The true discount on a bill due 9 months hence at `6%` per annum is Rs. 180. Find the amount of the bill and its present worth-

Options

A. Rs. 3000, Rs. 3180
B. Rs. 4000, Rs. 4180
C. Rs. 4500, Rs. 4680
D. None of these

Answer

To solve the problem step by step, we will use the formula for true discount and follow the logical steps to find both the amount of the bill and its present worth.

### Step 1: Understand the Given Information
- True Discount (TD) = Rs. 180
- Rate (R) = 6% per annum
- Time (T) = 9 months = \( \frac{9}{12} \) years = \( \frac{3}{4} \) years

### Step 2: Use the Formula for True Discount
The formula for true discount is given by:

\[ TD = \frac{A \times R \times T}{100 + R \times T} \]

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 values into the formula:

\[ 180 = \frac{A \times 6 \times \frac{3}{4}}{100 + 6 \times \frac{3}{4}} \]

### Step 4: Simplify the Equation
Calculate \( R \times T \):

\[ R \times T = 6 \times \frac{3}{4} = \frac{18}{4} = \frac{9}{2} \]

Now substitute this back into the equation:

\[ 180 = \frac{A \times \frac{9}{2}}{100 + \frac{9}{2}} \]

Calculate \( 100 + \frac{9}{2} \):

\[ 100 + \frac{9}{2} = \frac{200}{2} + \frac{9}{2} = \frac{209}{2} \]

Now the equation becomes:

\[ 180 = \frac{A \times \frac{9}{2}}{\frac{209}{2}} \]

### Step 5: Cross Multiply to Solve for A
Cross multiplying gives:

\[ 180 \times \frac{209}{2} = A \times \frac{9}{2} \]

This simplifies to:

\[ 180 \times 209 = A \times 9 \]

### Step 6: Calculate A
Now, calculate \( 180 \times 209 \):

\[ 180 \times 209 = 37620 \]

Now, divide by 9 to find \( A \):

\[ A = \frac{37620}{9} = 4180 \]

### Step 7: Find the Present Worth (PW)
The present worth can be calculated using the formula:

\[ PW = A - TD \]

Substituting the values:

\[ PW = 4180 - 180 = 4000 \]

### Final Answers
- Amount of the bill (A) = Rs. 4180
- Present Worth (PW) = Rs. 4000