The compound interest on `Rs. 10,000` in 2 years at `4%` per annum, the interest being compounded half-yearly, is :

Options

A. `Rs.636.80`
B. `Rs. 824.32`
C. `Rs. 912. 86`
D. `Rs. 828. 82`

Answer

To find the compound interest on Rs. 10,000 in 2 years at a rate of 4% per annum, compounded half-yearly, we can follow these steps:

### Step 1: Determine the adjusted rate and time
Since the interest is compounded half-yearly, we need to adjust the rate and the time:
- The annual interest rate is 4%, so the half-yearly rate is:
\[ \text{Half-yearly rate} = \frac{4\%}{2} = 2\% \]
- The time period in years is 2 years, so the number of half-year periods is:
\[ \text{Number of periods} = 2 \times 2 = 4 \]

### Step 2: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \(A\) = Amount after time \(n\)
- \(P\) = Principal amount (initial investment)
- \(r\) = Rate of interest per period
- \(n\) = Number of periods

Substituting the values:
- \(P = 10,000\)
- \(r = 2\%\)
- \(n = 4\)

The formula becomes:
\[ A = 10,000 \left(1 + \frac{2}{100}\right)^4 \]

### Step 3: Calculate the amount
Now, we can calculate:
\[ A = 10,000 \left(1 + 0.02\right)^4 = 10,000 \left(1.02\right)^4 \]

Calculating \(1.02^4\):
\[ 1.02^4 = 1.08243216 \quad (\text{approximately}) \]
Now substituting back:
\[ A = 10,000 \times 1.08243216 \approx 10,824.32 \]

### Step 4: Calculate the compound interest
The compound interest (CI) is given by:
\[ CI = A - P \]
Substituting the values:
\[ CI = 10,824.32 - 10,000 = 824.32 \]

### Final Answer
The compound interest on Rs. 10,000 in 2 years at 4% per annum, compounded half-yearly, is approximately Rs. 824.32.

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