The mean marks of 45 students of division A of class X is 56, of 40 students of division B is 52 and of 40 students of division C is 50. What is the combined mean of marks of all the students of the three divisions of class X?

Introduction / Context:
This question is about finding a combined mean when we know the separate means and sizes of multiple groups. Here, three different divisions of a class have their own average marks and numbers of students. We must merge these to obtain the overall average for all students together. Such problems test the concept of weighted averages, where each group contributes in proportion to its size.

Given Data / Assumptions:
- Division A: 45 students, mean marks = 56
- Division B: 40 students, mean marks = 52
- Division C: 40 students, mean marks = 50
- Total number of students = 45 + 40 + 40 = 125
- We assume all marks are from the same examination and on the same scale.

Concept / Approach:
The combined mean is not the simple average of 56, 52 and 50, because each division has a different number of students. Instead, we calculate the total marks in each division using total = mean × number of students, sum these totals to get grand total marks, and then divide by the total number of students. This process gives the correct weighted average that reflects contributions of all three divisions accurately.

Step-by-Step Solution:
Step 1: Total marks of division A = 45 × 56 = 2520.Step 2: Total marks of division B = 40 × 52 = 2080.Step 3: Total marks of division C = 40 × 50 = 2000.Step 4: Grand total marks for all students = 2520 + 2080 + 2000 = 6600.Step 5: Total number of students = 125, so combined mean = 6600 ÷ 125 = 52.8.

Verification / Alternative check:
We can quickly check reasonableness. Most students are in division A and B with means 56 and 52, while division C has a slightly smaller mean of 50. Therefore the combined mean should be a bit above 52 rather than near 50 or 56. The computed value 52.8 lies between the highest and lowest divisional means and is slightly closer to 56 because division A has the largest number of students, which is consistent with intuition.

Why Other Options Are Wrong:
An answer like 51.4 or 52.1 is too low given that two of the three divisions have means 52 and 56. On the other hand, 54.2 is too high, as that would ignore the influence of division C with average 50. The option 53.6 also does not match the exact weighted calculation. Only 52.8 coincides with the correctly computed combined mean of all 125 students.

Common Pitfalls:
A frequent error is to take the simple average of 56, 52 and 50, which would be 52.67, without accounting for different numbers of students in each division. Another mistake is miscounting the total number of students or miscalculating one of the partial totals. Careful multiplication and addition followed by division by the correct total size avoids these issues.

Final Answer:
The combined mean of marks of all students in the three divisions is 52.8 marks.