A man borrows money at 3% per annum interest payable yearly and lend it immediately at 5% interest (compound) payable half-yearly and therebygains₹330 at the end of the year. The sum borrowed is

Options

A. ₹ 17,000
B. ₹ 16,500
C. ₹ 15,000
D. ₹ 16,000

Answer

To solve the problem step by step, we will first define the variables and then calculate the interest earned and paid to find the principal amount borrowed.

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

### Step 2: Calculate Simple Interest Paid
The man borrows money at a simple interest rate of 3% per annum. The interest paid at the end of the year can be calculated using the formula for simple interest:
\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]
Where:
- \( R = 3\% \)
- \( T = 1 \) year

So, the interest paid at the end of the year is:
\[ \text{SI} = \frac{P \times 3 \times 1}{100} = \frac{3P}{100} \]

### Step 3: Calculate Compound Interest Earned
The man lends the money at a compound interest rate of 5% per annum, compounded half-yearly. The effective rate for half-yearly compounding is:
\[ \text{Rate per half year} = \frac{5}{2} = 2.5\% \]
Since the interest is compounded twice in a year, we can use the compound interest formula:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( r = 5\% \)
- \( n = 2 \) (since it is compounded half-yearly)

The total amount after one year will be:
\[ A = P \left(1 + \frac{5}{100}\right)^2 = P \left(1 + 0.025\right)^2 = P (1.025)^2 \]
Calculating \( (1.025)^2 \):
\[ (1.025)^2 = 1.050625 \]
Thus, the total amount after one year is:
\[ A = P \times 1.050625 \]

### Step 4: Calculate Compound Interest Earned
The compound interest earned can be calculated as:
\[ \text{Compound Interest} = A - P = P \times 1.050625 - P = P(1.050625 - 1) = P \times 0.050625 \]

### Step 5: Calculate the Gain
According to the problem, the gain from this transaction is ₹330. Therefore, we can set up the equation:
\[ \text{Compound Interest} - \text{Simple Interest} = 330 \]
Substituting the values we calculated:
\[ P \times 0.050625 - \frac{3P}{100} = 330 \]

### Step 6: Solve for \( P \)
Now, let's simplify the equation:
\[ P \times 0.050625 - \frac{3P}{100} = 330 \]
Convert \( \frac{3P}{100} \) to a decimal:
\[ \frac{3P}{100} = 0.03P \]
Thus, the equation becomes:
\[ P \times 0.050625 - 0.03P = 330 \]
Combining like terms:
\[ P(0.050625 - 0.03) = 330 \]
Calculating \( 0.050625 - 0.03 \):
\[ 0.050625 - 0.03 = 0.020625 \]
So, we have:
\[ P \times 0.020625 = 330 \]
Now, solving for \( P \):
\[ P = \frac{330}{0.020625} \approx 16000 \]

### Final Answer
The sum borrowed is ₹16,000.

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