Out of 10 teachers of a school, one teacher retires and at his place a new teacher of age 25 years joins. As a result of it, the average age of the teachers is reduced by 3 years. The age of the retired teacher is:

Options

A. 60 years
B. 58 years
C. 56 years
D. 55 years

Answer

To find the age of the retired teacher, we can follow these steps:

### Step 1: Define the variables
Let the age of the retired teacher be \( x \). The total number of teachers is 10.

### Step 2: Calculate the initial total age of the teachers
The average age of the teachers before the retirement can be expressed as:
\[ \text{Average age} = \frac{\text{Total age of teachers}}{\text{Number of teachers}} \]
Let’s denote the average age of the teachers before the retirement as \( A \). Therefore, the total age of the teachers before the retirement is:
\[ \text{Total age} = 10A \]

### Step 3: Calculate the total age after the retirement
When the teacher retires and a new teacher of age 25 joins, the new total age becomes:
\[ \text{New total age} = 10A - x + 25 \]

### Step 4: Set up the equation for the new average
After the new teacher joins, the average age of the teachers is reduced by 3 years, so the new average age is \( A - 3 \). The total number of teachers remains 10, thus:
\[ A - 3 = \frac{10A - x + 25}{10} \]

### Step 5: Clear the fraction by multiplying through by 10
\[ 10(A - 3) = 10A - x + 25 \]
Expanding this gives:
\[ 10A - 30 = 10A - x + 25 \]

### Step 6: Simplify the equation
Subtract \( 10A \) from both sides:
\[ -30 = -x + 25 \]
Rearranging gives:
\[ x - 30 = 25 \]

### Step 7: Solve for \( x \)
Adding 30 to both sides:
\[ x = 55 \]

### Conclusion
The age of the retired teacher is \( 55 \) years.

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