The banker's gain on a bill due 2 years hence at `5%` is Rs.8 , find the present worth of the bill -----

Options

A. Rs. 800
B. Rs.650
C. Rs.750
D. Rs.850

Answer

To solve the problem of finding the present worth of a bill given the banker's gain, we can follow these steps:

### Step 1: Understand the given information
We know:
- Banker's Gain (BG) = Rs. 8
- Rate (R) = 5%
- Time (T) = 2 years

### Step 2: Calculate the True Discount (TD)
The formula for True Discount is given by:
\[ TD = \frac{BG \times 100}{R \times T} \]

Substituting the known values:
\[ TD = \frac{8 \times 100}{5 \times 2} \]

Calculating this:
\[ TD = \frac{800}{10} = 80 \]

### Step 3: Use the True Discount to find the Present Worth (PW)
The relationship between True Discount, Present Worth, and Banker's Gain is given by:
\[ TD = BG + PW \]

Rearranging this gives:
\[ PW = TD - BG \]

Substituting the values we have:
\[ PW = 80 - 8 = 72 \]

### Step 4: Calculate Present Worth using the formula
Alternatively, we can use the formula:
\[ TD = \sqrt{BG \times PW} \]

Since we already calculated TD as 80 and BG as 8, we can rearrange this formula to find PW:
\[ 80 = \sqrt{8 \times PW} \]

Squaring both sides:
\[ 80^2 = 8 \times PW \]
\[ 6400 = 8 \times PW \]

Now, divide both sides by 8:
\[ PW = \frac{6400}{8} = 800 \]

### Final Answer
The present worth of the bill is Rs. 800.

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