The true discount on a bill due 8 months hence at `12%` per annum is Rs. 240. Find the amount of the bill and its present worth-

Options

A. Rs. 3000, Rs. 3240
B. Rs. 2000, Rs. 2240
C. Rs. 2100, Rs. 2340
D. None of these

Answer

To solve the problem, we need to find the amount of the bill and its present worth given the true discount, rate, and time. Here’s a step-by-step solution:

### Step 1: Identify the given values
- True Discount (TD) = Rs. 240
- Rate (R) = 12% per annum
- Time (T) = 8 months = \( \frac{8}{12} \) years = \( \frac{2}{3} \) years

### Step 2: Use the formula for True Discount
The formula for True Discount is:
\[ TD = \frac{A \times R \times T}{100 + R \times T} \]
Where:
- \( A \) = Amount of the bill
- \( R \) = Rate of interest
- \( T \) = Time in years

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula:
\[ 240 = \frac{A \times 12 \times \frac{2}{3}}{100 + 12 \times \frac{2}{3}} \]

### Step 4: Simplify the equation
Calculate \( R \times T \):
\[ R \times T = 12 \times \frac{2}{3} = 8 \]
Now substitute this back into the equation:
\[ 240 = \frac{A \times 8}{100 + 8} \]
This simplifies to:
\[ 240 = \frac{A \times 8}{108} \]

### Step 5: Cross-multiply to solve for \( A \)
Cross-multiplying gives:
\[ 240 \times 108 = A \times 8 \]
Calculating \( 240 \times 108 \):
\[ 240 \times 108 = 25920 \]
So we have:
\[ 25920 = A \times 8 \]
Now, divide both sides by 8:
\[ A = \frac{25920}{8} = 3240 \]

### Step 6: Calculate the Present Worth (PW)
The Present Worth is calculated as:
\[ PW = A - TD \]
Substituting the values:
\[ PW = 3240 - 240 = 3000 \]

### Final Answers
- Amount of the bill (A) = Rs. 3240
- Present Worth (PW) = Rs. 3000