The hotel bill for a number of persons for overnight stay is Rs 4,800 If there were four more persons, the bill each person had to pay would have reduced by Rs 200. Find the number of persons staying overnight.

Answer

To solve the problem, we need to find the number of persons (let's call it \( X \)) staying overnight at the hotel, given that the total bill is Rs 4,800. If there were four more persons, the cost per person would decrease by Rs 200.

### Step-by-Step Solution:

1. **Define the Variables:**
Let \( X \) be the number of persons staying overnight.

2. **Calculate the Cost per Person:**
The total bill for \( X \) persons is Rs 4,800. Therefore, the cost per person is:
\[ \text{Cost per person} = \frac{4800}{X} \]

3. **Calculate the New Cost per Person with Additional Persons:**
If there were 4 more persons, the number of persons would be \( X + 4 \). The new cost per person would be:
\[ \text{New cost per person} = \frac{4800}{X + 4} \]

4. **Set Up the Equation:**
According to the problem, the difference in cost per person when 4 more persons are added is Rs 200. Therefore, we can set up the equation:
\[ \frac{4800}{X} - \frac{4800}{X + 4} = 200 \]

5. **Simplify the Equation:**
To simplify, we can take \( 4800 \) common:
\[ 4800 \left( \frac{1}{X} - \frac{1}{X + 4} \right) = 200 \]
Dividing both sides by 4800 gives:
\[ \frac{1}{X} - \frac{1}{X + 4} = \frac{200}{4800} \]
Simplifying \( \frac{200}{4800} \) gives:
\[ \frac{1}{X} - \frac{1}{X + 4} = \frac{1}{24} \]

6. **Combine the Fractions:**
The left-hand side can be combined:
\[ \frac{(X + 4) - X}{X(X + 4)} = \frac{4}{X(X + 4)} \]
Setting this equal to \( \frac{1}{24} \):
\[ \frac{4}{X(X + 4)} = \frac{1}{24} \]

7. **Cross Multiply:**
Cross multiplying gives:
\[ 4 \cdot 24 = X(X + 4) \]
Simplifying this gives:
\[ 96 = X^2 + 4X \]

8. **Rearrange the Equation:**
Rearranging the equation gives:
\[ X^2 + 4X - 96 = 0 \]

9. **Factor the Quadratic Equation:**
To factor, we look for two numbers that multiply to -96 and add to 4. These numbers are 12 and -8:
\[ (X - 8)(X + 12) = 0 \]

10. **Solve for \( X \):**
Setting each factor to zero gives:
\[ X - 8 = 0 \quad \Rightarrow \quad X = 8 \]
\[ X + 12 = 0 \quad \Rightarrow \quad X = -12 \quad (\text{not valid since } X \text{ cannot be negative}) \]

11. **Conclusion:**
The number of persons staying overnight is \( X = 8 \).

### Verification:
- If there are 8 persons, the cost per person is \( \frac{4800}{8} = 600 \).
- If there are 12 persons (8 + 4), the cost per person is \( \frac{4800}{12} = 400 \).
- The difference in cost per person is \( 600 - 400 = 200 \), which matches the problem statement.

### Final Answer:
The number of persons staying overnight is **8**.