = 23 years. - When the coach is included, the group size becomes 13. - New average age of 13 persons = 23 + 2 = 25 years. - We assume all ages are in whole years, although that is not strictly required for the calculation.

Concept / Approach:
The core idea is that total age = average age * number of persons. We first find the total age of the 12 players. Then we find the total age of the group of 13 (players plus coach) using the new average. The difference between these two totals equals the age of the coach. This simple linear relation is used very frequently in average problems.

Step-by-Step Solution:
Step 1: Total age of 12 players = 23 * 12. Step 2: Compute the product: 23 * 12 = 276 years. Step 3: When the coach is included, number of persons = 13 and new average age = 25 years. Step 4: Total age of 13 persons = 25 * 13. Step 5: Compute the product: 25 * 13 = 325 years. Step 6: Age of the coach = total age of 13 persons minus total age of 12 players. Step 7: Age of coach = 325 - 276 = 49 years.

Verification / Alternative check:
We can verify by reversing the logic. Assume the coach is 49 years old. Then total age of players plus coach = 276 + 49 = 325. Dividing by 13 gives 325 / 13 = 25 years as the new average. This matches the stated increase from 23 to 25 years, confirming that 49 years is correct.

Why Other Options Are Wrong:
Option A (41): If the coach were 41, the new total would be 276 + 41 = 317 and the average would be 317 / 13, which is less than 25. Option B (47): Total would be 276 + 47 = 323, giving an average less than 25 years. Option D (51): Total would be 276 + 51 = 327, giving an average slightly more than 25 years. Option E (45): Also fails to produce the exact average of 25 years.

Common Pitfalls:
Some learners mistakenly add 2 years directly to the total of 12 players, or they add 2 years to 23 and assume that is the coach's age. Others forget to multiply the average by the number of persons and instead treat the average as a simple additive quantity. Always remember that average age is total age divided by number of persons, so when the group size changes, total age and average must both be handled carefully.

Final Answer:
The age of the coach is 49 years.