A man borrows some money at 3% simple interest per annum and lends it to somebody at 5% interest to be compounded annually. By this he makes a profit of Rs. 541 at the end of 3 years. The money he borrowed was

Options

A. ₹ 8,000
B. ₹ 6,762
C. ₹ 6,000
D. ₹ 8,070

Answer

To solve the problem step by step, we need to find out how much money the man borrowed, given that he makes a profit of Rs. 541 after lending it at a higher interest rate.

### Step 1: Define the variables
Let the amount of money borrowed be \( P \).

### Step 2: Calculate the simple interest on the borrowed amount
The simple interest (SI) for 3 years at a rate of 3% per annum can be calculated using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
where \( R = 3\% \) and \( T = 3 \) years.

Substituting the values:
\[ SI = \frac{P \times 3 \times 3}{100} = \frac{9P}{100} \]

### Step 3: Calculate the total amount to be paid after 3 years
The total amount to be paid back after 3 years is the principal plus the simple interest:
\[ \text{Total Amount Paid} = P + SI = P + \frac{9P}{100} = P \left(1 + \frac{9}{100}\right) = P \left(\frac{109}{100}\right) \]

### Step 4: Calculate the compound interest earned from lending the money
When the man lends the money at 5% compounded annually, the amount after 3 years can be calculated using the formula:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
where \( R = 5\% \) and \( T = 3 \) years.

Substituting the values:
\[ A = P \left(1 + \frac{5}{100}\right)^3 = P \left(1.05\right)^3 \]
Calculating \( (1.05)^3 \):
\[ (1.05)^3 = 1.157625 \]
So, the total amount received after 3 years is:
\[ A = P \times 1.157625 \]

### Step 5: Calculate the profit made
The profit made by the man is the difference between the amount received from lending and the amount paid back:
\[ \text{Profit} = A - \text{Total Amount Paid} \]
Substituting the values:
\[ \text{Profit} = P \times 1.157625 - P \left(\frac{109}{100}\right) \]
Factoring out \( P \):
\[ \text{Profit} = P \left(1.157625 - \frac{109}{100}\right) \]

### Step 6: Simplify the profit equation
Convert \( \frac{109}{100} \) to decimal:
\[ \frac{109}{100} = 1.09 \]
Thus, the profit equation becomes:
\[ \text{Profit} = P \left(1.157625 - 1.09\right) = P \times 0.067625 \]

### Step 7: Set the profit equal to Rs. 541
According to the problem, the profit is Rs. 541:
\[ P \times 0.067625 = 541 \]

### Step 8: Solve for \( P \)
\[ P = \frac{541}{0.067625} \]
Calculating \( P \):
\[ P \approx 8000 \]

### Conclusion
The amount of money the man borrowed is Rs. 8000.

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