If for a certain amount and same interest rate the simple interest is Rs. 600 and compound interest is Rs. 612 of two years, then annual rate of interest is :

Question

Accepted Answer

To determine the annual rate of interest, we need to understand the relationship between simple interest (SI) and compound interest (CI) over two years. Given that the simple interest is Rs. 600 and the compound interest is Rs. 612 for two years, we can use these values to find the rate of interest.

### Explanation for Option 3 (0.04) as the Correct Answer:

1. **Simple Interest Formula:**
   $$
   \text{SI} = \frac{P \times R \times T}{100}
   $$
   Where $ P $ is the principal, $ R $ is the rate of interest per annum, and $ T $ is the time in years. Given that SI is Rs. 600 for two years, we have:
   $$
   600 = \frac{P \times R \times 2}{100}
   $$
   Simplifying, we get:
   $$
   P \times R = 30000
   $$

2. **Compound Interest Formula:**
   $$
   \text{CI} = P \left(1 + \frac{R}{100}\right)^T - P
   $$
   Given that CI is Rs. 612 for two years, we have:
   $$
   612 = P \left(1 + \frac{R}{100}\right)^2 - P
   $$
   Simplifying, we get:
   $$
   612 = P \left(\left(1 + \frac{R}{100}\right)^2 - 1\right)
   $$

3. **Equating and Solving:**
   From the simple interest equation, we know $ P \times R = 30000 $. Substituting this into the compound interest equation:
   $$
   612 = \frac{30000}{R} \left(\left(1 + \frac{R}{100}\right)^2 - 1\right)
   $$
   Solving this equation for $ R $, we find that $ R = 4\% $ or 0.04.

### Explanation for Why Other Options are Incorrect:

1. **Option 1 (0.03):**
   If the rate were 3%, the compound interest calculation would not yield Rs. 612. The difference between compound and simple interest would be less than Rs. 12, which contradicts the given data.

2. **Option 2 (0.06):**
   A 6% rate would result in a much higher compound interest than Rs. 612 due to the compounding effect over two years, making this option incorrect.

3. **Option 4 (0.08):**
   An 8% rate would significantly increase the compound interest beyond Rs. 612, as the compounding effect would be even more pronounced, thus making this option incorrect.

In conclusion, the calculations and logical reasoning confirm that the annual rate of interest is indeed 4%, making Option 3 the correct choice.