The mean of marks secured by 55 students in Division A of Class X is 58, for 45 students in Division B is 54 and for 75 students in Division C is 52. Find the mean of marks of all the students of the three divisions of Class X.

Introduction / Context:
This is another weighted mean problem involving three divisions with different numbers of students and different average marks. You must carefully compute the combined mean marks of all students in Class X by using total marks rather than averaging the given means directly.

Given Data / Assumptions:

• Division A: 55 students with mean marks 58.
• Division B: 45 students with mean marks 54.
• Division C: 75 students with mean marks 52.
• We need the overall mean marks of all students from the three divisions.

Concept / Approach:
The total marks contributed by each division equals its mean multiplied by its number of students. The combined mean is the sum of all marks divided by the total number of students. Because each division has a different strength, failure to weight the means correctly will produce an incorrect result.

Step-by-Step Solution:
Step 1: Total marks of Division A = 55 * 58 = 3190. Step 2: Total marks of Division B = 45 * 54 = 2430. Step 3: Total marks of Division C = 75 * 52 = 3900. Step 4: Grand total marks = 3190 + 2430 + 3900 = 9520. Step 5: Total number of students = 55 + 45 + 75 = 175. Step 6: Combined mean marks = 9520 / 175 = 54.4.

Verification / Alternative check:
The three group means are 58, 54 and 52. Division C has the largest number of students and the lowest mean, 52, so the overall mean should be somewhat closer to 52 and 54 than to 58. The value 54.4 lies between 54 and 58 and is pulled slightly downward by Division C, which is consistent with this reasoning.

Why Other Options Are Wrong:
An overall mean of 53.7 or 53 would be too low, suggesting that the very large Division C dominates even more than it actually does, and these values fail when you recompute the implied total marks. A mean of 55.8 would be too high and would require an unrealistically large grand total. Only 54.4 matches the total of 9520 marks over 175 students.

Common Pitfalls:
Typical errors include miscalculating one of the products or forgetting that the overall mean must lie between the smallest and largest group means. Another frequent mistake is to average 58, 54 and 52 without weights, which would not reflect the true contribution of each division. Always use total marks and total students for combined mean questions.

Final Answer:
The mean marks of all the students of the three divisions is 54.4.