What is the difference between compound Interest on `Rs. 5,000` for `1(1)/2`years at `4%` per annum according as the interest is compounded yearly or halfyearly?

Options

A. `Rs. 2.04`
B. `Rs. 3.06`
C. `Rs. 8.30 `
D. `Rs. 4.80 `

Answer

To solve the problem of finding the difference between compound interest on Rs. 5,000 for 1.5 years at 4% per annum, compounded yearly and half-yearly, we will follow these steps:

### Step 1: Calculate Compound Interest when Compounded Yearly
1. **Identify the principal (P)**: Rs. 5,000
2. **Identify the rate (R)**: 4% per annum
3. **Identify the time (T)**: 1.5 years
4. **Calculate the amount (A)** using the formula for compound interest:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Here, we need to split the time into 1 year and 0.5 years:
\[ A = 5000 \left(1 + \frac{4}{100}\right)^{1.5} = 5000 \left(1.04\right)^{1.5} \]

5. **Calculate \( (1.04)^{1.5} \)**:
\[ (1.04)^{1.5} \approx 1.0608 \]
6. **Calculate the total amount (A)**:
\[ A \approx 5000 \times 1.0608 \approx 5304 \]

7. **Calculate Compound Interest (CI)**:
\[ CI = A - P = 5304 - 5000 = 304 \]

### Step 2: Calculate Compound Interest when Compounded Half-Yearly
1. **Identify the half-yearly rate**: Since the annual rate is 4%, the half-yearly rate is \( \frac{4}{2} = 2\% \).
2. **Identify the number of compounding periods**: For 1.5 years, there are 3 half-year periods.
3. **Calculate the amount (A)** using the formula:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Where \( n \) is the number of compounding periods:
\[ A = 5000 \left(1 + \frac{2}{100}\right)^3 = 5000 \left(1.02\right)^3 \]

4. **Calculate \( (1.02)^3 \)**:
\[ (1.02)^3 \approx 1.061208 \]
5. **Calculate the total amount (A)**:
\[ A \approx 5000 \times 1.061208 \approx 5306.04 \]

6. **Calculate Compound Interest (CI)**:
\[ CI = A - P = 5306.04 - 5000 = 306.04 \]

### Step 3: Find the Difference in Compound Interest
1. **Calculate the difference**:
\[ \text{Difference} = CI_{\text{half-yearly}} - CI_{\text{yearly}} = 306.04 - 304 = 2.04 \]

### Final Answer
The difference between the compound interest when compounded yearly and half-yearly is **Rs. 2.04**.

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