In a class of 80 boys, the average age is 15 years. The average age of one 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 in the class?

Introduction / Context:
This is an average age problem where a large class is divided into several groups with different average ages. You know the overall average of the entire class and the averages of two subgroups, and you are asked to find the average of the remaining boys. It is a good test of your ability to break down averages into totals and then recombine them correctly.

Given Data / Assumptions:

Total number of boys in the class = 80.
Average age of the whole class = 15 years.
One group has 15 boys with average age 16 years.
Another group has 25 boys with average age 14 years.
The remaining boys form a third group whose average age we need to find.

Concept / Approach:
The basic principle is that total age = average age * number of boys. We first calculate the total age of the whole class. Then we find the total ages of the two known groups. Subtracting these totals from the overall total age gives the combined total age of the remaining boys. Dividing that by the number of remaining boys yields their average age.

Step-by-Step Solution:
Step 1: Compute total age of all 80 boys. Total age of class = 15 * 80 = 1200 years. Step 2: Compute total age of the first group of 15 boys. Total age of first group = 16 * 15 = 240 years. Step 3: Compute total age of the second group of 25 boys. Total age of second group = 14 * 25 = 350 years. Step 4: Compute total age of the remaining boys. Total age of known groups = 240 + 350 = 590 years. Total age of remaining boys = 1200 - 590 = 610 years. Step 5: Compute the number of remaining boys. Remaining boys = 80 - (15 + 25) = 80 - 40 = 40. Step 6: Compute average age of remaining boys. Average age = 610 / 40 = 15.25 years.

Verification / Alternative Check:
We can check reasonableness: one group is slightly older than the overall average (16 vs 15), another is younger (14), and the remaining group has an average of 15.25, which is just above the overall average. Because the sizes of the groups differ, this slightly higher average for the remaining group balances the two other groups to produce the overall average of 15 years over 80 boys. The arithmetic also confirms this correctly.

Why Other Options Are Wrong:
12.24 years and 13.25 years are far too low, and would make the overall average of 80 boys much lower than 15 years.
16 years would make the remaining group too old on average, pushing the overall class average above 15 years when combined with the other two groups.

Common Pitfalls:
Some students try to average the three group averages directly without taking into account the number of boys in each group, which leads to incorrect results. Others forget to compute or subtract the total ages of the known groups before working on the remaining group. Always handle such problems by converting averages into totals, performing additions or subtractions on totals, and then returning to averages.

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