In a school, the mean marks of 65 students in Division A of Class X is 54, for 30 students in Division B is 50 and for 55 students in Division C is 48. What is the combined mean marks of all the students of the three divisions of Class X?

Introduction / Context:
This aptitude question tests the concept of weighted mean, where different groups have different sizes and different averages. Instead of simply averaging the three given means, you must correctly account for the number of students in each division to find the overall mean for the full class.

Given Data / Assumptions:

• Division A of Class X has 65 students with mean marks 54.
• Division B has 30 students with mean marks 50.
• Division C has 55 students with mean marks 48.
• We need the combined mean marks of all students in the three divisions together.

Concept / Approach:
When several groups have different sizes, the overall mean is a weighted mean. We first convert each group mean to a total marks figure by multiplying mean by number of students. Then we add all these totals to get the grand total. Finally, we divide the grand total by the total number of students in all divisions. Taking a simple average of the three means would ignore the different group sizes and give a wrong answer.

Step-by-Step Solution:
Step 1: Total marks of Division A = 65 * 54 = 3510. Step 2: Total marks of Division B = 30 * 50 = 1500. Step 3: Total marks of Division C = 55 * 48 = 2640. Step 4: Grand total marks of all students = 3510 + 1500 + 2640 = 7650. Step 5: Total number of students = 65 + 30 + 55 = 150. Step 6: Overall mean marks = 7650 / 150 = 51.

Verification / Alternative check:
As a quick reasonableness check, note that Division A has the highest mean and also a relatively large strength. Division C has the lowest mean but not the largest strength. Therefore the overall mean must lie between the extremes 48 and 54, and it should be a little closer to 54 because Division A has many students. The value 51 fits this intuition well.

Why Other Options Are Wrong:
Option 50.3 and option 49.6 are too close to the lowest group mean of 48 and ignore the strong pull of Division A with mean 54. Option 52.4 is too high and would require several more students scoring near 54 or above, which is not supported by the given data. Only 51 matches the correct weighted mean calculation.

Common Pitfalls:
A very common mistake is to simply average 54, 50 and 48 without considering the different numbers of students. That simple average is (54 + 50 + 48) / 3 = 50.67, which does not match any given choice and is incorrect. Always remember that when group sizes differ, total marks and total students must be used to compute the overall mean.

Final Answer:
The combined mean marks of all three divisions of Class X is 51.