A trader owes a merchant Rs. 901 due 1 year’s hence. However, the trader wants to settle the account after 3 months. How mach cash should he pay. If rate of interest is `8%` per annum-

Options

A. Rs. 870
B. Rs. 850
C. Rs. `828cdot92`
D. Rs. `846cdot94`

Answer

To find out how much cash the trader should pay to settle the account after 3 months instead of 1 year, we need to calculate the present value of the amount due using the formula for true discount.

### Step-by-Step Solution:

1. **Identify the Amount Due and Time Frame:**
- Amount due (A) = Rs. 901
- Time until due = 1 year
- Time until payment = 3 months = 0.25 years
- Remaining time until due = 1 year - 0.25 years = 0.75 years = 9 months

2. **Determine the Rate of Interest:**
- Rate of interest (R) = 8% per annum

3. **Calculate the True Discount:**
- The formula for calculating the present value (PV) when you know the future value (FV) is:
\[ PV = \frac{FV}{(1 + r \cdot t)} \]
- Where:
- \( FV \) = Future Value (Amount due)
- \( r \) = Rate of interest (as a decimal)
- \( t \) = Time in years until the amount is due

4. **Convert the Rate of Interest:**
- Convert 8% to decimal:
\[ r = \frac{8}{100} = 0.08 \]

5. **Calculate the Present Value:**
- Substitute the values into the formula:
\[ PV = \frac{901}{(1 + 0.08 \cdot 0.75)} \]
- Calculate \( 0.08 \cdot 0.75 = 0.06 \)
- Thus, the equation becomes:
\[ PV = \frac{901}{1 + 0.06} = \frac{901}{1.06} \]
- Now, calculate \( PV \):
\[ PV = \frac{901}{1.06} \approx 849.05 \]

6. **Round Off the Present Value:**
- The trader should pay approximately Rs. 849.05. Since cash transactions are typically rounded to the nearest whole number, the trader should pay Rs. 849.

### Final Answer:
The trader should pay Rs. 849 to settle the account after 3 months.