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

Introduction / Context:
This is another weighted average problem involving three divisions of a class. Each division has a different number of students and a different mean mark. We are asked to find the overall mean for all students in Class X across these three divisions. The combined mean must account for the size of each division, and not just average the three given means directly.

Given Data / Assumptions:

• Division A: 30 students, mean marks = 67.
• Division B: 55 students, mean marks = 63.
• Division C: 40 students, mean marks = 61.
• All students in Class X belong to one of these divisions.
• We need the combined mean marks of all students.

Concept / Approach:
For several groups with different sizes and means, the overall mean is found by computing total marks for each group (mean multiplied by group size), adding these totals, and dividing by the total number of students. This approach automatically weights each division by its size. It is crucial not to simply average 67, 63 and 61, as that would ignore the different numbers of students in each division.

Step-by-Step Solution:
Step 1: Compute the total marks for Division A. Total for A = 30 * 67 = 2010. Step 2: Compute the total marks for Division B. Total for B = 55 * 63. 55 * 63 = 55 * (60 + 3) = 3300 + 165 = 3465. Step 3: Compute the total marks for Division C. Total for C = 40 * 61 = 2440. Step 4: Find the grand total of marks. Grand total = 2010 + 3465 + 2440 = 7915. Step 5: Compute the total number of students. Total students = 30 + 55 + 40 = 125. Step 6: Calculate the combined mean. Combined mean = grand total / total students = 7915 / 125. 125 * 60 = 7500, remainder 415. 125 * 3 = 375, remainder 40, so 7915 / 125 = 63 + 40 / 125. 40 / 125 = 0.32, hence combined mean = 63.32.
Verification / Alternative check:
We can check approximate consistency. Division B has the largest number of students and a mean of 63, so the overall mean should be near 63, slightly pulled upward by Division A and downward by Division C. The value 63.32 fits this expectation.
Why Other Options Are Wrong:
Values such as 62.62, 61.92, 64.72 or 60.12 correspond to different total marks and would arise only if the group sizes or means were misused. For example, 62.62 would produce a grand total of 62.62 * 125, which is not equal to 7915.
Common Pitfalls:
One common mistake is to compute the simple average (67 + 63 + 61) / 3, ignoring the different numbers of students in each division. Another is to miscalculate one of the products, especially 55 * 63, which affects the final mean directly.
Final Answer:
The mean of marks of all the students of the three divisions together is 63.32.