The average age of 15 students of a class is 15 years. Out of these, the average age of 5 students is 14 years and the average age of another 9 students is 16 years. What is the age of the remaining fifteenth student?

Introduction / Context:
This question is another application of averages and total sums. The group is divided into two overlapping subgroups, and we are given the average age of each subgroup as well as the overall average. We need to find the age of the one student who is not counted in either of the two specified groups of 5 and 9 students.

Given Data / Assumptions:

There are 15 students in the class.
The average age of all 15 students is 15 years.
The average age of 5 selected students is 14 years.
The average age of another 9 students is 16 years.
The ages are all in years and we need the age of the remaining one student.

Concept / Approach:
We use the relation total age = average age * number of students for each group. We first compute the total age of all 15 students using the overall average. Then we compute the total ages of the groups of 5 and 9 students separately. The remaining student is the fifteenth one whose age is not included in those two sums. Therefore, age of the fifteenth student equals the overall total minus the sum of the totals of the two subgroups.

Step-by-Step Solution:
Step 1: Compute total age of all 15 students. Total age of 15 students = 15 * 15 = 225 years. Step 2: Compute total age of the 5 students. Total age of 5 students = 14 * 5 = 70 years. Step 3: Compute total age of the 9 students. Total age of 9 students = 16 * 9 = 144 years. Step 4: Use these totals to find the age of the fifteenth student. Sum of ages of the 5 and 9 students together = 70 + 144 = 214 years. Age of fifteenth student = total age of 15 students - 214 = 225 - 214 = 11 years.

Verification / Alternative Check:
We can quickly verify: if the single remaining student is 11 years old, then the three groups are consistent. The 5 students sum to 70, the 9 students to 144 and the last student 11 gives a total of 70 + 144 + 11 = 225. Dividing 225 by 15 yields an average of exactly 15 years, which matches the given information.

Why Other Options Are Wrong:
Option 12 would give a total of 70 + 144 + 12 = 226, which would produce an overall average higher than 15 years.
Option 13 leads to a total of 227, again not matching the required total 225 for an average of 15.
Option 14 would take the total to 228, making the average 228 / 15, which is not equal to 15 years, so it is incorrect.

Common Pitfalls:
Some students mistakenly average 14 and 16 or try to treat the 5 and 9 students as disjoint without considering the overall total properly. Another error is miscalculating the totals for the subgroups or forgetting that there is exactly one remaining student whose age must account for the difference between the partial sums and the whole class sum.

Final Answer:
The age of the fifteenth student is 11 years.