In a group of 36 students, the average age is 14 years. When the age of the teacher is also included in this group, the average age increases to 15 years. What is the age of the teacher in years?

Introduction / Context:
This aptitude question is based on the concept of average age and how it changes when a new person is added to an existing group. Such problems are very common in competitive exams because they test your understanding of averages, total sum, and simple linear reasoning in one short statement. By understanding how to move between average and total sum, you can solve a wide variety of age and marks problems quickly in the exam.

Given Data / Assumptions:

There are 36 students in a group.
The average age of these 36 students is 14 years.
When the teacher is added, there are 37 persons in total.
The new average age of all 37 persons is 15 years.
All ages are measured in complete years and we assume simple arithmetic average.

Concept / Approach:
The key idea is that average = total sum / number of terms. Therefore, total sum = average * number of terms. First we find the total age of the 36 students using their average. Then we find the total age of the 37 persons using the new average. The difference between these two totals must be the age of the teacher, because only the teacher has been added to the group.

Step-by-Step Solution:
Step 1: Compute total age of 36 students. Total age of 36 students = average age * number of students = 14 * 36 = 504 years. Step 2: Compute total age of 37 persons including the teacher. Total age of 37 persons = new average age * total number of persons = 15 * 37 = 555 years. Step 3: Find the age of the teacher. Age of teacher = total age of 37 persons - total age of 36 students = 555 - 504 = 51 years.

Verification / Alternative Check:
If the teacher is 51 years old, then new total age = 504 + 51 = 555 years. Dividing this by 37 persons gives 555 / 37 = 15 years exactly. This matches the given average after including the teacher, so the calculation is consistent and confirms that 51 years is correct.

Why Other Options Are Wrong:
Option 31 years would give new total 504 + 31 = 535, and 535 / 37 is not 15, so it is incorrect.
Option 36 years would give total 540, and 540 / 37 is also not equal to 15, so it is wrong.
Option 54 years would give total 504 + 54 = 558, and 558 / 37 does not give an average of 15, so this option is also wrong.

Common Pitfalls:
A common mistake is to simply average 14 and 15, which has no direct meaning here. Another mistake is to forget that when the average changes, the total sum must be recalculated with the new number of persons. Some students also try to reason by guesswork instead of using the clear formula total = average * number of terms, which can lead to calculation errors.

Final Answer:
The age of the teacher is 51 years.