The difference between the simple interest and the true discount on a certain sum for 6 months at `4%` is Rs. 15. Find the sum-

Options

A. Rs. 32850
B. Rs. 28250
C. Rs. 38250
D. Rs. 38350

Answer

To solve the problem, we need to find the sum (P) given that the difference between the simple interest (SI) and the true discount (TD) for 6 months at a rate of 4% is Rs. 15.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Rate (R) = 4%
- Time (T) = 6 months = 0.5 years
- Difference between SI and TD = Rs. 15

2. **Formulas:**
- Simple Interest (SI) = \( \frac{P \times R \times T}{100} \)
- True Discount (TD) = \( \frac{TD \times R \times T}{100 + R \times T} \)

3. **Setting Up the Equation:**
- From the problem, we know:
\[ SI - TD = 15 \]
- Substitute the formulas for SI and TD:
\[ \frac{P \times R \times T}{100} - \frac{P \times R \times T}{100 + R \times T} = 15 \]

4. **Calculate SI and TD:**
- Calculate \( R \times T \):
\[ R \times T = 4 \times 0.5 = 2 \]
- Substitute \( R \) and \( T \) into the equation:
\[ \frac{P \times 4 \times 0.5}{100} - \frac{P \times 4 \times 0.5}{100 + 2} = 15 \]
- This simplifies to:
\[ \frac{2P}{100} - \frac{2P}{102} = 15 \]

5. **Finding a Common Denominator:**
- The common denominator is 10200 (which is 100 × 102):
\[ \frac{2P \times 102 - 2P \times 100}{10200} = 15 \]
- This simplifies to:
\[ \frac{2P \times (102 - 100)}{10200} = 15 \]
- Which further simplifies to:
\[ \frac{2P \times 2}{10200} = 15 \]

6. **Cross-Multiplying to Solve for P:**
- Cross-multiply:
\[ 4P = 15 \times 10200 \]
- Calculate \( 15 \times 10200 = 153000 \):
\[ 4P = 153000 \]
- Now divide by 4:
\[ P = \frac{153000}{4} = 38250 \]

7. **Final Answer:**
- The sum (P) is Rs. 38250.