-  Total ages of 30 boys = 14 x 30 = 420 years
 
Total age when class teacher is included = 15 x 31 = 465 years
 
Age of class teacher = 465 – 420 = 45 years

Let the age of the class teacher be x years.

We are given that the average age of 30 boys in the class is 14 years. Therefore, the total age of the 30 boys is:

30 * 14 = 420

We are also given that when the age of the class teacher is included, the average age becomes 15 years. Therefore, the total age of the 30 boys and the teacher is:

(30 * 14) + x

The total number of people in the class is 30 boys + 1 teacher = 31.

We can use the formula for the average:

average = total sum / number of elements

We can set up the equation using the two given averages:

15 = (420 + x) / 31

Multiplying both sides by 31, we get:

465 = 420 + x

Subtracting 420 from both sides, we get:

x = 45

Therefore, the age of the class teacher is 45 years.

To summarize the solution:

Let the age of the class teacher be x years.

The total age of the 30 boys is 30 * 14 = 420.

The total age of the 30 boys and the teacher is (30 * 14) + x.

The total number of people in the class is 30 boys + 1 teacher = 31.

Using the formula for the average, we can set up the equation 15 = (420 + x) / 31.

Solving for x, we get x = 45.

Therefore, the age of the class teacher is 45 years.