The compound interest on Rs. 2000 for 1 yr at the rate of 8% per annum, when the interest is compounded semi-annually is

Options

A. Rs. 163.20
B. Rs. 2163.20
C. Rs. 2000
D. None of these

Answer

To find the compound interest on Rs. 2000 for 1 year at the rate of 8% per annum, compounded semi-annually, we can follow these steps:

### Step 1: Identify the parameters
- Principal (P) = Rs. 2000
- Rate of interest (R) = 8% per annum
- Time (T) = 1 year

### Step 2: Adjust the rate and time for semi-annual compounding
Since the interest is compounded semi-annually:
- The time period in terms of compounding periods = 2 (because 1 year has 2 half-year periods)
- The rate per compounding period = 8% / 2 = 4%

### Step 3: 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 \(T\)
- \(P\) = Principal
- \(r\) = Rate of interest per compounding period
- \(n\) = Number of compounding periods

Substituting the values:
- \(P = 2000\)
- \(r = 4\%\)
- \(n = 2\)

\[ A = 2000 \left(1 + \frac{4}{100}\right)^2 \]

### Step 4: Simplify the expression
Calculate \(1 + \frac{4}{100}\):
\[ 1 + \frac{4}{100} = 1 + 0.04 = 1.04 \]

Now substitute this back into the formula:
\[ A = 2000 \times (1.04)^2 \]

### Step 5: Calculate \((1.04)^2\)
\[ (1.04)^2 = 1.0816 \]

### Step 6: Calculate the amount \(A\)
\[ A = 2000 \times 1.0816 = 2163.20 \]

### Step 7: Calculate the compound interest
The compound interest (CI) is given by:
\[ CI = A - P \]
Substituting the values:
\[ CI = 2163.20 - 2000 = 163.20 \]

### Final Answer
The compound interest on Rs. 2000 for 1 year at the rate of 8% per annum, compounded semi-annually, is **Rs. 163.20**.

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