A small hotel has 15 rooms which are all occupied. If each room is occupied by either one or two guests and there are 27 guests in total, how many rooms are occupied by two guests?

Answer

To solve the problem, we need to determine how many rooms in the hotel are occupied by two guests, given that there are a total of 15 rooms and 27 guests.

### Step-by-Step Solution:

1. **Define Variables**:
Let \( x \) be the number of rooms occupied by two guests, and \( y \) be the number of rooms occupied by one guest.

2. **Set Up Equations**:
From the problem, we know:
- The total number of rooms is 15:
\[ x + y = 15 \quad \text{(1)} \]
- The total number of guests is 27:
\[ 2x + y = 27 \quad \text{(2)} \]
Here, \( 2x \) accounts for the guests in the rooms with two guests, and \( y \) accounts for the guests in the rooms with one guest.

3. **Solve the System of Equations**:
We can solve these equations simultaneously. First, we can rearrange equation (1) to express \( y \) in terms of \( x \):
\[ y = 15 - x \quad \text{(3)} \]
Now, substitute equation (3) into equation (2):
\[ 2x + (15 - x) = 27 \]
Simplifying this gives:
\[ 2x + 15 - x = 27 \]
\[ x + 15 = 27 \]
\[ x = 27 - 15 \]
\[ x = 12 \]

4. **Find \( y \)**:
Now, substitute \( x = 12 \) back into equation (3) to find \( y \):
\[ y = 15 - 12 = 3 \]

5. **Conclusion**:
The number of rooms occupied by two guests is \( x = 12 \).

### Final Answer:
**12 rooms are occupied by two guests.**