The mean of marks secured by 30 students in Division A of Class X is 67, for 55 students in Division B is 63 and for 40 students in Division C is 61. What is the mean of marks of all the students of the three divisions of Class X?

Introduction / Context:
Here again we work with a weighted mean for several groups that have different sizes. Each division of Class X has its own mean marks and its own number of students. To find the overall mean marks for all divisions combined, we must take into account both the means and the group sizes.

Given Data / Assumptions:

• Division A: 30 students with mean marks 67.
• Division B: 55 students with mean marks 63.
• Division C: 40 students with mean marks 61.
• We must compute the combined mean marks of all students in these three divisions.

Concept / Approach:
As in any weighted average problem, the correct approach is to convert each group mean to a total marks figure for that group, sum these totals, and then divide by the total number of students. Simply averaging 67, 63 and 61 would ignore how many students are in each division and would lead to an incorrect result.

Step-by-Step Solution:
Step 1: Total marks of Division A = 30 * 67 = 2010. Step 2: Total marks of Division B = 55 * 63 = 3465. Step 3: Total marks of Division C = 40 * 61 = 2440. Step 4: Grand total marks = 2010 + 3465 + 2440 = 7915. Step 5: Total number of students = 30 + 55 + 40 = 125. Step 6: Combined mean marks = 7915 / 125 = 63.32.

Verification / Alternative check:
Notice that Division B has the largest number of students with mean 63, while Division A has a slightly higher mean and Division C a slightly lower mean. The overall mean should therefore lie close to 63, pulled a little upward by Division A and a little downward by Division C. The computed value 63.32 fits this expectation and matches the exact arithmetic.

Why Other Options Are Wrong:
An overall mean of 62.62 or 61.92 would place the combined mean below the main group in Division B, which is not reasonable given that Division A has a higher mean and still contributes many marks. A value like 64.72 is too high and would require a much higher total than 7915. Only 63.32 aligns with the correct weighted calculation.

Common Pitfalls:
A common mistake is to take the simple average (67 + 63 + 61) / 3, which does not account for the different group sizes. Another error is miscalculating individual products such as 55 * 63. It helps to check these intermediate products and to verify that the final average lies within the range of the group means.

Final Answer:
The combined mean marks of the students of the three divisions is 63.32.