In an exam of 100 marks, the average marks of a class of 40 students is 76. If the top 3 scorers of the class leave, the average score falls down by 1.If the other two toppers except the highest topper scored not more than 85 marks, then what is the minimum score did topper can score?

Options

A. 86
B. 98
C. 95
D. 92

Answer

To solve the problem step by step, we will follow the logical flow of calculations based on the information provided in the question.

### Step 1: Calculate the total marks of the class
The average marks of the class of 40 students is 76. To find the total marks of the class, we multiply the average by the number of students.

\[ \text{Total marks} = \text{Average} \times \text{Number of students} = 76 \times 40 = 3040 \]

### Step 2: Calculate the new average after the top 3 scorers leave
When the top 3 scorers leave, the average score falls by 1, making the new average 75. We can now calculate the total marks of the remaining 37 students.

\[ \text{Total marks of 37 students} = \text{New average} \times \text{Number of remaining students} = 75 \times 37 = 2775 \]

### Step 3: Calculate the total marks of the top 3 scorers
To find the total marks of the top 3 scorers, we subtract the total marks of the remaining 37 students from the total marks of the class.

\[ \text{Total marks of top 3 students} = \text{Total marks of 40 students} - \text{Total marks of 37 students} = 3040 - 2775 = 265 \]

### Step 4: Set up the equation for the top 3 scorers
Let the scores of the top 3 students be \(x\) (the highest scorer), \(y\), and \(z\) (the other two scorers). According to the problem, \(y\) and \(z\) cannot be more than 85.

We have:
\[ x + y + z = 265 \]

### Step 5: Maximize the scores of \(y\) and \(z\)
To find the minimum score of the highest scorer \(x\), we will assume that \(y\) and \(z\) scored the maximum possible, which is 85.

\[ y + z = 85 + 85 = 170 \]

### Step 6: Solve for \(x\)
Now we can substitute \(y + z\) back into the equation for the total marks of the top 3 students:

\[ x + 170 = 265 \]

Subtracting 170 from both sides gives us:

\[ x = 265 - 170 = 95 \]

### Conclusion
The minimum score that the highest scorer (topper) can achieve is **95**.