The average age of 80 boys in a class is 15 years. The average age of a group of 15 boys is 16 years, and the average age of another group of 25 boys is 14 years. What is the average age of the remaining boys?

Introduction / Context:
This question tests your understanding of weighted averages. We know the average age of the entire class and the average ages of two subgroups of boys. Using this information, we need to find the average age of the remaining boys in the class.

Given Data / Assumptions:

- Total number of boys in the class = 80.
- Overall average age = 15 years.
- First subgroup: 15 boys with an average age of 16 years.
- Second subgroup: 25 boys with an average age of 14 years.
- The remaining boys form the third group whose average age we must find.

Concept / Approach:
Average age is equal to total age divided by the number of boys. We first compute the total age of all 80 boys using the overall average. Then we calculate the total ages of the first and second subgroups separately. Subtracting these from the total age of the class gives the total age of the remaining boys. Finally, we divide this remaining total by the number of remaining boys to get their average age.

Step-by-Step Solution:
Step 1: Total age of all 80 boys = 80 × 15 = 1200 years. Step 2: Total age of the first subgroup (15 boys) = 15 × 16 = 240 years. Step 3: Total age of the second subgroup (25 boys) = 25 × 14 = 350 years. Step 4: Total number of boys in the first two groups = 15 + 25 = 40 boys. Step 5: Remaining number of boys = 80 − 40 = 40 boys. Step 6: Total age of the remaining 40 boys = 1200 − (240 + 350) = 1200 − 590 = 610 years. Step 7: Average age of the remaining boys = 610 / 40 = 15.25 years.

Verification / Alternative check:
We can recheck using a weighted average. Combine the subgroups: first group has total age 240, second group 350, and remaining group 610, giving 240 + 350 + 610 = 1200 years total for 80 boys. Dividing 1200 by 80 again gives 15 years, the original overall average. This confirms that the intermediate calculations, including the 15.25 years for the remaining boys, are consistent.

Why Other Options Are Wrong:
Values such as 12.24 years, 13.25 years, 14.5 years, or 16 years do not satisfy the total age constraint when combined with the averages of the other two groups. Plugging them back into the total will not restore the overall average of 15 years for the 80 boys, so they must be rejected.

Common Pitfalls:
A common mistake is to treat the average of the remaining group as the average of the two known subgroup averages, which is incorrect because the group sizes differ. Another error is miscalculating the total ages or forgetting to subtract both subgroups from the class total. Carefully using the formula total = average × number avoids these problems.

Final Answer:
The average age of the remaining boys in the class is 15.25 years.