The mean ago of 25 teachers in a schools is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place . If now the mean age of the teachers in this schools is 39 years, then the age (in years) of the newly appointed teacher is

Options

A. 35
B. 30
C. 40
D. 25

Answer

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

### Step 1: Calculate the total age of the 25 teachers.
The mean age of the 25 teachers is given as 40 years. We can use the formula for mean:

\[ \text{Mean} = \frac{\text{Sum of ages}}{\text{Number of teachers}} \]

Let \( S \) be the sum of the ages of the 25 teachers. Therefore,

\[ 40 = \frac{S}{25} \]

Multiplying both sides by 25 gives:

\[ S = 40 \times 25 = 1000 \text{ years} \]

### Step 2: Calculate the sum of the ages after one teacher retires.
When a teacher who is 60 years old retires, the sum of the ages of the remaining 24 teachers becomes:

\[ S' = S - 60 = 1000 - 60 = 940 \text{ years} \]

### Step 3: Calculate the new total age with the new teacher.
After the retirement, a new teacher is appointed, and the mean age of the 25 teachers is now 39 years. Thus, the new sum of the ages of all 25 teachers is:

\[ \text{New Mean} = \frac{\text{New Sum of ages}}{25} \]

Let \( S'' \) be the new sum of ages:

\[ 39 = \frac{S''}{25} \]

Multiplying both sides by 25 gives:

\[ S'' = 39 \times 25 = 975 \text{ years} \]

### Step 4: Calculate the age of the newly appointed teacher.
Now, we know the sum of the ages of the 24 remaining teachers is 940 years, and the new sum of all 25 teachers is 975 years. Therefore, the age of the newly appointed teacher \( x \) can be calculated as:

\[ x = S'' - S' = 975 - 940 = 35 \text{ years} \]

### Conclusion:
The age of the newly appointed teacher is **35 years**.

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