The difference between the simple interest and compound interest(interest is compounded half yearly) on a sum at the rate of 25% per annum for one year is ₹ 4375. Whatwill be the principal?

Options

A. ₹ 280000
B. ₹ 85000
C. ₹ 80000
D. ₹ 75000

Answer

To solve the problem, we need to find the principal amount (P) given the difference between simple interest (SI) and compound interest (CI) for one year at a rate of 25% per annum, compounded half-yearly.

### Step-by-Step Solution:

1. **Identify the formulas**:
- Simple Interest (SI) formula:
\[ SI = \frac{P \times R \times T}{100} \]
- Compound Interest (CI) formula (compounded half-yearly):
\[ CI = P \left(1 + \frac{R}{200}\right)^{2T} - P \]

2. **Calculate Simple Interest**:
- Given:
- Rate (R) = 25%
- Time (T) = 1 year
- Substitute the values into the SI formula:
\[ SI = \frac{P \times 25 \times 1}{100} = \frac{25P}{100} = \frac{P}{4} \]

3. **Calculate Compound Interest**:
- Since the interest is compounded half-yearly, the time period will be 2 (half-yearly for 1 year).
- Substitute the values into the CI formula:
\[ CI = P \left(1 + \frac{25}{200}\right)^{2} - P \]
- Simplifying:
\[ CI = P \left(1 + \frac{1}{8}\right)^{2} - P = P \left(\frac{9}{8}\right)^{2} - P \]
- Calculate \(\left(\frac{9}{8}\right)^{2}\):
\[ CI = P \left(\frac{81}{64}\right) - P = \frac{81P}{64} - P \]
- Rewrite \(P\) as \(\frac{64P}{64}\):
\[ CI = \frac{81P}{64} - \frac{64P}{64} = \frac{17P}{64} \]

4. **Find the difference between CI and SI**:
- Given that the difference between CI and SI is ₹4375:
\[ CI - SI = 4375 \]
- Substitute the values:
\[ \frac{17P}{64} - \frac{P}{4} = 4375 \]
- Rewrite \(\frac{P}{4}\) as \(\frac{16P}{64}\):
\[ \frac{17P}{64} - \frac{16P}{64} = 4375 \]
- Simplifying gives:
\[ \frac{P}{64} = 4375 \]

5. **Solve for P**:
- Multiply both sides by 64:
\[ P = 4375 \times 64 \]
- Calculate:
\[ P = 280000 \]

### Final Answer:
The principal amount (P) is ₹28000.