In an examination paper of five questions, 5% of the candidates answered all of them and 5% answered none. Of the rest, 25% candidates answered only one question and 20% answered 4 questions. If 396 candidates answered either 2 questions or 3 questions, the number of candidates that appeared for the examination was

Options

A. 800
B. 1000
C. 850
D. 900

Answer

To solve the problem step by step, we will break down the information given and calculate the total number of candidates who appeared for the examination.

### Step 1: Define the total number of candidates
Let the total number of candidates who appeared for the examination be \( A \).

### Step 2: Calculate the number of candidates who answered all questions
5% of the candidates answered all 5 questions. Therefore, the number of candidates who answered all questions is:
\[ \text{Candidates who answered all} = \frac{5}{100} \times A = \frac{A}{20} \]

### Step 3: Calculate the number of candidates who answered none
Similarly, 5% of the candidates answered none of the questions. Therefore, the number of candidates who answered none is also:
\[ \text{Candidates who answered none} = \frac{5}{100} \times A = \frac{A}{20} \]

### Step 4: Calculate the remaining candidates
The remaining candidates are those who answered either 1, 2, 3, or 4 questions. We can find this by subtracting the candidates who answered all and none from the total:
\[ \text{Remaining candidates} = A - \left(\frac{A}{20} + \frac{A}{20}\right) = A - \frac{2A}{20} = A - \frac{A}{10} = \frac{9A}{10} \]

### Step 5: Calculate the number of candidates who answered only one question
25% of the remaining candidates answered only one question:
\[ \text{Candidates who answered only one} = \frac{25}{100} \times \frac{9A}{10} = \frac{9A}{40} \]

### Step 6: Calculate the number of candidates who answered four questions
20% of the remaining candidates answered four questions:
\[ \text{Candidates who answered four} = \frac{20}{100} \times \frac{9A}{10} = \frac{9A}{50} \]

### Step 7: Calculate the number of candidates who answered either 2 or 3 questions
We know that 396 candidates answered either 2 or 3 questions. Thus, we can express this as:
\[ \text{Candidates who answered 2 or 3} = \frac{9A}{10} - \left(\frac{9A}{40} + \frac{9A}{50}\right) \]

### Step 8: Find a common denominator and simplify
To simplify \( \frac{9A}{40} + \frac{9A}{50} \), we find the LCM of 40 and 50, which is 200:
\[ \frac{9A}{40} = \frac{9A \times 5}{200} = \frac{45A}{200} \]
\[ \frac{9A}{50} = \frac{9A \times 4}{200} = \frac{36A}{200} \]
Thus,
\[ \frac{9A}{40} + \frac{9A}{50} = \frac{45A + 36A}{200} = \frac{81A}{200} \]

### Step 9: Set up the equation
Now we can set up the equation for the candidates who answered either 2 or 3 questions:
\[ \frac{9A}{10} - \frac{81A}{200} = 396 \]
Converting \( \frac{9A}{10} \) to have a denominator of 200 gives:
\[ \frac{9A \times 20}{200} = \frac{180A}{200} \]
Thus, the equation becomes:
\[ \frac{180A}{200} - \frac{81A}{200} = 396 \]
\[ \frac{99A}{200} = 396 \]

### Step 10: Solve for \( A \)
Now, multiply both sides by 200:
\[ 99A = 396 \times 200 \]
\[ 99A = 79200 \]
Now, divide by 99:
\[ A = \frac{79200}{99} = 800 \]

### Conclusion
The total number of candidates that appeared for the examination is \( \boxed{800} \).