The ratio of length and perimeter of any rectangle is 5 : 16, then the ratio of its length and breadth is :

Question

Accepted Answer

To solve the problem and justify why Option 4 (5:3) is the correct answer, let's analyze the given information and apply logical reasoning.

### Explanation:

1. **Given Information:**
   - The ratio of the length (L) to the perimeter (P) of a rectangle is 5:16.

2. **Perimeter of a Rectangle:**
   - The formula for the perimeter of a rectangle is $ P = 2(L + B) $, where L is the length and B is the breadth.

3. **Setting Up the Equation:**
   - According to the given ratio, $\frac{L}{P} = \frac{5}{16}$.
   - Substituting the formula for the perimeter, we have $\frac{L}{2(L + B)} = \frac{5}{16}$.

4. **Solving for the Ratio of Length to Breadth:**
   - Cross-multiply to solve for L and B:
     $$
     16L = 10(L + B)
     $$
   - Simplify the equation:
     $$
     16L = 10L + 10B
     $$
   - Rearrange to find the relationship between L and B:
     $$
     6L = 10B
     $$
   - Simplify the ratio:
     $$
     \frac{L}{B} = \frac{10}{6} = \frac{5}{3}
     $$

5. **Conclusion:**
   - The ratio of the length to the breadth of the rectangle is 5:3, which corresponds to Option 4.

### Why Other Options are Incorrect:

- **Option 1 (5:11):** 
  - This ratio does not satisfy the equation derived from the given perimeter ratio. When substituting back into the equation, it does not hold true.

- **Option 2 (5:8):**
  - Similarly, this ratio does not satisfy the equation $\frac{L}{B} = \frac{5}{3}$ derived from the given conditions.

- **Option 3 (5:4):**
  - This ratio also fails to satisfy the equation derived from the given perimeter ratio. It does not simplify to the correct ratio of 5:3.

By following the logical steps and calculations, we confirm that Option 4 (5:3) is indeed the correct answer, as it is the only ratio that satisfies the conditions given in the problem.