The average age of $(X + 14)$ students in a class is $X$ years. When the ages of two teachers are added, the average age increases by 1 year. If the combined age of the two teachers is 64 years, what is the value of $X$?

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**Solution:**

1. Let the number of students be $(X + 14)$ and their average age be $X$ years.

2. Total age of all students = $(X + 14) 	imes X = X^2 + 14X$

3. When two teachers join, the total number of people becomes $(X + 16)$.

4. The new average age is $(X + 1)$ years.

5. Total age including teachers = New average × New number of people = $(X + 1)(X + 16)$

6. The sum of the teachers' ages is given as 64 years.

7. Therefore, total age of students + total age of teachers = total age including teachers:

$$X^2 + 14X + 64 = (X + 1)(X + 16)$$

8. Expanding the right side:

$$X^2 + 14X + 64 = X^2 + 16X + X + 16 = X^2 + 17X + 16$$

9. Subtract $X^2$ from both sides:

$$14X + 64 = 17X + 16$$

10. Rearranging terms:

$$64 - 16 = 17X - 14X$$

$$48 = 3X$$

11. Solving for $X$:

$$X = \frac{48}{3} = 16$$

**Answer:** $16$

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