The average age of 40 students in a class is 16 years. The average age of 24 of these students is 15.5 years, and the average age of another 15 students is 16 2/3 years. What is the age of the 40th student?

Introduction / Context:
This question involves average ages and missing data for one student. We know the overall average age of 40 students and the average ages of two subgroups within the class. Using this information, we must find the age of the remaining (40th) student. This is a common type of problem in aptitude tests involving weighted averages and totals.

Given Data / Assumptions:

- Total number of students in the class = 40.
- Average age of all 40 students = 16 years.
- Average age of 24 students = 15.5 years.
- Average age of another 15 students = 16 2/3 years (i.e., 16.666... years or 50/3 years).
- We must find the age of the 40th student (the one not included in the two described subgroups).

Concept / Approach:
Average age is total age divided by number of students. We can compute the total age of all 40 students using the overall average. Then, we calculate the total ages of the two specified subgroups from their averages. Subtracting these known totals from the total for 40 students leaves the age of the remaining student. This method is a direct application of the weighted average concept.

Step-by-Step Solution:
Step 1: Total age of all 40 students = 40 × 16 = 640 years. Step 2: Total age of the first group of 24 students = 24 × 15.5 = 24 × 15.5 = 372 years. Step 3: Convert 16 2/3 years to an improper fraction: 16 2/3 = 50/3 years. Step 4: Total age of the second group of 15 students = 15 × (50/3) = 250 years. Step 5: Total number of students accounted for in the two groups = 24 + 15 = 39 students. Step 6: Total age of these 39 students = 372 + 250 = 622 years. Step 7: Age of the 40th student = total age of 40 students − total age of 39 students = 640 − 622 = 18 years.

Verification / Alternative check:
To verify, include the 40th student's age back into the sum. The new total is 622 + 18 = 640 years, and dividing by 40 gives an average age of 16 years, which matches the given data. This confirms that the missing student must be 18 years old for all averages to remain consistent.

Why Other Options Are Wrong:
If the 40th student's age were 16, 16.5, 17 or 19 years, the overall total age would be different and the overall average for the 40 students would not remain exactly 16 years. Only 18 years preserves the overall average once the two subgroup totals are accounted for.

Common Pitfalls:
Students may forget to convert the mixed fraction 16 2/3 to a proper fractional or decimal form correctly, or they might accidentally average the two subgroup averages directly instead of using totals. Always convert each average to a total by multiplying by the number of students in that group and then work with the totals to avoid such errors.

Final Answer:
The age of the 40th student is 18 years.