If the difference between the compound interest, compounded every six months, and the simple interest on a certain sum of money at the rate of `12%` per annum for one year is 36, the sum is :

Options

A. `Rs. 10,000 `
B. `Rs.12,000`
C. `Rs. 15,000`
D. `Rs. 9,000 `

Answer

To solve the problem step by step, we need to find the sum of money (principal) given the difference between the compound interest (CI) and simple interest (SI) is 36 when compounded semi-annually at a rate of 12% per annum.

### Step 1: Understand the Interest Rates and Time Periods
- The annual interest rate is 12%.
- Since the interest is compounded every six months, we need to halve the rate and double the time period:
- Half-yearly rate = \( \frac{12\%}{2} = 6\% \)
- Time period for compound interest = \( 2 \) years (since 1 year is compounded twice).

### Step 2: Calculate the Compound Interest
- The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where:
- \( A \) = total amount after time \( n \)
- \( P \) = principal amount
- \( r \) = rate of interest
- \( n \) = number of compounding periods

- For our case:
- \( r = 6\% \)
- \( n = 2 \) (since it's compounded twice in one year)

Thus, the amount after 1 year will be:
\[ A = P \left(1 + \frac{6}{100}\right)^2 = P \left(1 + 0.06\right)^2 = P \left(1.06\right)^2 = P \times 1.1236 \]

- The compound interest (CI) can be calculated as:
\[ CI = A - P = P \times 1.1236 - P = P(1.1236 - 1) = P \times 0.1236 \]

### Step 3: Calculate the Simple Interest
- The formula for simple interest is:
\[ SI = \frac{P \times r \times t}{100} \]
- Here, \( r = 12\% \) and \( t = 1 \) year, so:
\[ SI = \frac{P \times 12 \times 1}{100} = \frac{12P}{100} = 0.12P \]

### Step 4: Set Up the Equation for the Difference
- According to the problem, the difference between CI and SI is 36:
\[ CI - SI = 36 \]
Substituting the values we found:
\[ P \times 0.1236 - 0.12P = 36 \]

### Step 5: Simplify the Equation
- Combine like terms:
\[ (0.1236P - 0.12P) = 36 \]
\[ 0.0036P = 36 \]

### Step 6: Solve for Principal (P)
- To find \( P \):
\[ P = \frac{36}{0.0036} = 10000 \]

### Conclusion
The sum of money (principal) is **10,000**.

---