Students of three sections of a class, having 30, 30 and 40 students appeared for a test of 100 marks. The arithmetic means of the marks of the three sections are `72.2, 69.0 and 64.1` in that order. What is the arithmetic mean of the marks of all the students of the three sections?

Options

A. 66.6
B. 67.3
C. 68
D. 70.6

Explanation

To find the arithmetic mean of the marks of all the students from the three sections, we will follow these steps:

### Step 1: Calculate the total marks for each section
1. **Section 1**:
- Number of students = 30
- Arithmetic mean = 72.2
- Total marks for Section 1 = Number of students × Arithmetic mean
- Total marks for Section 1 = 30 × 72.2 = 2166

2. **Section 2**:
- Number of students = 30
- Arithmetic mean = 69.0
- Total marks for Section 2 = Number of students × Arithmetic mean
- Total marks for Section 2 = 30 × 69.0 = 2070

3. **Section 3**:
- Number of students = 40
- Arithmetic mean = 64.1
- Total marks for Section 3 = Number of students × Arithmetic mean
- Total marks for Section 3 = 40 × 64.1 = 2564

### Step 2: Calculate the total marks of all sections
- Total marks = Total marks of Section 1 + Total marks of Section 2 + Total marks of Section 3
- Total marks = 2166 + 2070 + 2564 = 6800

### Step 3: Calculate the total number of students
- Total number of students = Number of students in Section 1 + Number of students in Section 2 + Number of students in Section 3
- Total number of students = 30 + 30 + 40 = 100

### Step 4: Calculate the combined arithmetic mean
- Combined arithmetic mean = Total marks / Total number of students
- Combined arithmetic mean = 6800 / 100 = 68.0

Thus, the arithmetic mean of the marks of all the students of the three sections is **68.0**.