A path of uniform width, 2.5 m, runs around the inside of a rectangular field 30 m by 27 m. Find the area of the path.

Answer

To find the area of the path that runs around the inside of a rectangular field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the dimensions of the outer rectangle**:
The dimensions of the rectangular field are given as:
- Length = 30 m
- Width = 27 m

2. **Calculate the area of the outer rectangle**:
The area of a rectangle is calculated using the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
So, for the outer rectangle:
\[ \text{Area}_{\text{outer}} = 30 \, \text{m} \times 27 \, \text{m} = 810 \, \text{m}^2 \]

3. **Determine the dimensions of the inner rectangle**:
The path has a uniform width of 2.5 m running around the inside of the rectangle. Therefore, we need to subtract twice the width of the path from each dimension of the outer rectangle (once for each side):
- Length of inner rectangle = \(30 \, \text{m} - 2 \times 2.5 \, \text{m} = 30 \, \text{m} - 5 \, \text{m} = 25 \, \text{m}\)
- Width of inner rectangle = \(27 \, \text{m} - 2 \times 2.5 \, \text{m} = 27 \, \text{m} - 5 \, \text{m} = 22 \, \text{m}\)

4. **Calculate the area of the inner rectangle**:
Using the same area formula:
\[ \text{Area}_{\text{inner}} = 25 \, \text{m} \times 22 \, \text{m} = 550 \, \text{m}^2 \]

5. **Find the area of the path**:
The area of the path is the difference between the area of the outer rectangle and the area of the inner rectangle:
\[ \text{Area}_{\text{path}} = \text{Area}_{\text{outer}} - \text{Area}_{\text{inner}} = 810 \, \text{m}^2 - 550 \, \text{m}^2 = 260 \, \text{m}^2 \]

### Final Answer:
The area of the path is \(260 \, \text{m}^2\).

---