A bill for Rs. 5656 is drawn on july 14 at 5 months. It is discounted on Oct. 5th at `5%`-----
True discount -----

Options

A. Rs.50
B. Rs.54.56
C. Rs.56
D. None of these

Answer

To solve the problem of finding the true discount on a bill of Rs. 5656 that is drawn on July 14 for 5 months and discounted on October 5 at a rate of 5%, we will follow these steps:

### Step 1: Determine the Due Date of the Bill
The bill is drawn on July 14 for a duration of 5 months. Therefore, we need to calculate the due date.

- **Due Date Calculation**:
- July 14 + 5 months = December 14

### Step 2: Calculate the Legal Due Date
The legal due date is typically a few days after the nominal due date.

- **Legal Due Date**:
- December 14 + 3 days = December 17

### Step 3: Calculate the Unexpired Time
Next, we need to find out how much time is left from the date of discounting (October 5) to the legal due date (December 17).

- **Days Calculation**:
- From October 5 to October 31: 26 days
- November: 30 days
- From December 1 to December 17: 17 days
- Total unexpired days = 26 + 30 + 17 = 73 days

### Step 4: Convert Unexpired Days to Years
To calculate the true discount, we need to convert the unexpired days into years.

- **Conversion to Years**:
- \( \text{Unexpired Time} = \frac{73}{365} \) years

### Step 5: Calculate the Banker's Discount
The banker's discount can be calculated using the formula:
\[ \text{Banker's Discount} = \text{Amount} \times \text{Rate} \times \text{Time} \]
Where:
- Amount = Rs. 5656
- Rate = 5%
- Time = \( \frac{73}{365} \)

Substituting the values:
\[ \text{Banker's Discount} = 5656 \times \frac{5}{100} \times \frac{73}{365} \]
Calculating this gives:
\[ \text{Banker's Discount} = 5656 \times 0.05 \times 0.2 = 56.56 \]

### Step 6: Calculate the True Discount
The true discount can be calculated using the formula:
\[ \text{True Discount} = \frac{\text{Amount} \times \text{Rate} \times \text{Time}}{100 + \text{Rate} \times \text{Time}} \]
Substituting the values:
\[ \text{True Discount} = \frac{5656 \times 5 \times \frac{73}{365}}{100 + 5 \times \frac{73}{365}} \]
Calculating the denominator:
\[ 100 + 5 \times \frac{73}{365} = 100 + 1 = 101 \]
Thus, the true discount becomes:
\[ \text{True Discount} = \frac{56.56}{101} \approx 56 \]

### Final Answer
The true discount is approximately Rs. 56.

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