Problems on Ages — “The average age of 30 students is 15 years. Including the teacher, the average becomes 16 years. Find the teacher’s age.”

Introduction / Context:
Average problems often require reconstructing the total sum from the average and the number of members. When a new member joins and the average changes, we compare the new total to the old one to isolate the newcomer’s age.

Given Data / Assumptions:

• 30 students have average 15 ⇒ sum_students = 30 * 15.
• With teacher added: members = 31, average = 16 ⇒ new_total = 31 * 16.
• Teacher’s age = new_total − sum_students.

Concept / Approach:
Compute both totals explicitly and subtract. This avoids any algebraic confusion and works reliably for all average-with-addition problems.

Step-by-Step Solution:

Verification / Alternative check:
Recalculate average with teacher: (450 + 46)/31 = 496/31 = 16 ✓.

Why Other Options Are Wrong:
35/40/45/56 yield totals that do not produce the required 16 average when added to 450 and divided by 31.

Common Pitfalls:
Accidentally averaging the two averages; multiplying the teacher’s age by 31; or forgetting that the old total uses 30 while the new total uses 31.

Final Answer:
46 years