The compound interest on a certain sum of money for 2 years at 5% per annum is Rs. 410 . The simple interest on the same sum at the same rate and for the same time is

Options

A. Rs. 400
B. Rs. 300
C. Rs. 350
D. Rs . 405

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 5% per annum is Rs. 410.

### Step-by-Step Solution:

1. **Understand the Relationship Between Compound Interest and Principal**:
We know that the compound interest (CI) for 2 years is Rs. 410. The formula for compound interest for 2 years is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where \(A\) is the total amount after \(n\) years, \(P\) is the principal, \(r\) is the rate of interest, and \(n\) is the number of years.

2. **Calculate the Amount (A)**:
The compound interest for 2 years can also be expressed as:
\[ CI = A - P \]
Therefore, we can write:
\[ A = CI + P \]
Given that \(CI = 410\), we need to express \(A\) in terms of \(P\).

3. **Express A in Terms of P**:
Using the formula for compound interest:
\[ A = P \left(1 + \frac{5}{100}\right)^2 = P \left(1 + 0.05\right)^2 = P \left(1.05\right)^2 \]
Simplifying further:
\[ A = P \times 1.1025 \]

4. **Set Up the Equation**:
Now we can set up the equation:
\[ CI = A - P \implies 410 = P \times 1.1025 - P \]
This simplifies to:
\[ 410 = P(1.1025 - 1) = P \times 0.1025 \]

5. **Solve for Principal (P)**:
Rearranging gives us:
\[ P = \frac{410}{0.1025} = 4000 \]

6. **Calculate Simple Interest (SI)**:
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the values we have:
\[ SI = \frac{4000 \times 5 \times 2}{100} \]
Simplifying this:
\[ SI = \frac{40000}{100} = 400 \]

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