The mean of marks secured by 60 students in Division A of Class X is 64, by 40 students in Division B is 60, and by 60 students in Division C is 58. Find the mean of the marks of all the students of the three divisions of Class X together.

Introduction / Context:
This final problem in the set is another weighted average question involving three divisions of a class. Each division has a known number of students and a known mean mark. The goal is to calculate the overall mean of marks when all three divisions are combined. Solving it reinforces the systematic method for combining multiple group means with different group sizes.

Given Data / Assumptions:

• Division A: 60 students, mean marks = 64.
• Division B: 40 students, mean marks = 60.
• Division C: 60 students, mean marks = 58.
• We must determine the mean of the marks of all students across these three divisions.

Concept / Approach:
The combined mean is found by summing the total marks from each division and dividing by the total number of students. For each division, total marks are equal to the division mean multiplied by the number of students. Once we compute these totals for all three divisions, we add them together and divide by the total student count. This correctly weights each division in proportion to its size.

Step-by-Step Solution:
Step 1: Compute total marks for Division A. Total for A = 60 * 64. 60 * 64 = 3840. Step 2: Compute total marks for Division B. Total for B = 40 * 60 = 2400. Step 3: Compute total marks for Division C. Total for C = 60 * 58. 60 * 58 = 60 * (60 − 2) = 3600 − 120 = 3480. Step 4: Add these to find the grand total of marks. Grand total = 3840 + 2400 + 3480 = 9720. Step 5: Compute the total number of students. Total students = 60 + 40 + 60 = 160. Step 6: Find the combined mean. Combined mean = grand total / total students = 9720 / 160. Divide numerator and denominator by 10: 972 / 16. 16 * 60 = 960, remainder 12. 12 / 16 = 0.75, so 972 / 16 = 60.75.
Verification / Alternative check:
Division A and C each have 60 students, while Division B has 40. Their means 64, 60 and 58 suggest an overall mean between 58 and 64, leaning a bit toward the larger groups A and C. The value 60.75 fits naturally in this range and matches the calculation.
Why Other Options Are Wrong:
60.05 and 59.35 are too low given that one large division has mean 64. 62.15 and 61.4 are higher than the correctly computed 60.75 and do not correspond to the ratio 9720 / 160.
Common Pitfalls:
One pitfall is to average 64, 60 and 58 directly without weighting, which does not account for the different group sizes. Another is to miscalculate one of the products, especially 60 * 58, resulting in an incorrect grand total and combined mean.
Final Answer:
The mean of marks of all the students of the three divisions together is 60.75.