If simple interest is 60 and CI is 66 for 2 years at certain rate of interest on some amount then find the amount and rate of interest?

Options

A. ₹ 120, 50%
B. ₹ 150, 20%
C. ₹ 150, 30%
D. ₹ 100, 30%

Answer

To solve the problem, we need to find the principal amount and the rate of interest given the simple interest (SI) and compound interest (CI) for 2 years.

### Step-by-Step Solution:

1. **Identify the given values:**
- Simple Interest (SI) for 2 years = 60
- Compound Interest (CI) for 2 years = 66

2. **Calculate the difference between CI and SI:**
\[ \text{Difference} = \text{CI} - \text{SI} = 66 - 60 = 6 \]

3. **Understand the relationship between SI and CI:**
- The difference between CI and SI for 2 years is due to the interest on the interest earned in the first year. This difference can be represented as:
\[ \text{Difference} = \text{SI for 1 year} \times \frac{r}{100} \]
where \( r \) is the rate of interest.

4. **Calculate SI for 1 year:**
- Since SI for 2 years is 60, SI for 1 year is:
\[ \text{SI for 1 year} = \frac{60}{2} = 30 \]

5. **Set up the equation to find the rate of interest (r):**
\[ 6 = 30 \times \frac{r}{100} \]
Rearranging gives:
\[ r = \frac{6 \times 100}{30} = 20 \]

6. **Now we have the rate of interest:**
- Rate of interest \( r = 20\% \).

7. **Let the principal amount be \( X \). Calculate the Compound Interest (CI) using the formula:**
\[ \text{CI} = X \left(1 + \frac{r}{100}\right)^2 - X \]
Substituting \( r = 20 \):
\[ \text{CI} = X \left(1 + \frac{20}{100}\right)^2 - X = X \left(1.2^2\right) - X = X(1.44) - X = 0.44X \]

8. **Set up the equation using the given CI:**
\[ 0.44X = 66 \]
Solving for \( X \):
\[ X = \frac{66}{0.44} = 150 \]

9. **Final results:**
- Principal amount \( X = 150 \)
- Rate of interest \( r = 20\% \)

### Summary:
- The principal amount is **150**.
- The rate of interest is **20%**.