The average age of 19 boys in a class is 21 years. When the age of the teacher is included, the average increases to 22 years. What is the age of the teacher in years?

Introduction / Context:
This is a typical average age problem where a teacher is added to a group of students and the average age of the combined group changes. From the change in average, we can compute the teacher's age. Such questions help reinforce the link between average, total sum, and number of individuals in a group.

Given Data / Assumptions:
- There are 19 boys in a class.
- The average age of these 19 boys is 21 years.
- When the teacher is included, the average age becomes 22 years.
- Ages are given in years.
- 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 compute the total age of the 19 boys using their average. Then, using the new average when the teacher is included, we compute the total age of the 20 individuals. The difference between these totals gives the teacher's age directly.

Step-by-Step Solution:
Step 1: Total age of the 19 boys is 19 * 21 years.Step 2: Compute this total: 19 * 21 = 399 years.Step 3: When the teacher is included, there are 20 individuals with average age 22 years, so the new total age is 20 * 22 years.Step 4: Compute this new total: 20 * 22 = 440 years.Step 5: The teacher's age is the difference between the total age of the group including the teacher and the total age of only the boys.Step 6: Teacher's age = 440 - 399 = 41 years.

Verification / Alternative check:
If we add a teacher aged 41 years to the group of 19 boys whose total age is 399 years, the new total becomes 399 + 41 = 440 years. Dividing by 20 individuals gives an average of 440 / 20 = 22 years, which matches the problem statement. This confirms that the calculated age of the teacher is correct.

Why Other Options Are Wrong:
If the teacher were 39, 40, 44, or 47 years old, the total age with the teacher would be 399 plus that value. Dividing those totals by 20 would not give an exact average of 22 years. For example, with a 39 year old teacher, the total would be 438 and the average would be 438 / 20 = 21.9 years, which is not equal to 22 years. Therefore, those options are incorrect.

Common Pitfalls:
Common mistakes include forgetting to multiply the average by the number of individuals to get total age, or using 19 instead of 20 when computing the new total. Some learners also try to adjust the average directly without working with totals, which can cause confusion. The safe approach is always to convert averages to sums and then back to averages as needed.

Final Answer:
The age of the teacher is 41 years.