The banker's gain on a bill due 1 year 4 months hence at `7 1/2 %` per annum simple interest is Rs.16. Find the sum -----

Options

A. Rs.1760
B. Rs.1560
C. Rs.1660
D. Rs.1860

Answer

To solve the problem step by step, we will follow the method outlined in the video transcript.

### Step 1: Identify the Given Values
- Banker's Gain (BG) = Rs. 16
- Rate of Interest (R) = 7.5% per annum = \( \frac{15}{2} \% \)
- Time (T) = 1 year 4 months = \( 1 \frac{1}{3} \) years = \( \frac{4}{3} \) years

### Step 2: Calculate the Total Discount (TD)
The formula for Total Discount is given by:

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

Substituting the values:

\[ TD = \frac{16 \times 100}{\frac{15}{2} \times \frac{4}{3}} \]

### Step 3: Simplify the Calculation
First, calculate the denominator:

\[ R \times T = \frac{15}{2} \times \frac{4}{3} = \frac{15 \times 4}{2 \times 3} = \frac{60}{6} = 10 \]

Now substitute back into the Total Discount formula:

\[ TD = \frac{16 \times 100}{10} = \frac{1600}{10} = 160 \]

### Step 4: Use the Total Discount to Find the Sum (Principal Amount)
The formula relating Total Discount (TD), Principal Amount (A), Rate (R), and Time (T) is:

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

Substituting the known values:

\[ 160 = \frac{A \times \frac{15}{2} \times \frac{4}{3}}{100 + 10} \]

### Step 5: Simplify Further
First, calculate \(100 + R \times T\):

\[ 100 + 10 = 110 \]

Now substitute this into the equation:

\[ 160 = \frac{A \times 10}{110} \]

### Step 6: Solve for A
Cross-multiply to solve for A:

\[ 160 \times 110 = A \times 10 \]

\[ 17600 = A \times 10 \]

Now divide both sides by 10:

\[ A = \frac{17600}{10} = 1760 \]

### Final Answer
The sum (Principal Amount) is Rs. 1760.

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