The compound interest on a certain sum of money for 2 years at `10%` per annum isRs. 420. The simple interest on the same sum at the same rate and for the same time will be

Options

A. Rs.350
B. Rs.375
C. Rs.380
D. Rs.400

Answer

To solve the problem, we need to find the simple interest on a certain sum of money, given that the compound interest for 2 years at 10% per annum is Rs. 420.

### Step-by-Step Solution:

1. **Understand the relationship between Compound Interest (CI) and Simple Interest (SI):**
- The formula for Compound Interest for 2 years is:
\[ CI = P \left(1 + \frac{r}{100}\right)^n - P \]
where \( P \) is the principal, \( r \) is the rate of interest, and \( n \) is the number of years.
- The formula for Simple Interest is:
\[ SI = \frac{P \cdot r \cdot n}{100} \]

2. **Given Values:**
- Compound Interest (CI) = Rs. 420
- Rate (r) = 10%
- Time (n) = 2 years

3. **Calculate the Principal (P) using the CI formula:**
- Rearranging the CI formula:
\[ CI = P \left(1 + \frac{r}{100}\right)^n - P \]
\[ 420 = P \left(1 + \frac{10}{100}\right)^2 - P \]
\[ 420 = P \left(1.1^2 - 1\right) \]
\[ 1.1^2 = 1.21 \Rightarrow 1.21 - 1 = 0.21 \]
\[ 420 = P \cdot 0.21 \]
\[ P = \frac{420}{0.21} = 2000 \]

4. **Now calculate the Simple Interest (SI) using the SI formula:**
\[ SI = \frac{P \cdot r \cdot n}{100} \]
\[ SI = \frac{2000 \cdot 10 \cdot 2}{100} \]
\[ SI = \frac{40000}{100} = 400 \]

5. **Final Answer:**
The simple interest on the same sum at the same rate and for the same time is Rs. 400.