If difference between true discount and banker’s discount on a sum due 4 months hence is Rs 20. Find true discount, banker’s discount and amount of bill, the rate of simple interest charged being 5%p.a.

Answer

To solve the problem step by step, we will find the True Discount (Td), Banker's Discount (Bd), and the amount of the bill (face value) given the difference between True Discount and Banker's Discount is Rs. 20, and the rate of simple interest is 5% per annum.

### Step 1: Understand the relationship between True Discount, Banker's Discount, and Banker's Gain
We know that:
- Banker's Gain (Bg) = Banker's Discount (Bd) - True Discount (Td)
- Given that the difference between True Discount and Banker's Discount is Rs. 20, we can write:
\[ Bg = Bd - Td = 20 \]

### Step 2: Relate Banker's Gain to True Discount
Banker's Gain can also be expressed as the interest on the True Discount for the time period until the amount is due:
\[ Bg = Td \times \frac{r \times n}{100} \]
Where:
- \( r = 5\% \) (rate of interest)
- \( n = \frac{4}{12} = \frac{1}{3} \) years (time period)

### Step 3: Set up the equation for True Discount
Substituting the values into the equation:
\[ 20 = Td \times \frac{5 \times \frac{1}{3}}{100} \]
This simplifies to:
\[ 20 = Td \times \frac{5}{300} = Td \times \frac{1}{60} \]
Now, solving for True Discount (Td):
\[ Td = 20 \times 60 = 1200 \]

### Step 4: Calculate Banker's Discount
Using the relationship from Step 1:
\[ Bd = Td + Bg = 1200 + 20 = 1220 \]

### Step 5: Calculate the Amount of the Bill
The Banker's Discount is also defined as:
\[ Bd = FV \times \frac{r \times n}{100} \]
Where FV is the face value (amount of the bill). We can rearrange this to find FV:
\[ 1220 = FV \times \frac{5 \times \frac{1}{3}}{100} \]
This simplifies to:
\[ 1220 = FV \times \frac{5}{300} \]
Now, solving for FV:
\[ FV = 1220 \times \frac{300}{5} = 1220 \times 60 = 73200 \]

### Summary of Results
- True Discount (Td) = Rs. 1200
- Banker's Discount (Bd) = Rs. 1220
- Amount of Bill (FV) = Rs. 73200