The ratio between the length and the perimeter of a rectangular plot is `1 : 3`. The ratio between the length and breadth of the plot is `k : 1`. Find `k`.

Answer

To solve the problem, we need to find the value of \( k \) given the ratios involving the length \( L \) and breadth \( B \) of a rectangular plot.

### Step-by-Step Solution:

1. **Understanding the Ratios**:
- We know that the ratio of the length \( L \) to the perimeter \( P \) of the rectangle is \( 1:3 \).
- The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(L + B) \]
- Therefore, we can express the ratio as:
\[ \frac{L}{P} = \frac{1}{3} \]

2. **Expressing the Perimeter in Terms of Length**:
- From the ratio, we can write:
\[ L = \frac{1}{3} P \]
- Substituting the formula for perimeter:
\[ L = \frac{1}{3} \times 2(L + B) \]
- Simplifying this gives:
\[ L = \frac{2}{3}(L + B) \]

3. **Eliminating the Fraction**:
- To eliminate the fraction, multiply both sides by 3:
\[ 3L = 2(L + B) \]

4. **Expanding and Rearranging**:
- Expanding the right side:
\[ 3L = 2L + 2B \]
- Rearranging gives:
\[ 3L - 2L = 2B \quad \Rightarrow \quad L = 2B \]

5. **Finding the Ratio of Length to Breadth**:
- We want to express the ratio \( \frac{L}{B} \):
\[ \frac{L}{B} = \frac{2B}{B} = 2 \]
- Therefore, the ratio of length to breadth is:
\[ L:B = 2:1 \]

6. **Identifying \( k \)**:
- Since the ratio of length to breadth is \( k:1 \), we can see that \( k = 2 \).

### Final Answer:
The value of \( k \) is \( 2 \).

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