A bill of Rs.8,000 drawn on 5th January 2019 for 8 months was discounted for Rs.7,680 on a certain date. Find the date on which it was discounted at 10% p.a.

Answer

To find the date on which the bill was discounted, we can follow these steps:

### Step 1: Identify the given values
- Face value of the bill (FV) = Rs. 8000
- Cash value when discounted = Rs. 7680
- Rate of interest (r) = 10% per annum
- Duration of the bill = 8 months
- Date of drawing the bill = 5th January 2019

### Step 2: Calculate the Bankers Discount (BD)
The Bankers Discount can be calculated using the formula:
\[ \text{BD} = \text{Face Value} - \text{Cash Value} \]

Substituting the values:
\[ \text{BD} = 8000 - 7680 = 320 \text{ Rs.} \]

### Step 3: Determine the nominal due date
The nominal due date is the date after 8 months from the drawing date (5th January 2019).
- Adding 8 months to 5th January 2019 gives us:
- January (26 days remaining) + February (28 days, 2019 is not a leap year) + March (31 days) + April (30 days) + May (31 days) + June (30 days) + July (31 days) + August (31 days) + September (5 days) = 8th September 2019.

### Step 4: Calculate the legal due date
The legal due date is 3 days after the nominal due date:
- Legal due date = 8th September 2019 + 3 days = 11th September 2019.

### Step 5: Set up the equation for Bankers Discount
The formula for Bankers Discount is:
\[ \text{BD} = \text{Face Value} \times \frac{r}{100} \times \frac{n}{365} \]
Where \( n \) is the number of days before the legal due date when the bill was discounted.

Substituting the known values:
\[ 320 = 8000 \times \frac{10}{100} \times \frac{n}{365} \]

### Step 6: Solve for \( n \)
Rearranging the equation:
\[ 320 = 800 \times \frac{n}{365} \]
\[ \frac{320 \times 365}{800} = n \]
\[ n = \frac{116800}{800} = 146 \text{ days} \]

### Step 7: Calculate the discount date
To find the discount date, we subtract \( n \) days from the legal due date (11th September 2019):
- Start from 11th September 2019 and go back 146 days.

Counting backwards:
- September has 11 days left.
- August has 31 days.
- July has 31 days.
- June has 30 days.
- May has 31 days.
- April has 30 days.

Calculating the total:
- From September 11 to August 31 = 11 days
- August = 31 days
- July = 31 days
- June = 30 days
- May = 31 days
- April = 15 days (to complete 146 days)

Adding these gives:
11 (September) + 31 (August) + 31 (July) + 30 (June) + 31 (May) + 15 (April) = 146 days.

Thus, the bill was discounted on **15th April 2019**.