Averages — “The average age of 19 boys is 21 years. Including the teacher, the class average becomes 22 years. What is the teacher’s age?”

Introduction / Context:
Average (arithmetic mean) problems are solved by converting averages to totals (sum = average * count). When a new member is added and the average changes, we compare the new total to the old total to find the newcomer’s value.

Given Data / Assumptions:

• 19 boys, average 21 ⇒ total of boys = 19 * 21.
• After including the teacher: 20 persons, average 22 ⇒ new total = 20 * 22.
• Teacher’s age = new total − boys’ total.

Concept / Approach:
Compute both totals explicitly, then subtract to isolate the teacher’s age. This avoids confusion from trying to average the two averages, which is a common mistake.

Step-by-Step Solution:

Verification / Alternative check:
(399 + 41) / 20 = 440 / 20 = 22, exactly matching the stated new average; hence the computation is correct.

Why Other Options Are Wrong:
39/40/44/38 produce totals that do not yield an overall average of 22 when combined with 399 and divided by 20.

Common Pitfalls:
Averaging 21 and 22 directly; forgetting to change the count from 19 to 20; arithmetic slip while multiplying 19 * 21 or 20 * 22.

Final Answer:
41 years