What is the difference (in Rs.) between the compound interest, when interest is compound 6-monthly, and the simple interest on a sum of Rs. 10,000 for `1 1/2` years at 10% p.a?

Options

A. 102.25
B. 87
C. 76.25
D. 91.5

Answer

To solve the problem, we need to find the difference between the compound interest (when compounded semi-annually) and the simple interest on a sum of Rs. 10,000 for 1.5 years at a rate of 10% per annum.

### Step-by-Step Solution:

1. **Identify the Principal, Rate, and Time:**
- Principal (P) = Rs. 10,000
- Rate (R) = 10% per annum
- Time (T) = 1.5 years

2. **Convert the Rate for Semi-annual Compounding:**
- Since the interest is compounded semi-annually, we divide the annual rate by 2.
- Semi-annual Rate = 10% / 2 = 5% per 6 months.

3. **Convert Time for Semi-annual Compounding:**
- The time of 1.5 years is equivalent to 3 half-year periods (since 1.5 years = 1.5 * 2 = 3).

4. **Calculate Simple Interest (SI):**
- The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
- Substituting the values:
\[ SI = \frac{10,000 \times 10 \times 1.5}{100} = \frac{150,000}{100} = 1500 \]

5. **Calculate Compound Interest (CI):**
- The formula for Compound Interest is:
\[ CI = P \left(1 + \frac{R}{100}\right)^n - P \]
- Where \( n \) is the number of compounding periods.
- Substituting the values:
\[ CI = 10,000 \left(1 + \frac{5}{100}\right)^3 - 10,000 \]
- Simplifying:
\[ CI = 10,000 \left(1.05\right)^3 - 10,000 \]
- Calculating \( (1.05)^3 \):
\[ (1.05)^3 = 1.157625 \]
- Now substituting back:
\[ CI = 10,000 \times 1.157625 - 10,000 = 11576.25 - 10,000 = 1576.25 \]

6. **Calculate the Difference between CI and SI:**
- Difference = CI - SI
- Substituting the values:
\[ Difference = 1576.25 - 1500 = 76.25 \]

### Final Answer:
The difference between the compound interest and the simple interest is Rs. 76.25.