A bill for RS 10200 is drawn on July 14 at 5 months. It is discounted on 5th October at 10%. Find the banker's discount, true discount, banker's gain and the money that the holder of the bill receives.

Options

A. X 204, X 200, X 4 and X 9996
B. X 100, X 200, X 2 and X 4989
C. X 121, X 172, X 132 and X 4046
D. X 136, X 132, X 138 and X 3649

Answer

To solve the problem step by step, we will calculate the Banker's Discount (BD), True Discount (TD), Banker's Gain (BG), and the amount the holder of the bill receives.

### Step 1: Determine the Nominal Due Date
The bill is drawn on July 14 for 5 months. Therefore, the nominal due date is:
- **Due Date**: July 14 + 5 months = December 14.

### Step 2: Determine the Legal Date of Discounting
The bill is discounted on October 5. The legal date is the date when the bill is discounted:
- **Legal Date**: October 5.

### Step 3: Calculate the Time Remaining Until Due Date
We need to calculate the number of days from the legal date (October 5) to the nominal due date (December 14):
- Days remaining in October: 31 - 5 = 26 days
- Days in November: 30 days
- Days in December until the due date: 14 days
- **Total Days Remaining**: 26 + 30 + 14 = 70 days.

### Step 4: Convert Days to Years
To find the time in years:
- **Time in years** = 70 days / 365 days = 70/365 = 0.1918 years (approximately).

### Step 5: Calculate the Banker's Discount (BD)
The formula for Banker's Discount is:
\[ \text{BD} = \text{Bill Amount} \times \text{Rate} \times \text{Time} \]
Where:
- Bill Amount = Rs. 10,200
- Rate = 10% = 0.10
- Time = 5 months = 5/12 years

Calculating BD:
\[ \text{BD} = 10200 \times 0.10 \times \frac{5}{12} \]
\[ \text{BD} = 10200 \times 0.10 \times 0.4167 \]
\[ \text{BD} = 10200 \times 0.04167 = 425 \]

### Step 6: Calculate the True Discount (TD)
The formula for True Discount is:
\[ \text{TD} = \frac{\text{Bill Amount} \times \text{Rate} \times \text{Time}}{100 + (\text{Rate} \times \text{Time})} \]

Calculating TD:
\[ \text{TD} = \frac{10200 \times 0.10 \times \frac{5}{12}}{100 + (0.10 \times \frac{5}{12})} \]
\[ \text{TD} = \frac{425}{100 + 4.167} \]
\[ \text{TD} = \frac{425}{104.167} \approx 4.08 \]

### Step 7: Calculate the Banker's Gain (BG)
Banker's Gain is calculated as:
\[ \text{BG} = \text{BD} - \text{TD} \]
\[ \text{BG} = 425 - 4.08 \approx 420.92 \]

### Step 8: Calculate the Amount Received by the Holder
The amount received by the holder of the bill is:
\[ \text{Amount Received} = \text{Bill Amount} - \text{TD} \]
\[ \text{Amount Received} = 10200 - 4.08 \approx 10195.92 \]

### Summary of Results
- Banker's Discount (BD) = Rs. 425
- True Discount (TD) = Rs. 4.08
- Banker's Gain (BG) = Rs. 420.92
- Amount Received by Holder = Rs. 10195.92