Five question were asked in an examination. `5%` of the total examinees answered all 5 questions and `5%` of total examinees did not answered any question. `25%` of the rest of the examinees answered only 1 question and `20%` answered 4 questions. `24 (1)/( 2)%` of the total examinees answered only 2 questions. If 200 examinees answered 3 questions, then find the total number of examinees.

Answer

To solve the problem step by step, let's denote the total number of examinees as \( N \).

### Step 1: Calculate the number of examinees who answered all 5 questions and those who did not answer any questions.
- **5% of total examinees answered all 5 questions**:
\[ \text{Number of examinees who answered all 5 questions} = 0.05N \]
- **5% of total examinees did not answer any questions**:
\[ \text{Number of examinees who did not answer any questions} = 0.05N \]

### Step 2: Calculate the remaining examinees.
- Total examinees who answered either all or none:
\[ \text{Total answered or did not answer} = 0.05N + 0.05N = 0.1N \]
- Remaining examinees:
\[ \text{Remaining examinees} = N - 0.1N = 0.9N \]

### Step 3: Calculate the number of examinees who answered only 1 question.
- **25% of the remaining examinees answered only 1 question**:
\[ \text{Number of examinees who answered only 1 question} = 0.25 \times 0.9N = 0.225N \]

### Step 4: Calculate the number of examinees who answered only 4 questions.
- **20% of the remaining examinees answered only 4 questions**:
\[ \text{Number of examinees who answered only 4 questions} = 0.20 \times 0.9N = 0.18N \]

### Step 5: Calculate the number of examinees who answered only 2 questions.
- **24.5% of the total examinees answered only 2 questions**:
\[ \text{Number of examinees who answered only 2 questions} = 0.245N \]

### Step 6: Calculate the number of examinees who answered 3 questions.
- We know that **200 examinees answered 3 questions**.

### Step 7: Set up the equation.
- The total percentage of examinees who answered questions is:
\[ 0.225N + 0.18N + 0.245N + 200 = 0.9N \]
- Combine the terms:
\[ 0.225N + 0.18N + 0.245N = 0.65N \]
- Thus, we have:
\[ 0.65N + 200 = 0.9N \]

### Step 8: Solve for \( N \).
- Rearranging the equation gives:
\[ 200 = 0.9N - 0.65N \]
\[ 200 = 0.25N \]
- Dividing both sides by 0.25:
\[ N = \frac{200}{0.25} = 800 \]

### Final Answer:
The total number of examinees is \( \boxed{800} \).

---