What is the difference between the compound interests on Rs. 5000 for `1(1)/(2)` year at 4% per annum compounded yearly and half-yearly?

Options

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

Answer

To solve the problem of finding the difference between the compound interests on Rs. 5000 for 1.5 years at 4% per annum compounded yearly and half-yearly, we can follow these steps:

### Step 1: Calculate Compound Interest Compounded Yearly
1. **Identify the principal (P)**: Rs. 5000
2. **Identify the rate of interest (R)**: 4% per annum
3. **Identify the time (T)**: 1.5 years
4. **Use the formula for compound interest**:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the values:
\[ A = 5000 \left(1 + \frac{4}{100}\right)^{1.5} \]
\[ A = 5000 \left(1 + 0.04\right)^{1.5} \]
\[ A = 5000 \left(1.04\right)^{1.5} \]

5. **Calculate \( (1.04)^{1.5} \)**:
\[ (1.04)^{1.5} \approx 1.0604 \]

6. **Calculate the amount (A)**:
\[ A \approx 5000 \times 1.0604 \approx 5302 \]

7. **Calculate the compound interest (CI)**:
\[ CI = A - P = 5302 - 5000 = 302 \]

### Step 2: Calculate Compound Interest Compounded Half-Yearly
1. **Identify the principal (P)**: Rs. 5000
2. **Identify the rate of interest (R)**: 4% per annum (which is 2% per half-year)
3. **Identify the time (T)**: 1.5 years = 3 half-year periods
4. **Use the formula for compound interest**:
\[ A = P \left(1 + \frac{R/2}{100}\right)^{2T} \]
Substituting the values:
\[ A = 5000 \left(1 + \frac{2}{100}\right)^{3} \]
\[ A = 5000 \left(1 + 0.02\right)^{3} \]
\[ A = 5000 \left(1.02\right)^{3} \]

5. **Calculate \( (1.02)^{3} \)**:
\[ (1.02)^{3} \approx 1.061208 \]

6. **Calculate the amount (A)**:
\[ A \approx 5000 \times 1.061208 \approx 5306.04 \]

7. **Calculate the compound interest (CI)**:
\[ CI = A - P = 5306.04 - 5000 = 306.04 \]

### Step 3: Find the Difference Between the Two Compound Interests
1. **Calculate the difference**:
\[ \text{Difference} = CI_{\text{half-yearly}} - CI_{\text{yearly}} = 306.04 - 302 = 4.04 \]

### Final Answer
The difference between the compound interests on Rs. 5000 for 1.5 years at 4% per annum compounded yearly and half-yearly is approximately **Rs. 4.04**.

---