The average age of a class of 39 students is 15 years. When the age of the teacher is included, the average increases by 3 months. What is the age of the teacher in years?

Introduction / Context:
This problem again involves average ages, but here the average increases by a fraction of a year, namely 3 months. The scenario describes a class of students with a known average age, and then the teacher is added. The new average allows us to find the teacher's age. This type of question tests careful handling of fractional years and understanding of averages.

Given Data / Assumptions:
- There are 39 students in a class.
- The average age of the 39 students is 15 years.
- When the teacher is included, the average age increases by 3 months, which is 0.25 years, so the new average is 15.25 years.
- Ages are in years, but fractions of years are allowed for the average.
- We must find the age of the teacher.

Concept / Approach:
The average age is equal to the total of all ages divided by the number of individuals. We first calculate the total age of the 39 students. Then, using the new average and the fact that there are now 40 individuals including the teacher, we compute the new total age. The teacher's age is obtained by subtracting the original total from the new total. Careful conversion of 3 months into years is essential.

Step-by-Step Solution:
Step 1: Total age of the 39 students is 39 * 15 years.Step 2: Compute this total: 39 * 15 = 585 years.Step 3: When the teacher is added, there are 40 individuals and the new average age is 15 years + 0.25 years = 15.25 years.Step 4: New total age including the teacher is 40 * 15.25 years.Step 5: Compute this new total: 40 * 15.25 = 610 years.Step 6: The teacher's age is the difference between the new total age and the original total age of the students, that is 610 - 585.Step 7: Teacher's age = 610 - 585 = 25 years.

Verification / Alternative check:
If the teacher is 25 years old, the combined total age is 585 + 25 = 610 years. Dividing by 40 individuals gives an average of 610 / 40 = 15.25 years, which corresponds to 15 years and 3 months. This matches the condition that the average increases by 3 months when the teacher is included. The solution is therefore consistent.

Why Other Options Are Wrong:
If the teacher were 30, 35, 40, or 28 years old, the new total age would be 615, 620, 625, or 613 years respectively. Dividing each by 40 would give averages that differ from 15.25 years. For instance, 620 / 40 = 15.5 years, not 15.25 years. Therefore, none of those values fit the given information.

Common Pitfalls:
Some learners do not convert 3 months into 0.25 years and instead treat it as 3 years or ignore the fractional part. Others mistakenly use 39 instead of 40 when multiplying by the new average. Procedural errors like these lead to incorrect teacher ages. Correctly handling units and remembering that the number of individuals changes from 39 to 40 are crucial steps.

Final Answer:
The age of the teacher is 25 years.