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

Introduction / Context:
This problem involves combining means from several groups with different sizes. We are given the mean marks and number of students in three divisions of Class X and asked to find the overall mean mark of all students combined. This is a direct application of the concept of weighted average, where each division's mean is weighted by the number of students in that division.

Given Data / Assumptions:

• Division A: 65 students, mean marks = 54.
• Division B: 30 students, mean marks = 50.
• Division C: 55 students, mean marks = 48.
• All students belong to exactly one of these three divisions.
• We must find the mean marks for all students together.

Concept / Approach:
The combined mean of several groups is equal to the total of all marks divided by the total number of students. For each division, total marks = mean * number of students. We compute the total marks for each division, add them, and then divide by the combined strength of all three divisions. This gives a weighted average that correctly accounts for the different sizes of the divisions.

Step-by-Step Solution:
Step 1: Compute total marks for Division A. Total for A = 65 * 54. 65 * 54 = 65 * 50 + 65 * 4 = 3250 + 260 = 3510. Step 2: Compute total marks for Division B. Total for B = 30 * 50 = 1500. Step 3: Compute total marks for Division C. Total for C = 55 * 48. 55 * 48 = 55 * 40 + 55 * 8 = 2200 + 440 = 2640. Step 4: Add all totals to find the grand total of marks. Grand total = 3510 + 1500 + 2640 = 7650. Step 5: Compute the total number of students. Total students = 65 + 30 + 55 = 150. Step 6: Find the combined mean. Combined mean = grand total / total students = 7650 / 150. 7650 / 150 = 51.
Verification / Alternative check:
We can simplify by dividing numerator and denominator by 10, giving 765 / 15. 15 * 50 = 750 and 15 * 51 = 765, confirming that 765 / 15 = 51. Thus, the combined mean is consistent with all given group means and sizes.
Why Other Options Are Wrong:
Means such as 50.3, 49.6 or 52.4 correspond to different grand totals and would result from incorrect calculations of total marks or student counts. A value like 53 would imply a much higher grand total than 7650 when multiplied by 150 students.
Common Pitfalls:
A common mistake is to simple average the three means (54, 50 and 48) without considering the different numbers of students in each division. Another error is mismultiplying or adding the totals for each division, which leads to an incorrect numerator for the combined mean.
Final Answer:
The mean of the marks of all students in the three divisions is 51.