Explanation:

- Simple Interest Calculation:

- Principal doubles in 8 years under simple interest (SI).

- SI formula: \( \text{SI} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100} \).

- 2 = 1 + \((\frac{\text{Rate} \times 8}{100})\) gives the rate of simple interest, \( R_s \).

- \( R_s = 12.5\% \).

- Compound Interest Calculation:

- Principal quadruples in 2 years under compound interest (CI).

- CI formula for 4 times: \( P(1 + \frac{R_c}{100})^2 = 4P \).

- Solves to find the rate of compound interest, \( R_c \).

- \( R_c \approx 100\% \).

- Ratio Calculation:

- Ratio of rates of SI to CI: \( \frac{12.5}{100} = \frac{1}{8} \).

- Thus, Option 4 matches \(1 : 8\).

- Option Verification:

- Option 1: \(2 : 3\)

- Option 2: \(3 : 5\)

- Option 3: \(5 : 3\)

- Option 4: \(1 : 8\)