A man borrowed some money from a private organisation at `5%` simple interest per annum. He lended `50%` of this money to another person at `10%` compound interest per annum and thereby the man made a profit of `Rs. 3,205` in 4 years. The man borrowed.

Options

A. `Rs. 80,000`
B. `Rs. 1,00,000 `
C. `Rs. 1,20,000 `
D. `Rs. 1,50,000 `

Answer

To solve the problem step by step, we will follow the information provided and apply the formulas for simple and compound interest.

### Step 1: Define the Variables
Let the amount borrowed by the man be \( P \).

### Step 2: Calculate Simple Interest
The man borrowed money at a simple interest rate of \( 5\% \) per annum for \( 4 \) years. The formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
where:
- \( R = 5\% \)
- \( T = 4 \) years

Substituting the values:
\[ SI = \frac{P \times 5 \times 4}{100} = \frac{20P}{100} = \frac{P}{5} \]

### Step 3: Calculate the Amount Paid Back
The total amount paid back after \( 4 \) years is:
\[ \text{Total Amount} = P + SI = P + \frac{P}{5} = \frac{5P + P}{5} = \frac{6P}{5} \]

### Step 4: Calculate Compound Interest
The man lent \( 50\% \) of the borrowed amount, which is \( \frac{P}{2} \), at a compound interest rate of \( 10\% \) per annum for \( 4 \) years. The formula for compound interest (CI) is:
\[ CI = A - P \]
where \( A \) is the amount after \( T \) years, calculated as:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the values:
\[ A = \frac{P}{2} \left(1 + \frac{10}{100}\right)^4 = \frac{P}{2} \left(1.1\right)^4 \]

Calculating \( (1.1)^4 \):
\[ (1.1)^4 = 1.4641 \]
Thus,
\[ A = \frac{P}{2} \times 1.4641 = 0.73205P \]

Now, the compound interest earned is:
\[ CI = A - \frac{P}{2} = 0.73205P - \frac{P}{2} \]
Converting \( \frac{P}{2} \) to a decimal:
\[ \frac{P}{2} = 0.5P \]
So,
\[ CI = 0.73205P - 0.5P = 0.23205P \]

### Step 5: Set Up the Profit Equation
According to the problem, the profit made by the man is \( Rs. 3205 \). Therefore, we can set up the equation:
\[ 0.23205P - \frac{P}{5} = 3205 \]
Substituting \( \frac{P}{5} \) as \( 0.2P \):
\[ 0.23205P - 0.2P = 3205 \]
This simplifies to:
\[ 0.03205P = 3205 \]

### Step 6: Solve for \( P \)
To find \( P \), we divide both sides by \( 0.03205 \):
\[ P = \frac{3205}{0.03205} = 100,000 \]

### Final Answer
The man borrowed \( Rs. 100,000 \).

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