Find compound interest on Rs. 5000 for 2 years at 10% per annum, compounded half-yearly.

Options

A. Rs. 1077.5
B. Rs. 1072.5
C. Rs. 1000
D. Rs. 1100

Answer

To find the compound interest on Rs. 5000 for 2 years at 10% per annum, compounded half-yearly, we will follow these steps:

### Step 1: Determine the half-yearly interest rate
Since the annual interest rate is 10%, the half-yearly interest rate will be half of that:
\[ \text{Half-yearly interest rate} = \frac{10\%}{2} = 5\% \]

### Step 2: Convert the interest rate to a decimal
To use the interest rate in calculations, we convert it from a percentage to a decimal:
\[ \text{Half-yearly interest rate in decimal} = \frac{5}{100} = 0.05 \]

### Step 3: Determine the number of compounding periods
Since the interest is compounded half-yearly and the total time is 2 years, we need to find the total number of compounding periods:
\[ \text{Number of compounding periods} = 2 \text{ years} \times 2 = 4 \text{ periods} \]

### Step 4: Use the compound interest formula
The formula for compound interest is given by:
\[ A = P \left(1 + r\right)^n \]
Where:
- \(A\) = the amount after time \(n\)
- \(P\) = principal amount (initial investment)
- \(r\) = interest rate per period
- \(n\) = number of compounding periods

Substituting the values:
\[ A = 5000 \left(1 + 0.05\right)^4 \]

### Step 5: Calculate the amount
First, calculate \(1 + 0.05\):
\[ 1 + 0.05 = 1.05 \]
Now raise it to the power of 4:
\[ 1.05^4 = 1.21550625 \]
Now multiply by the principal:
\[ A = 5000 \times 1.21550625 \approx 6077.53125 \]

### Step 6: Calculate the compound interest
The compound interest (CI) is calculated by subtracting the principal from the total amount:
\[ CI = A - P \]
Substituting the values:
\[ CI = 6077.53125 - 5000 \approx 1077.53125 \]

### Step 7: Round off the compound interest
Rounding off to two decimal places, we get:
\[ CI \approx 1077.53 \]

### Final Answer
The compound interest on Rs. 5000 for 2 years at 10% per annum, compounded half-yearly, is approximately **Rs. 1077.53**.

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