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

Introduction / Context:
This is a typical problem where adding one more person to a group changes the average, and you must work backwards to find that person's age. It tests your skill in converting averages to total sums and then using the new average to deduce the missing individual value from the changed total.

Given Data / Assumptions:

• Number of students = 36.

• Average age of these students = 14 years.

• When the teacher is added, total persons = 37.

• New average age of the group = 15 years.

• We need to find the teacher's age.

Concept / Approach:
Average age is the total of all ages divided by the number of people. First we compute the total age of the 36 students using their average. Then we compute the total age of the 37 people using the new average. The difference between these two totals must be the teacher's age, because the teacher is the only additional person contributing to the increased total.

Step-by-Step Solution:
Total age of 36 students = 36 * 14 = 504 years. After adding the teacher, number of people = 37. New average age = 15 years. Total age of 37 people = 37 * 15 = 555 years. Teacher's age = total age of 37 people − total age of 36 students. Teacher's age = 555 − 504 = 51 years.

Verification / Alternative check:
We can verify by recalculating the average. With teacher age 51 added, new total = 504 + 51 = 555. Dividing by 37 gives 555 / 37 = 15 years, which matches the given new average. The original total and average for 36 students remain unchanged, so all conditions are satisfied, confirming that the teacher is 51 years old.

Why Other Options Are Wrong:
If the teacher were 35, 45 or 54 years old, the total ages and resulting new average would change. For example, a teacher age of 45 would produce a total of 504 + 45 = 549, giving an average of 549 / 37, which is not 15. Only an age of 51 gives exactly the new average described in the problem.

Common Pitfalls:
Some students miscalculate the total ages due to multiplication errors, or they forget that the new average applies to 37 people, not 36. Others try to reason about the difference in averages without translating into totals, which can be confusing. Sticking to the method of total = average * number of people is the most reliable approach.

Final Answer:
The teacher's age is 51 years.