The difference between simple and compound interest compounded annually, on a certain sum of money for 2 years at 4% per annum is Re 1. The sum (in Rs.) is:

Options

A. 650
B. 630
C. 625
D. 640

Answer

To solve the problem, we need to find the sum of money (P) for which the difference between simple interest (SI) and compound interest (CI) compounded annually for 2 years at a rate of 4% per annum is Re 1.

### Step-by-Step Solution:

1. **Understanding Simple Interest (SI)**:
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount (the sum of money)
- \( R \) = Rate of interest (4% in this case)
- \( T \) = Time (2 years)

Substituting the values:
\[ SI = \frac{P \times 4 \times 2}{100} = \frac{8P}{100} = \frac{2P}{25} \]

2. **Understanding Compound Interest (CI)**:
The formula for Compound Interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where \( A \) is the amount after time \( T \).

Substituting the values:
\[ A = P \left(1 + \frac{4}{100}\right)^2 = P \left(1 + 0.04\right)^2 = P \left(1.04\right)^2 \]
Calculating \( (1.04)^2 \):
\[ (1.04)^2 = 1.0816 \]
Therefore,
\[ A = 1.0816P \]
The Compound Interest (CI) is then:
\[ CI = A - P = 1.0816P - P = 0.0816P \]

3. **Finding the Difference**:
The difference between Compound Interest and Simple Interest is given as Re 1:
\[ CI - SI = 1 \]
Substituting the expressions we found:
\[ 0.0816P - \frac{2P}{25} = 1 \]

4. **Finding a Common Denominator**:
The LCM of 25 and 1 (the coefficient of \( P \) in \( 0.0816P \)) is 25. We convert \( 0.0816 \) to a fraction:
\[ 0.0816 = \frac{816}{10000} = \frac{204}{2500} \]
Now, we rewrite the equation:
\[ \frac{204P}{2500} - \frac{2P}{25} = 1 \]
Converting \( \frac{2P}{25} \) to have a denominator of 2500:
\[ \frac{2P}{25} = \frac{200P}{2500} \]
Thus, we have:
\[ \frac{204P - 200P}{2500} = 1 \]
Simplifying gives:
\[ \frac{4P}{2500} = 1 \]

5. **Solving for P**:
Cross-multiplying gives:
\[ 4P = 2500 \]
Dividing both sides by 4:
\[ P = \frac{2500}{4} = 625 \]

### Final Answer:
The sum of money is **Rs. 625**.