The difference between simple interest (compounded annually) on a sum of money for 2 years at 10% per annum Is Rs 65. The sum is-

Options

A. Rs 65,065
B. Rs 6,565
C. Rs 6,500
D. Rs 65,650

Answer

To solve the problem, we need to find the principal amount (P) given that the difference between simple interest (SI) and compound interest (CI) for 2 years at a rate of 10% per annum is Rs 65.

### Step-by-Step Solution:

1. **Define the Principal Amount**:
Let the principal amount be \( P \).

2. **Calculate Simple Interest (SI)**:
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Here, \( R = 10\% \) and \( T = 2 \) years.
\[ SI = \frac{P \times 10 \times 2}{100} = \frac{20P}{100} = \frac{P}{5} \]

3. **Calculate Compound Interest (CI)**:
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Here, \( R = 10\% \) and \( T = 2 \) years.
\[ A = P \left(1 + \frac{10}{100}\right)^2 = P \left(\frac{11}{10}\right)^2 = P \times \frac{121}{100} \]
The compound interest (CI) is then:
\[ CI = A - P = \frac{121P}{100} - P = \frac{121P}{100} - \frac{100P}{100} = \frac{21P}{100} \]

4. **Set Up the Equation for the Difference**:
According to the problem, the difference between CI and SI is Rs 65:
\[ CI - SI = 65 \]
Substituting the values we found:
\[ \frac{21P}{100} - \frac{P}{5} = 65 \]
Convert \(\frac{P}{5}\) to a fraction with a denominator of 100:
\[ \frac{P}{5} = \frac{20P}{100} \]
Now the equation becomes:
\[ \frac{21P}{100} - \frac{20P}{100} = 65 \]
Simplifying gives:
\[ \frac{P}{100} = 65 \]

5. **Solve for P**:
To find \( P \):
\[ P = 65 \times 100 = 6500 \]

### Final Answer:
The sum of money (principal amount) is Rs 6500.

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