The difference between the compound interest and the simple interest on a certain sum at 12% per annum for 2 yr Is Rs. 90. The sum Is equal to

Options

A. ₹ 6260
B. ₹ 6350
C. ₹ 6250
D. ₹ 6255

Answer

To solve the problem, we need to find the principal amount (P) given that the difference between the compound interest (CI) and simple interest (SI) for 2 years at a rate of 12% per annum is Rs. 90.

### Step-by-Step Solution:

1. **Understanding the Formulas**:
- The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
- The formula for Compound Interest (CI) is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]

2. **Given Values**:
- Rate (R) = 12%
- Time (T) = 2 years
- Difference between CI and SI = Rs. 90

3. **Calculating Simple Interest**:
- Substitute the values into the SI formula:
\[ SI = \frac{P \times 12 \times 2}{100} = \frac{24P}{100} = \frac{6P}{25} \]

4. **Calculating Compound Interest**:
- Substitute the values into the CI formula:
\[ CI = P \left(1 + \frac{12}{100}\right)^2 - P \]
- Simplifying the term inside the parentheses:
\[ CI = P \left(1 + 0.12\right)^2 - P = P \left(1.12\right)^2 - P \]
- Calculating \((1.12)^2\):
\[ (1.12)^2 = 1.2544 \]
- Therefore:
\[ CI = P \times 1.2544 - P = 0.2544P \]

5. **Finding the Difference**:
- The difference between CI and SI is given as Rs. 90:
\[ CI - SI = 90 \]
- Substitute the expressions for CI and SI:
\[ 0.2544P - \frac{6P}{25} = 90 \]

6. **Finding a Common Denominator**:
- Convert \(\frac{6P}{25}\) to a decimal:
\[ \frac{6P}{25} = 0.24P \]
- Now we have:
\[ 0.2544P - 0.24P = 90 \]
- Simplifying:
\[ 0.0144P = 90 \]

7. **Solving for P**:
- Divide both sides by 0.0144:
\[ P = \frac{90}{0.0144} = 6250 \]

### Final Answer:
The principal amount (P) is Rs. 6250.