The average age of 36 students in a group is 14 years. When the age of the teacher is included, the average increases by 1 year. What is the age of the teacher in years?

Introduction / Context:
This question is another example of average age calculation where a teacher is added to a group of students. The average age of the students is known, and including the teacher increases the average by a fixed amount. Using the relationship between average, total age, and number of individuals, we can compute the teacher's age. This type of problem is common in quantitative aptitude tests.

Given Data / Assumptions:
- There are 36 students in the group.
- The average age of these 36 students is 14 years.
- When the teacher's age is included, the average age increases by 1 year, becoming 15 years.
- Ages are in complete years.
- We are required to find the age of the teacher.

Concept / Approach:
The average age equals total age divided by the number of people. We first find the total age of the 36 students using their average. Then we compute the new total age when the teacher is included, using the new average and the new number of individuals, which is 37. The difference between the two totals is the age of the teacher.

Step-by-Step Solution:
Step 1: Total age of the 36 students is 36 * 14 years.Step 2: Compute this total: 36 * 14 = 504 years.Step 3: After including the teacher, there are 37 individuals with an average age of 15 years.Step 4: The new total age is 37 * 15 years.Step 5: Compute this new total: 37 * 15 = 555 years.Step 6: The teacher's age is equal to the new total age minus the original total age of the students, that is 555 - 504.Step 7: Teacher's age = 555 - 504 = 51 years.

Verification / Alternative check:
If the teacher is 51 years old, then adding this to the students' total age of 504 years gives 555 years. Dividing by 37 individuals gives an average of 555 / 37 = 15 years, which matches the increased average given in the question. This confirms that the calculated age of the teacher is correct and consistent with the data.

Why Other Options Are Wrong:
If the teacher's age were 31, 36, 41, or 45 years, the combined total age would be 535, 540, 545, or 549 years respectively. Dividing those totals by 37 would give average ages different from 15 years. For example, 540 / 37 is not equal to 15. Therefore, none of those ages can produce the specified new average and must be rejected.

Common Pitfalls:
Some learners mistakenly use 36 instead of 37 when calculating the new total from the new average. Others try to add 1 year directly to each student's age, which is not what the average increase means here. The correct approach is always to convert averages into totals by multiplying by the number of individuals, and then compare the totals before and after the change.

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