A moneylender 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

Answer

To solve the problem step by step, we will follow these calculations:

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

### Step 2: Convert the Annual Rate to Half-Yearly Rate
Since the lending is compounded half-yearly, we need to divide the annual interest rate by 2:
- Half-yearly rate = 6% / 2 = 3%

### Step 3: Calculate the Effective Annual Rate for Compound Interest
The formula for the effective annual rate when compounded half-yearly is:
\[ \text{Effective Rate} = \left(1 + \frac{r}{100}\right)^n - 1 \]
Where \( r \) is the half-yearly rate and \( n \) is the number of compounding periods in a year (which is 2 for half-yearly).
\[ \text{Effective Rate} = \left(1 + \frac{3}{100}\right)^2 - 1 \]
Calculating this:
\[ \text{Effective Rate} = (1.03)^2 - 1 = 1.0609 - 1 = 0.0609 \text{ or } 6.09\% \]

### Step 4: Set Up the Equation for Gain
The gain from the moneylender’s transactions is given as Rs. 104.50. The difference in interest earned and interest paid can be expressed as:
\[ \text{Interest Earned} - \text{Interest Paid} = 104.50 \]
Let \( P \) be the principal amount borrowed. The interest paid on borrowing is:
\[ \text{Interest Paid} = P \times \frac{4}{100} = 0.04P \]
The interest earned from lending is:
\[ \text{Interest Earned} = P \times \frac{6.09}{100} = 0.0609P \]
Thus, we can write:
\[ 0.0609P - 0.04P = 104.50 \]

### Step 5: Simplify the Equation
Combining the terms gives:
\[ (0.0609 - 0.04)P = 104.50 \]
\[ 0.0209P = 104.50 \]

### Step 6: Solve for \( P \)
To find \( P \), divide both sides by 0.0209:
\[ P = \frac{104.50}{0.0209} \approx 5000 \]

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

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