What will be the area of a 2 metre wide boundary around a reactangular field ?
I.The breadth of the field is one-fourth of its permeter. Area of the field is 144 sq. metre.
II. The ratio of the length and breadth is respectively `3:2.`
III Area of the field is 216 sq. metre.

Options

A. I, II and III
B. only I or II and III
C. I and II
D. Anyone of the three statements

Explanation

To find the area of a 2-meter wide boundary around a rectangular field, we need to analyze the given statements step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the area of a boundary that is 2 meters wide surrounding a rectangular field. The area of the boundary can be calculated by finding the area of the larger rectangle (field + boundary) and subtracting the area of the field.

2. **Area of the Field**:
- The area of the field is given in the statements.
- We will analyze each statement to determine if it provides sufficient information to find the dimensions of the field.

3. **Analyzing Statement I**:
- The breadth of the field is one-fourth of its perimeter.
- The area of the field is 144 sq. meters.
- Let the length be \( L \) and the breadth be \( B \).
- From the area, we have \( L \times B = 144 \).
- The perimeter \( P \) is given by \( P = 2(L + B) \).
- According to the statement, \( B = \frac{P}{4} \).
- Substituting \( P \) gives us \( B = \frac{2(L + B)}{4} \) or \( B = \frac{L + B}{2} \).
- Rearranging gives \( L = B \).
- Substituting \( L = B \) into the area equation \( L \times B = 144 \) gives \( L^2 = 144 \), hence \( L = 12 \) and \( B = 12 \).
- The dimensions of the field are \( 12 \times 12 \) (which is a square).
- The area of the larger rectangle (field + boundary) is \( (12 + 4) \times (12 + 4) = 16 \times 16 = 256 \) sq. meters.
- The area of the boundary is \( 256 - 144 = 112 \) sq. meters.

4. **Analyzing Statement II**:
- The ratio of the length and breadth is 3:2.
- Let \( L = 3x \) and \( B = 2x \).
- The area of the field is not provided in this statement, so we cannot calculate the exact dimensions or area of the field.
- Therefore, Statement II alone is insufficient.

5. **Analyzing Statement III**:
- The area of the field is given as 216 sq. meters.
- We can use the ratio from Statement II (3:2) to find the dimensions.
- Let \( L = 3x \) and \( B = 2x \).
- Then, \( 3x \times 2x = 216 \) gives \( 6x^2 = 216 \) or \( x^2 = 36 \), hence \( x = 6 \).
- Thus, \( L = 18 \) and \( B = 12 \).
- The area of the larger rectangle (field + boundary) is \( (18 + 4) \times (12 + 4) = 22 \times 16 = 352 \) sq. meters.
- The area of the boundary is \( 352 - 216 = 136 \) sq. meters.

### Conclusion:
- **Statement I** alone is sufficient to determine the area of the boundary as 112 sq. meters.
- **Statement II** is insufficient on its own.
- **Statement III** alone is also sufficient to determine the area of the boundary as 136 sq. meters.

### Final Answer:
The area of the 2-meter wide boundary around the rectangular field can be determined from Statement I or Statement III.