In an exam, the average marks obtained by the students was found to be 60. After omission of computational errors, the average marks of some 100 candidates had to be changed from 60 to 30 and the average with respect to all the examinees came down to 45 marks. The total number of candidates who took the exam, was

Options

A. 200
B. 210
C. 240
D. 180

Answer

To solve the problem step by step, we will use the information provided to find the total number of candidates who took the exam.

### Step 1: Define Variables
Let \( N \) be the total number of candidates who took the exam.

### Step 2: Calculate Total Marks Before Omission
The average marks of all students before the omission of errors was 60. Therefore, the total marks of all candidates can be expressed as:
\[ \text{Total Marks} = \text{Average} \times \text{Number of Candidates} = 60N \]

### Step 3: Calculate Total Marks After Omission
After the omission of computational errors, the average marks of the 100 candidates changed from 60 to 30. The total marks for these 100 candidates can now be calculated as:
\[ \text{Total Marks of 100 Candidates} = 30 \times 100 = 3000 \]

### Step 4: Calculate the New Total Marks
The total marks of the remaining \( N - 100 \) candidates (those not affected by the error) remain the same. Therefore, the new total marks for all candidates after correcting the errors is:
\[ \text{New Total Marks} = 3000 + \text{Total Marks of Remaining Candidates} \]

### Step 5: Calculate Total Marks of Remaining Candidates
The total marks of the remaining candidates can be calculated as:
\[ \text{Total Marks of Remaining Candidates} = 60N - 6000 \]
This is because the original total marks for all candidates was \( 60N \), and we are subtracting the incorrect total marks of the 100 candidates (which was \( 60 \times 100 = 6000 \)).

### Step 6: Calculate New Average for All Candidates
The new average for all candidates is given as 45. Therefore, we can set up the equation:
\[ \frac{3000 + (60N - 6000)}{N} = 45 \]

### Step 7: Simplify the Equation
Simplifying the equation gives:
\[ \frac{60N - 3000}{N} = 45 \]
Multiplying both sides by \( N \) results in:
\[ 60N - 3000 = 45N \]

### Step 8: Solve for \( N \)
Rearranging the equation gives:
\[ 60N - 45N = 3000 \]
\[ 15N = 3000 \]
\[ N = \frac{3000}{15} = 200 \]

### Conclusion
The total number of candidates who took the exam is \( N = 200 \).

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