Find the compound interest on Rs 18360 for 1 year and 6 months at the rate of 8% if interest is compounded semi-annually?

Options

A. ₹ 2222.92
B. ₹ 2227.92
C. ₹ 2292.22
D. ₹ 2292.27

Answer

To find the compound interest on Rs 18,360 for 1 year and 6 months at the rate of 8% compounded semi-annually, we can follow these steps:

### Step 1: Identify the Principal, Rate, and Time
- **Principal (P)** = Rs 18,360
- **Rate (R)** = 8% per annum
- **Time (T)** = 1 year and 6 months = 1.5 years

### Step 2: Convert Time to Semi-Annual Periods
Since the interest is compounded semi-annually, we need to convert the time into semi-annual periods:
- Total semi-annual periods = 1.5 years × 2 = 3 semi-annual periods

### Step 3: Calculate the Semi-Annual Interest Rate
- Semi-annual interest rate = Annual rate / 2 = 8% / 2 = 4%

### Step 4: 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 period
- \( n \) = Number of compounding periods

Substituting the values:
- \( P = 18,360 \)
- \( r = 4 \) (semi-annual rate)
- \( n = 3 \) (number of semi-annual periods)

\[ A = 18360 \left(1 + \frac{4}{100}\right)^3 \]
\[ A = 18360 \left(1 + 0.04\right)^3 \]
\[ A = 18360 \left(1.04\right)^3 \]

### Step 5: Calculate \( (1.04)^3 \)
Calculating \( (1.04)^3 \):
\[ (1.04)^3 = 1.124864 \]

### Step 6: Calculate the Amount \( A \)
Now substitute back to find \( A \):
\[ A = 18360 \times 1.124864 \]
\[ A \approx 20605.55 \]

### Step 7: Calculate the Compound Interest
Compound Interest (CI) is given by:
\[ CI = A - P \]
\[ CI = 20605.55 - 18360 \]
\[ CI \approx 2245.55 \]

### Final Answer
The compound interest on Rs 18,360 for 1 year and 6 months at the rate of 8% compounded semi-annually is approximately **Rs 2245.55**.

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