The difference of S.I and C.I on a sum at 10% for `1(1)/(2)` years is Rs. 244. Find the sum, If compounded half yearly.

Answer

To solve the problem, we need to find the principal sum (P) given that the difference between the Simple Interest (SI) and Compound Interest (CI) for a period of 1.5 years at a rate of 10% compounded half-yearly is Rs. 244.

### Step-by-Step Solution:

1. **Identify the Rate and Time:**
- The rate (R) is 10% per annum.
- The time (T) is 1.5 years, which can be expressed as 1.5 = 3/2 years.

2. **Convert Time for Half-Yearly Compounding:**
- Since the interest is compounded half-yearly, we need to convert the time into half-year periods:
- Total half-year periods = 1.5 years × 2 = 3 half-years.

3. **Adjust the Rate for Half-Yearly Compounding:**
- The effective rate for each half-year period = 10% / 2 = 5%.

4. **Calculate Simple Interest (SI):**
- SI for 1.5 years can be calculated using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
- Here, \(T = 1.5\) years, so:
\[ SI = \frac{P \times 10 \times 1.5}{100} = \frac{15P}{100} = 0.15P \]

5. **Calculate Compound Interest (CI):**
- The formula for CI when compounded half-yearly is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
- Where \(n\) is the number of compounding periods. Thus:
\[ A = P \left(1 + \frac{5}{100}\right)^3 = P \left(1.05\right)^3 \]
- Calculate \( (1.05)^3 \):
\[ (1.05)^3 = 1.157625 \]
- Therefore, the amount \(A\) becomes:
\[ A = P \times 1.157625 \]
- The CI is given by:
\[ CI = A - P = P \times 1.157625 - P = P(1.157625 - 1) = P \times 0.157625 \]

6. **Find the Difference Between CI and SI:**
- The difference between CI and SI is given as Rs. 244:
\[ CI - SI = P \times 0.157625 - 0.15P = P(0.157625 - 0.15) = P \times 0.007625 \]
- Setting this equal to Rs. 244:
\[ P \times 0.007625 = 244 \]

7. **Solve for P:**
- Rearranging gives:
\[ P = \frac{244}{0.007625} \approx 32000 \]

### Final Answer:
The principal sum (P) is Rs. 32,000.

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