From a class of 42 boys, a boy aged 10 years goes away and in his place, a new boy is admitted. If on account of this change, the average age of the boys in that class increases by 2 months, the age of the newcomer is :

Options

A. 19 years
B. 17 years
C. 10 years 6 months
D. 12 years 2 months

Answer

To solve the problem, we need to determine the age of the newcomer boy based on the information given about the average age of the class.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- There are 42 boys in the class.
- One boy aged 10 years leaves the class.

2. **Determine the Change in Average Age**:
- The average age of the boys increases by 2 months when the new boy is admitted.
- Since 2 months is equivalent to \( \frac{2}{12} \) years, this means the average age increases by \( \frac{1}{6} \) years.

3. **Calculate the Total Age of the Class Before the Change**:
- Let the initial average age of the 42 boys be \( A \) years.
- Therefore, the total age of the boys before the change is \( 42A \).

4. **Calculate the Total Age After One Boy Leaves**:
- When the 10-year-old boy leaves, the total age becomes \( 42A - 10 \).

5. **Introduce the New Boy**:
- Let the age of the newcomer be \( x \) years.
- The new total age of the class after the newcomer joins is \( 42A - 10 + x \).

6. **Calculate the New Average Age**:
- The new average age of the class with 42 boys is \( \frac{42A - 10 + x}{42} \).
- According to the problem, this new average age is \( A + \frac{1}{6} \).

7. **Set Up the Equation**:
- We can set up the equation based on the new average:
\[ \frac{42A - 10 + x}{42} = A + \frac{1}{6} \]

8. **Clear the Denominator**:
- Multiply both sides by 42 to eliminate the fraction:
\[ 42A - 10 + x = 42A + 7 \]

9. **Simplify the Equation**:
- Subtract \( 42A \) from both sides:
\[ -10 + x = 7 \]
- Add 10 to both sides:
\[ x = 17 \]

10. **Conclusion**:
- The age of the newcomer is **17 years**.