What is the compound interest (in Rs.) for 1 year on a sum of `Rs. 20000` at the rate of `40%` per annum compounded half yearly?

Options

A. `8000 `
B. `8650`
C. ` 8750 `
D. `8800`

Answer

To calculate the compound interest for 1 year on a sum of Rs. 20,000 at the rate of 40% per annum compounded half-yearly, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the principal amount (P)**:
- The principal amount is given as Rs. 20,000.

2. **Determine the annual interest rate (R)**:
- The annual interest rate is 40%.

3. **Calculate the half-yearly interest rate**:
- Since the interest is compounded half-yearly, we divide the annual interest rate by 2.
\[ \text{Half-yearly rate} = \frac{40\%}{2} = 20\% \]

4. **Determine the time period in half-yearly terms (n)**:
- The time period is 1 year, which is equivalent to 2 half-year periods.
\[ n = 2 \]

5. **Use the compound interest formula**:
- The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
where \(A\) is the amount after time \(n\), \(P\) is the principal, \(R\) is the rate of interest, and \(n\) is the number of compounding periods.

6. **Substitute the values into the formula**:
\[ A = 20000 \left(1 + \frac{20}{100}\right)^2 \]
\[ A = 20000 \left(1 + 0.2\right)^2 \]
\[ A = 20000 \left(1.2\right)^2 \]
\[ A = 20000 \times 1.44 \]
\[ A = 28800 \]

7. **Calculate the compound interest (CI)**:
- The compound interest is the total amount minus the principal.
\[ CI = A - P \]
\[ CI = 28800 - 20000 \]
\[ CI = 8800 \]

### Final Answer:
The compound interest for 1 year on a sum of Rs. 20,000 at the rate of 40% per annum compounded half-yearly is **Rs. 8,800**.

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