Student 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 students of the three sections. .What is the arithmetic mean of the marks of all the sections?

Options

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

Answer

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

### Step 1: Calculate the total marks for each section
- For Section A (30 students, mean = 72.2):
\[ \text{Total Marks A} = 30 \times 72.2 = 2166 \]

- For Section B (30 students, mean = 69.0):
\[ \text{Total Marks B} = 30 \times 69.0 = 2070 \]

- For Section C (40 students, mean = 64.1):
\[ \text{Total Marks C} = 40 \times 64.1 = 2564 \]

### Step 2: Calculate the combined total marks
Now, we add the total marks from all three sections:
\[ \text{Combined Total Marks} = \text{Total Marks A} + \text{Total Marks B} + \text{Total Marks C} \]
\[ \text{Combined Total Marks} = 2166 + 2070 + 2564 = 6800 \]

### Step 3: Calculate the total number of students
Next, we find the total number of students:
\[ \text{Total Students} = 30 + 30 + 40 = 100 \]

### Step 4: Calculate the combined arithmetic mean
Finally, we calculate the combined arithmetic mean by dividing the combined total marks by the total number of students:
\[ \text{Combined Mean} = \frac{\text{Combined Total Marks}}{\text{Total Students}} = \frac{6800}{100} = 68.0 \]

### Final Answer
The arithmetic mean of the marks of all the sections is **68.0**.

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