A rectangular lawn is surrounded by a path of width 2 m on all sides . Now if the length of the lawn is reduced by 2m the lawn becomes a square lawn and the area of path becomes 13/11 times, what is the length of the original lawn ?

Options

A. a. 8 m
B. b. 9 m
C. c. 10 m
D. d. 12 m

Answer

To solve the problem step by step, we will denote the original length of the lawn as \( L \) and the original breadth as \( B \).

### Step 1: Understand the dimensions of the lawn and path
The rectangular lawn is surrounded by a path of width 2 m on all sides. Therefore, the dimensions of the area including the path will be:
- Length of the area including the path: \( L + 4 \) (2 m on each side)
- Breadth of the area including the path: \( B + 4 \)

### Step 2: Establish the relationship after reducing the length
According to the problem, if the length of the lawn is reduced by 2 m, the lawn becomes a square. This means:
\[ B = L - 2 \]

### Step 3: Calculate the area of the path
The area of the path can be calculated in two ways:
1. **New Area of the Path** after reducing the length:
\[ \text{New Area of Path} = (L + 4)(B + 4) - (L - 2)(B) \]
Substituting \( B = L - 2 \):
\[ \text{New Area of Path} = (L + 4)((L - 2) + 4) - (L - 2)(L - 2) \]
Simplifying this:
\[ = (L + 4)(L + 2) - (L - 2)^2 \]
\[ = (L^2 + 6L + 8) - (L^2 - 4L + 4) \]
\[ = L^2 + 6L + 8 - L^2 + 4L - 4 \]
\[ = 10L + 4 \]

2. **Original Area of the Path**:
\[ \text{Original Area of Path} = (L + 4)(B + 4) - LB \]
Substituting \( B = L - 2 \):
\[ = (L + 4)((L - 2) + 4) - L(L - 2) \]
\[ = (L + 4)(L + 2) - (L^2 - 2L) \]
\[ = (L^2 + 6L + 8) - (L^2 - 2L) \]
\[ = L^2 + 6L + 8 - L^2 + 2L \]
\[ = 8L + 8 \]

### Step 4: Set up the equation based on the area relationship
According to the problem, the new area of the path is \( \frac{13}{11} \) times the original area of the path:
\[ 10L + 4 = \frac{13}{11}(8L + 8) \]

### Step 5: Solve the equation
Cross-multiplying to eliminate the fraction:
\[ 11(10L + 4) = 13(8L + 8) \]
\[ 110L + 44 = 104L + 104 \]
Rearranging gives:
\[ 110L - 104L = 104 - 44 \]
\[ 6L = 60 \]
\[ L = 10 \]

### Conclusion
The length of the original lawn is \( L = 10 \) m.