In a class there are 20 boys whose average age is decreased by 2 months when one boy aged 18 years is replaced by a new boy the age of the new boy is

Options

A. A)14 years 8 months
B. B)15 years
C. C)16 years 4 months
D. D)17 years 10 months

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
We have a class of 20 boys, and their average age decreases by 2 months when one boy aged 18 years is replaced by a new boy. We need to find the age of the new boy.

### Step 2: Define Variables
Let the average age of the 20 boys be \( K \) years. Since the average age decreases by 2 months, the new average age becomes \( K - \frac{2}{12} \) years (converting months to years).

### Step 3: Calculate the Total Age Before Replacement
The total age of the 20 boys before the replacement can be calculated as:
\[ \text{Total Age Before} = 20K \]

### Step 4: Calculate the Total Age After Replacement
When the boy aged 18 years is replaced by a new boy aged \( N \) years, the total age becomes:
\[ \text{Total Age After} = 20K - 18 + N \]

### Step 5: Set Up the Equation
Since the average age decreases by 2 months, we can set up the equation:
\[ 20K - 18 + N = 20 \left( K - \frac{2}{12} \right) \]

### Step 6: Simplify the Equation
Expanding the right side:
\[ 20K - 18 + N = 20K - \frac{40}{12} \]
Now, we can simplify this:
\[ -18 + N = -\frac{40}{12} \]

### Step 7: Solve for \( N \)
Rearranging gives:
\[ N = -\frac{40}{12} + 18 \]
To combine the terms, convert 18 into a fraction with a denominator of 12:
\[ N = -\frac{40}{12} + \frac{216}{12} \]
This simplifies to:
\[ N = \frac{216 - 40}{12} = \frac{176}{12} \]
Now, simplifying \( \frac{176}{12} \):
\[ N = \frac{44}{3} \text{ years} \approx 14.67 \text{ years} \]
Converting \( \frac{44}{3} \) years into years and months:
\[ 14 \text{ years and } \left( \frac{44 - 42}{3} \times 12 \right) \text{ months} = 14 \text{ years and } 8 \text{ months} \]

### Final Answer
The age of the new boy is **14 years and 8 months**.

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