If the difference between the compound interest and the simple interest on a certain sum for 2 yr at 8% per annum is ? 32, then the sum is

Options

A. ? 5000
B. ? 5500
C. ? 6000
D. ? 5250

Answer

To solve the problem, we need to find the principal sum (P) given that the difference between the compound interest (CI) and the simple interest (SI) for 2 years at an interest rate of 8% per annum is ₹32.

### Step-by-Step Solution:

1. **Understand the Formulas**:
- Simple Interest (SI) is calculated using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
- Compound Interest (CI) for 2 years can be calculated using the formula:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
where \( A \) is the total amount after time \( T \), and then:
\[ CI = A - P \]

2. **Calculate Simple Interest**:
- Given \( R = 8\% \) and \( T = 2 \) years, we can substitute these values into the SI formula:
\[ SI = \frac{P \times 8 \times 2}{100} = \frac{16P}{100} = \frac{4P}{25} \]

3. **Calculate Compound Interest**:
- Using the compound interest formula for 2 years:
\[ A = P \left(1 + \frac{8}{100}\right)^2 = P \left(1 + 0.08\right)^2 = P \left(1.08\right)^2 \]
- Calculate \( (1.08)^2 \):
\[ (1.08)^2 = 1.1664 \]
- Therefore, the amount \( A \) becomes:
\[ A = 1.1664P \]
- Now, calculate the compound interest:
\[ CI = A - P = 1.1664P - P = 0.1664P \]

4. **Set Up the Equation for the Difference**:
- We know that the difference between CI and SI is ₹32:
\[ CI - SI = 32 \]
- Substitute the values we found:
\[ 0.1664P - \frac{4P}{25} = 32 \]

5. **Convert SI to a Decimal**:
- Convert \( \frac{4P}{25} \) to a decimal:
\[ \frac{4P}{25} = 0.16P \]
- Now, substitute this back into the equation:
\[ 0.1664P - 0.16P = 32 \]

6. **Simplify the Equation**:
- Simplifying gives:
\[ 0.0064P = 32 \]

7. **Solve for P**:
- Divide both sides by 0.0064:
\[ P = \frac{32}{0.0064} = 5000 \]

### Final Answer:
The sum (P) is ₹5000.