A class consists of 3 sections A, B and C with 35, 35 and 30 students respectively, The arithmetic means of the marks secured by students of sections A and B, who appeared for a test of 100 marks are 74 and 70 respectively, The arithmetic mean of the marks secured by students of section C, who appeared for a test in the same subject which carried 75 marks is 51. What is the average percentage of marks secured by all the 100 student of the three sections?

Options

A. 70
B. 70.8
C. 65
D. 67.5

Explanation

To find the average percentage of marks secured by all the students in sections A, B, and C, we can follow these steps:

### Step 1: Calculate the total marks secured by students in Section A
- The number of students in Section A = 35
- The arithmetic mean of marks in Section A = 74
- Total marks secured by Section A = Number of students × Mean marks
\[ \text{Total marks in A} = 35 \times 74 = 2590 \]

### Step 2: Calculate the total marks secured by students in Section B
- The number of students in Section B = 35
- The arithmetic mean of marks in Section B = 70
- Total marks secured by Section B = Number of students × Mean marks
\[ \text{Total marks in B} = 35 \times 70 = 2450 \]

### Step 3: Calculate the total marks secured by students in Section C
- The number of students in Section C = 30
- The arithmetic mean of marks in Section C = 51 (out of 75)
- To convert this to a score out of 100, we can use the following formula:
\[ \text{Equivalent score out of 100} = \left(\frac{\text{Mean marks in C}}{75}\right) \times 100 = \left(\frac{51}{75}\right) \times 100 = 68 \]
- Now, calculate the total marks secured by Section C:
\[ \text{Total marks in C} = 30 \times 68 = 2040 \]

### Step 4: Calculate the total marks secured by all students
- Total marks secured by all sections = Total marks in A + Total marks in B + Total marks in C
\[ \text{Total marks} = 2590 + 2450 + 2040 = 7080 \]

### Step 5: Calculate the average percentage of marks
- Total number of students = 35 (A) + 35 (B) + 30 (C) = 100
- Average percentage of marks = \(\frac{\text{Total marks}}{\text{Total number of students}}\)
\[ \text{Average percentage} = \frac{7080}{100} = 70.80 \]

### Final Answer
The average percentage of marks secured by all the 100 students of the three sections is **70.80%**.

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