A person borrows some money at 16% simple rate of interest and lends the amount at 16% compound rate of interest which is compounded semi-annually. In this way he earns 56 rs then find the amount?

Options

A. ₹9750
B. ₹8750
C. ₹ 8000
D. ₹7750

Answer

To solve the problem step by step, we need to find the principal amount (P) that a person borrows at a simple interest rate of 16% and lends at a compound interest rate of 16% compounded semi-annually, resulting in a profit of Rs. 56.

### Step 1: Understand the Interest Formulas
- **Simple Interest (SI)** is calculated using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal, \( R \) is the rate of interest, and \( T \) is the time in years.

- **Compound Interest (CI)** for semi-annual compounding is calculated using the formula:
\[ CI = P \left(1 + \frac{R}{200}\right)^{2T} - P \]
where \( R \) is the rate of interest, and \( T \) is the time in years.

### Step 2: Set Up the Equations
Given:
- Rate of interest \( R = 16\% \)
- Time \( T = 1 \) year (since the profit is calculated for one year)
- The profit earned is \( CI - SI = 56 \)

### Step 3: Calculate Simple Interest
Using the SI formula:
\[ SI = \frac{P \times 16 \times 1}{100} = \frac{16P}{100} = \frac{4P}{25} \]

### Step 4: Calculate Compound Interest
Using the CI formula for semi-annual compounding:
\[ CI = P \left(1 + \frac{16}{200}\right)^{2 \times 1} - P \]
\[ = P \left(1 + 0.08\right)^{2} - P \]
\[ = P \left(1.08\right)^{2} - P \]
Calculating \( (1.08)^{2} \):
\[ (1.08)^{2} = 1.1664 \]
Thus,
\[ CI = P \times 1.1664 - P = 0.1664P \]

### Step 5: Set Up the Profit Equation
Now we can set up the equation based on the profit:
\[ CI - SI = 56 \]
Substituting the values we found:
\[ 0.1664P - \frac{4P}{25} = 56 \]

### Step 6: Find a Common Denominator
To solve for \( P \), we need a common denominator. The common denominator for 25 and 1 is 25:
\[ 0.1664P = \frac{1664P}{10000} \]
Now converting \( \frac{4P}{25} \) to have a denominator of 10000:
\[ \frac{4P}{25} = \frac{1600P}{10000} \]
Thus, the equation becomes:
\[ \frac{1664P}{10000} - \frac{1600P}{10000} = 56 \]
\[ \frac{64P}{10000} = 56 \]

### Step 7: Solve for P
Multiply both sides by 10000:
\[ 64P = 560000 \]
Now divide by 64:
\[ P = \frac{560000}{64} = 8750 \]

### Conclusion
The amount (principal) borrowed is Rs. 8750.