The average age of 24 students and the principal is 15 years. When the principal’s age is excluded, the average age decreases by 1 year. What is the age of the principal?

Options

A. (a) 38
B. (b) 40
C. (c) 39
D. (d) 37

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We know that the average age of 24 students and the principal is 15 years. This means that the total age of the 24 students plus the principal's age is equal to the average age multiplied by the total number of individuals (students + principal).

### Step 2: Set up the equation for the total age
Let the age of the principal be \( X \). The total age of the 24 students can be represented as \( S \). Therefore, we can write the equation for the total age:
\[ \frac{S + X}{25} = 15 \]
Multiplying both sides by 25 gives:
\[ S + X = 375 \quad \text{(1)} \]

### Step 3: Calculate the new average when the principal is excluded
When the principal's age is excluded, the average age of the 24 students decreases by 1 year, making it 14 years. The equation for the average age of the students alone is:
\[ \frac{S}{24} = 14 \]
Multiplying both sides by 24 gives:
\[ S = 336 \quad \text{(2)} \]

### Step 4: Substitute the value of \( S \) into equation (1)
Now, we can substitute the value of \( S \) from equation (2) into equation (1):
\[ 336 + X = 375 \]

### Step 5: Solve for \( X \)
Now, we can solve for \( X \):
\[ X = 375 - 336 \]
\[ X = 39 \]

### Conclusion
The age of the principal is \( 39 \) years.

---