The difference between the simple and compound interest on a certain sum of money for 2 years at 4% per annum is Rs. 4. The sum is

Answer

To solve the problem, we need to find the principal sum (P) based on the given information about the difference between simple interest (SI) and compound interest (CI) over 2 years at an interest rate of 4% per annum.

### Step-by-Step Solution:

1. **Understand the Formula for Simple Interest (SI)**:
The formula for Simple Interest for 2 years is:
\[ \text{SI} = P \times \frac{R}{100} \times T \]
Where \( P \) is the principal, \( R \) is the rate of interest, and \( T \) is the time in years. For our case, \( R = 4\% \) and \( T = 2 \).

\[ \text{SI} = P \times \frac{4}{100} \times 2 = \frac{8P}{100} = 0.08P \]

2. **Understand the Formula for Compound Interest (CI)**:
The formula for Compound Interest for 2 years is:
\[ \text{CI} = P \left(1 + \frac{R}{100}\right)^T - P \]
For our case, substituting \( R = 4\% \) and \( T = 2 \):
\[ \text{CI} = P \left(1 + \frac{4}{100}\right)^2 - P = P \left(1.04^2\right) - P \]
Calculating \( 1.04^2 \):
\[ 1.04^2 = 1.0816 \]
Therefore:
\[ \text{CI} = P \times 1.0816 - P = 0.0816P \]

3. **Calculate the Difference Between CI and SI**:
The difference between Compound Interest and Simple Interest is given as Rs. 4:
\[ \text{Difference} = \text{CI} - \text{SI} = 0.0816P - 0.08P \]
Simplifying this:
\[ \text{Difference} = 0.0016P \]
Setting this equal to Rs. 4:
\[ 0.0016P = 4 \]

4. **Solve for Principal (P)**:
To find \( P \), we rearrange the equation:
\[ P = \frac{4}{0.0016} \]
Calculating \( P \):
\[ P = 2500 \]

### Conclusion:
The principal sum is Rs. 2500.