The difference between the simple interest and the compound interest at the same rate of interest on a sum of money at the end of the second year will be Rs. `4cdot00`. If the rate of interest is `5%` per annum, what is the sum?

Options

A. Rs. 1200
B. Rs. 1600
C. Rs. 2000
D. None of these

Answer

To solve the problem, we need to find the sum of money (principal) based on the given difference between simple interest (SI) and compound interest (CI) at the end of the second year. The difference is given as Rs. 4.00, and the rate of interest is 5% per annum.

### Step-by-Step Solution:

1. **Understand the formulas**:
- The formula for Simple Interest (SI) for 2 years is:
\[ SI = \frac{P \times R \times T}{100} \]
where \(P\) is the principal, \(R\) is the rate of interest, and \(T\) is the time in years.
- The formula for Compound Interest (CI) for 2 years is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
For 2 years, this can be simplified to:
\[ CI = P \left(1 + \frac{R}{100}\right)^2 - P \]

2. **Calculate SI for 2 years**:
- Given \(R = 5\%\) and \(T = 2\):
\[ SI = \frac{P \times 5 \times 2}{100} = \frac{10P}{100} = \frac{P}{10} \]

3. **Calculate CI for 2 years**:
- Using the formula for CI:
\[ CI = P \left(1 + \frac{5}{100}\right)^2 - P = P \left(1.05^2\right) - P \]
- Calculate \(1.05^2\):
\[ 1.05^2 = 1.1025 \]
- Therefore:
\[ CI = P \times 1.1025 - P = 0.1025P \]

4. **Find the difference between CI and SI**:
- The difference between CI and SI is:
\[ CI - SI = 0.1025P - \frac{P}{10} \]
- Convert \(\frac{P}{10}\) to a decimal:
\[ \frac{P}{10} = 0.1P \]
- Thus:
\[ CI - SI = 0.1025P - 0.1P = 0.0025P \]

5. **Set the difference equal to Rs. 4**:
- According to the problem:
\[ 0.0025P = 4 \]

6. **Solve for \(P\)**:
- To find \(P\), divide both sides by 0.0025:
\[ P = \frac{4}{0.0025} = 1600 \]

### Conclusion:
The sum of money (principal) is Rs. 1600.