What sum will discharge a debt of Rs. 5300 due a year and a half hence at `4%` per annum?

Options

A. Rs. 5000
B. Rs. 4500
C. Rs. 4200
D. Rs. 5250

Answer

To find the sum that will discharge a debt of Rs. 5300 due a year and a half hence at 4% per annum, we will use the formula for calculating the present value (sum) based on true discount.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Debt (Principal, P) = Rs. 5300
- Rate of interest (R) = 4% per annum
- Time (T) = 1.5 years (which is equal to \( \frac{3}{2} \) years)

2. **Convert Time into a Fraction:**
- Time in years = \( \frac{3}{2} \)

3. **Use the True Discount Formula:**
- The formula for calculating the present value (sum) is:
\[ \text{Sum} = \frac{100 \times P}{100 + (R \times T)} \]
- Here, \( R \times T = 4 \times \frac{3}{2} = 6 \).

4. **Calculate the Denominator:**
- Now, calculate \( 100 + (R \times T) \):
\[ 100 + 6 = 106 \]

5. **Substitute Values into the Formula:**
- Now substitute the values into the formula:
\[ \text{Sum} = \frac{100 \times 5300}{106} \]

6. **Calculate the Numerator:**
- Calculate \( 100 \times 5300 = 530000 \).

7. **Perform the Division:**
- Now divide \( 530000 \) by \( 106 \):
\[ \text{Sum} = \frac{530000}{106} \approx 5000 \]

8. **Conclusion:**
- The sum that will discharge the debt is Rs. 5000.

### Final Answer:
The sum required to discharge the debt of Rs. 5300 due a year and a half hence at 4% per annum is Rs. 5000.

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