The problem states:

• The average age of 30 boys is 15 years.
• When the ages of two teachers are included, the average age increases by 1 year.

We need to find the sum of the ages of the two teachers.

Step-by-step solution:

1. Total age of the 30 boys:

   • The average age of the boys is 15 years.
   • Total age of the 30 boys = $30 \times 15 = 450$ years.

2. New average age:

   • When two teachers are included, the average age increases by 1, making it 16 years.
   • The total number of people now is $30 + 2 = 32$.

3. Total age of all 32 people:

   • The new average is 16 years, so the total age of the 32 people is: $32 \times 16 = 512 \text{ years}.$

4. Sum of the ages of the two teachers:

   • The total age of the 32 people is 512 years, and the total age of the 30 boys is 450 years.
   • Therefore, the sum of the ages of the two teachers is: $512 - 450 = 62 \text{ years}.$

Conclusion:

The sum of the ages of the two teachers is 62 years, so the correct answer is:

C. 62 years

Would you like more details or have any questions about this solution?

Here are 5 related questions you might find useful:

1. How do you calculate the average age when more people are added?
2. What would happen if the average increased by 2 years instead of 1 year?
3. How would the solution change if the class had 40 boys?
4. What if the ages of the boys were not all the same?
5. How would you calculate the individual ages of the two teachers if additional data were provided?

Tip: When solving age problems, always calculate total age first, then use averages to find specific unknowns.