In a joint family of 24 members, the average age decreases by 4 months when a girl aged 24 years is replaced by a new boy. What is the age of the new boy?

Introduction / Context:
This problem is based on the concept of average age and how replacing one member of a group with another affects the overall average. We are told that a 24-year-old girl is replaced by a boy, and the average age of a 24-member joint family decreases by 4 months. Using this change in average, we must find the boy's age.

Given Data / Assumptions:

- There are 24 members in the joint family.
- One girl aged 24 years leaves the family.
- A new boy joins in her place.
- Due to this replacement, the average age of the 24 members decreases by 4 months.
- Four months is equal to 4 / 12 = 1 / 3 year.

Concept / Approach:
A change in average age multiplied by the number of members gives the change in the total sum of ages. Here, the average decreases by 4 months, so the total age of the group decreases. That total decrease must equal the difference between the age of the girl who left and the age of the boy who joined. We can write this as an equation and solve for the boy's age.

Step-by-Step Solution:
Step 1: Change in average age = 4 months = 1 / 3 year. Step 2: Number of family members = 24. Step 3: Decrease in total age of the family = change in average × number of members = (1 / 3) × 24 = 8 years. Step 4: The girl who left was 24 years old. Let the boy's age be B years. Step 5: The total age decreases when a 24-year-old is replaced by a B-year-old boy, so decrease in total = 24 − B. Step 6: This must equal 8 years, so 24 − B = 8. Step 7: Solving gives B = 24 − 8 = 16 years.

Verification / Alternative check:
If the girl is 24 years old and the boy is 16 years old, the family loses 8 years of total age (24 − 16 = 8). Dividing this total decrease of 8 years by 24 members yields 8 / 24 = 1 / 3 year, which is 4 months. This matches the stated decrease in average age, confirming that the boy's age is correctly found as 16 years.

Why Other Options Are Wrong:
A boy aged 12, 15, 18, or 20 years would produce decreases in total age of 12, 9, 6, or 4 years respectively, leading to average decreases of 12 / 24, 9 / 24, 6 / 24, or 4 / 24 years. None of these correspond to exactly 4 months, so those ages are incorrect.

Common Pitfalls:
Students sometimes confuse a decrease in average age with a decrease in individual ages or forget to convert months into a fraction of a year correctly. Another common error is to ignore the number of family members when calculating the change in total age. Always remember that total change in age = change in average × number of members.

Final Answer:
The age of the new boy is 16 years.