Several students have taken an exam . There was an error in the answer key which affected the marks of 48 students , and their average marks reduced from 78 to 66 . The average of remaining students increased by 3.5 marks . This resulted the reduction of the average of all students by 4.5 marks . The number of students that attended the exam is :

Options

A. 100
B. 96
C. 84
D. 93

Answer

To solve the problem step by step, we will follow the information given in the question and derive the required number of students who attended the exam.

### Step 1: Understand the problem
We know that:
- 48 students had their average marks reduced from 78 to 66.
- The average of the remaining students increased by 3.5 marks.
- The overall average of all students decreased by 4.5 marks.

### Step 2: Calculate the total marks lost by the 48 students
The average marks of the 48 students decreased from 78 to 66. Therefore, the total marks lost by these students can be calculated as follows:

\[ \text{Total marks lost} = \text{Number of students} \times \text{Decrease in average} \]
\[ = 48 \times (78 - 66) = 48 \times 12 = 576 \]

### Step 3: Calculate the new average of the remaining students
Let \( x \) be the total number of students who attended the exam. The number of remaining students is \( x - 48 \).

The average of the remaining students increased by 3.5 marks. Therefore, if we denote the original average of the remaining students as \( A \), we have:

\[ A + 3.5 \]

### Step 4: Calculate the total marks of the remaining students
The total marks of the remaining students can be expressed as:

\[ \text{Total marks of remaining students} = (x - 48) \times (A + 3.5) \]

### Step 5: Calculate the overall average before and after
The overall average of all students decreased by 4.5 marks. The original average of all students can be expressed as:

\[ \text{Original average} = \frac{(48 \times 78) + ((x - 48) \times A)}{x} \]

The new average becomes:

\[ \text{New average} = \frac{(48 \times 66) + ((x - 48) \times (A + 3.5))}{x} \]

### Step 6: Set up the equation based on the average decrease
Since the overall average decreased by 4.5 marks, we can set up the equation:

\[ \frac{(48 \times 66) + ((x - 48) \times (A + 3.5))}{x} = \frac{(48 \times 78) + ((x - 48) \times A)}{x} - 4.5 \]

### Step 7: Simplify the equation
Cross-multiplying and simplifying gives us:

\[ (48 \times 66) + ((x - 48) \times (A + 3.5)) = (48 \times 78) + ((x - 48) \times A) - 4.5x \]

### Step 8: Substitute known values
Substituting the values we calculated:

\[ (48 \times 66) + (x - 48)(A + 3.5) = (48 \times 78) + (x - 48)A - 4.5x \]

### Step 9: Solve for \( x \)
After solving the equation step by step, we will find that:

\[ x = 93 \]

### Conclusion
The total number of students that attended the exam is **93**.

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