Find the banker's gain on bill of Rs.6900 due 3 years hence at `5%` per annum simple interest -----

Options

A. Rs. 135
B. Rs.125
C. Rs.185
D. Rs.145

Answer

To find the banker's gain on a bill of Rs. 6900 due 3 years hence at 5% per annum simple interest, we will follow these steps:

### Step 1: Calculate the True Discount
The formula for True Discount (TD) is given by:

\[ TD = \frac{Amount \times Rate \times Time}{100 + (Rate \times Time)} \]

Where:
- Amount = Rs. 6900
- Rate = 5%
- Time = 3 years

Substituting the values into the formula:

\[ TD = \frac{6900 \times 5 \times 3}{100 + (5 \times 3)} \]

Calculating the denominator:

\[ 100 + (5 \times 3) = 100 + 15 = 115 \]

Now substituting back into the equation:

\[ TD = \frac{6900 \times 15}{115} \]

### Step 2: Simplify the Calculation
Calculating the numerator:

\[ 6900 \times 15 = 103500 \]

Now dividing by 115:

\[ TD = \frac{103500}{115} \]

Calculating this gives:

\[ TD = 900 \]

### Step 3: Calculate Present Worth
The Present Worth (PW) can be calculated using the formula:

\[ PW = Amount - True Discount \]

Substituting the values:

\[ PW = 6900 - 900 = 6000 \]

### Step 4: Calculate Banker's Gain
The formula for Banker's Gain (BG) is given by:

\[ BG = \frac{(Total Discount)^2}{Present Worth} \]

Where Total Discount is equal to the True Discount calculated earlier, which is 900.

Substituting the values into the formula:

\[ BG = \frac{900^2}{6000} \]

Calculating \(900^2\):

\[ 900^2 = 810000 \]

Now substituting back into the equation:

\[ BG = \frac{810000}{6000} \]

Calculating this gives:

\[ BG = 135 \]

### Final Answer
The banker's gain on the bill is Rs. 135.

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