A money-lender borrows money at `4%` per annum and pays the Interest at the end of the year. He lends it at 6% per annum compound interest compounded half yearly and receives the interest at the end of the year. In this way, he gains `Rs.104.50` a year. The amount of money he borrows, is

Options

A. `Rs. 6,000`
B. `Rs. 5,500`
C. `Rs. 5,000`
D. `Rs. 4,500`

Answer

To solve the problem, we will break it down into steps:

### Step 1: Understand the Interest Rates
The money-lender borrows money at a rate of 4% per annum and lends it at a rate of 6% per annum, compounded half-yearly.

### Step 2: Calculate the Effective Rate of Interest
Since the money-lender lends at 6% compounded half-yearly, we need to calculate the effective interest rate for one year.

- The half-yearly rate is:
\[ \text{Half-yearly rate} = \frac{6\%}{2} = 3\% \]

- The effective interest rate for one year (compounded) is calculated as:
\[ \text{Effective Rate} = \left(1 + \frac{3}{100}\right)^2 - 1 \]
\[ = \left(1.03\right)^2 - 1 \]
\[ = 1.0609 - 1 = 0.0609 \text{ or } 6.09\% \]

### Step 3: Calculate the Gain in Interest
The gain in interest is the difference between the effective interest rate received and the interest rate paid.

- The interest paid on borrowed money at 4% is:
\[ \text{Interest Paid} = 4\% \]

- The gain in interest is:
\[ \text{Gain} = \text{Effective Rate} - \text{Interest Paid} = 6.09\% - 4\% = 2.09\% \]

### Step 4: Set Up the Equation Based on Gain
According to the problem, the gain of 2.09% on the borrowed amount equals Rs. 104.50.

Let the amount borrowed be \( x \).

- Therefore, we can write:
\[ 2.09\% \text{ of } x = 104.50 \]
\[ \frac{2.09}{100} \times x = 104.50 \]

### Step 5: Solve for \( x \)
To find \( x \), we rearrange the equation:
\[ x = \frac{104.50 \times 100}{2.09} \]
Calculating this gives:
\[ x = \frac{10450}{2.09} \approx 5000 \]

### Conclusion
The amount of money he borrows is approximately Rs. 5000.

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