What sum will give `Rs. 244` as the difference between simple interest and compound interest at `10%` in `1(1)/2` years compounded half yearly?

Options

A. `Rs. 40,000`
B. `Rs. 36,000`
C. `Rs. 32,000 `
D. `Rs. 28,000 `

Answer

To solve the problem of finding the sum that will give Rs. 244 as the difference between simple interest (SI) and compound interest (CI) at a rate of 10% compounded half-yearly over a period of 1.5 years, we can follow these steps:

### Step 1: Determine the effective rate and time period
Since the interest is compounded half-yearly, we need to adjust the rate and time period:
- The nominal rate of interest is 10% per annum. Therefore, the effective rate for half-yearly compounding is:
\[ \text{Effective Rate} = \frac{10\%}{2} = 5\% \]
- The time period of 1.5 years is equivalent to:
\[ \text{Number of half-years} = 1.5 \times 2 = 3 \text{ half-years} \]

### Step 2: Calculate the amounts for CI and SI
Let's assume the principal amount (P) is Rs. 8000 (as per the video transcript's example).

**For Compound Interest (CI):**
- The formula for the amount (A) after n years with compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
- Here, \( P = 8000 \), \( r = 5\% \), and \( n = 3 \):
\[ A = 8000 \left(1 + \frac{5}{100}\right)^3 = 8000 \left(1.05\right)^3 \]
- Calculating \( (1.05)^3 \):
\[ (1.05)^3 \approx 1.157625 \]
- Therefore:
\[ A \approx 8000 \times 1.157625 \approx 9261 \]

**For Simple Interest (SI):**
- The formula for simple interest is:
\[ SI = \frac{P \times r \times t}{100} \]
- Here, \( P = 8000 \), \( r = 5\% \), and \( t = 3 \):
\[ SI = \frac{8000 \times 5 \times 3}{100} = \frac{120000}{100} = 1200 \]

### Step 3: Calculate the difference between CI and SI
- The difference between CI and SI is:
\[ \text{Difference} = CI - SI = 9261 - 8000 - 1200 = 1261 \]

### Step 4: Relate the difference to Rs. 244
According to the problem, the difference between SI and CI is Rs. 244. We can set up a proportion:
- From the example, we found that the difference of 1261 units corresponds to a principal of 8000 units. We can set up the equation:
\[ \frac{244}{1261} = \frac{P}{8000} \]
- To find the principal (P):
\[ P = \frac{244 \times 8000}{1261} \]

### Step 5: Calculate the principal
- Now we can calculate:
\[ P \approx \frac{1952000}{1261} \approx 1548.09 \]

### Conclusion
The sum that will give Rs. 244 as the difference between simple interest and compound interest at 10% in 1.5 years compounded half-yearly is approximately Rs. 1548.09.