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

Introduction / Context:
Once again, this problem asks for the combined mean of several groups with different sizes and means. We know the average marks in three divisions of a class and the number of students in each division. The objective is to compute the overall mean marks for all students from all three divisions. This reinforces the idea of weighted averages across multiple groups.

Given Data / Assumptions:

• Division A: 55 students, mean marks = 58.
• Division B: 45 students, mean marks = 54.
• Division C: 75 students, mean marks = 52.
• We must find the combined mean for all students of Class X in these divisions.

Concept / Approach:
The overall mean is the total of all marks divided by the total number of students. For each division, the total marks are the product of the division mean and the division size. Once we find the total for each division, we add them together and divide by the combined number of students. This approach gives a properly weighted overall average that reflects the different group sizes.

Step-by-Step Solution:
Step 1: Compute total marks for Division A. Total for A = 55 * 58. 55 * 58 = 55 * (60 − 2) = 3300 − 110 = 3190. Step 2: Compute total marks for Division B. Total for B = 45 * 54. 45 * 54 = 45 * (50 + 4) = 2250 + 180 = 2430. Step 3: Compute total marks for Division C. Total for C = 75 * 52. 75 * 52 = 75 * (50 + 2) = 3750 + 150 = 3900. Step 4: Find the grand total of marks. Grand total = 3190 + 2430 + 3900 = 9520. Step 5: Compute total number of students. Total students = 55 + 45 + 75 = 175. Step 6: Calculate the combined mean. Combined mean = 9520 / 175. 175 * 50 = 8750, remainder 770. 175 * 4 = 700, remainder 70. 175 * 0.4 = 70, so 770 / 175 = 4.4. Therefore, 9520 / 175 = 54.4.
Verification / Alternative check:
Since the largest group is Division C with mean 52, but Division A has a higher mean of 58 and a substantial size, the overall mean should be between 52 and 58, most likely a little above the middle of the three means. The value 54.4 fits logically in this range and is consistent with the computed grand total.
Why Other Options Are Wrong:
A mean of 53 or 53.7 would correspond to lower total marks than 9520, while 55.8 or 56.1 would correspond to higher totals. These values do not match the exact ratio 9520 / 175 and hence are incorrect.
Common Pitfalls:
Some may incorrectly average 58, 54 and 52 directly, ignoring the different sizes of the divisions, which does not produce the correct result. Others might miscalculate one of the products, especially 55 * 58 or 75 * 52, leading to a wrong total and mean.
Final Answer:
The combined mean of marks of all students of the three divisions is 54.4.