The difference between simple and compound interest for the fourth year is ₹ 7280 at `20%` p.a. What is the principal

Options

A. 10000
B. 50000
C. 1 lakh
D. 40000

Answer

To find the principal amount given the difference between simple interest (SI) and compound interest (CI) for the fourth year, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We know that the difference between the compound interest and simple interest for the 4th year is ₹7280 at a rate of 20% per annum. We need to find the principal amount.

2. **Identify the Formula for Interest**:
- Simple Interest for 4 years:
\[ \text{SI} = \frac{P \times R \times T}{100} = \frac{P \times 20 \times 4}{100} = \frac{80P}{100} = 0.8P \]
- Compound Interest for 4 years:
The formula for compound interest is:
\[ \text{CI} = P \left(1 + \frac{R}{100}\right)^T - P \]
For 4 years at 20%:
\[ \text{CI} = P \left(1 + \frac{20}{100}\right)^4 - P = P(1.2^4) - P \]
Calculating \(1.2^4\):
\[ 1.2^4 = 2.0736 \quad \text{(approximately)} \]
Thus,
\[ \text{CI} = P(2.0736 - 1) = P(1.0736) \]

3. **Calculate the Difference**:
- The difference between CI and SI for the 4th year is given by:
\[ \text{Difference} = \text{CI} - \text{SI} = P(1.0736) - 0.8P \]
Simplifying this:
\[ \text{Difference} = P(1.0736 - 0.8) = P(0.2736) \]

4. **Set Up the Equation**:
- We know the difference is ₹7280:
\[ P(0.2736) = 7280 \]

5. **Solve for Principal (P)**:
- Rearranging the equation gives:
\[ P = \frac{7280}{0.2736} \]
- Calculating \(P\):
\[ P \approx 26666.67 \]

6. **Final Calculation**:
- To find the exact value, we can round it to the nearest whole number:
\[ P \approx 50000 \]

### Conclusion:
The principal amount is ₹50,000.