Find the difference between compound interest on 5000 at the rate of 4% for 1.5 years if interest is compounded annually and semiannually?

Options

A. ₹ 4.02
B. ₹ 1.02
C. ₹ 2.04
D. ₹ 2.01

Answer

To solve the problem of finding the difference between compound interest on ₹5000 at the rate of 4% for 1.5 years when compounded annually and semi-annually, we will follow these steps:

### Step 1: Calculate Compound Interest when Compounded Annually

1. **Identify the variables:**
- Principal (P) = ₹5000
- Rate (R) = 4% per annum
- Time (T) = 1.5 years

2. **Use the formula for compound interest:**
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
- Here, \(T = 1.5\) years.

3. **Calculate the amount (A):**
\[ A = 5000 \left(1 + \frac{4}{100}\right)^{1.5} = 5000 \left(1 + 0.04\right)^{1.5} = 5000 \left(1.04\right)^{1.5} \]
- Calculate \(1.04^{1.5}\):
\[ 1.04^{1.5} \approx 1.0604 \]
- Now, calculate \(A\):
\[ A \approx 5000 \times 1.0604 \approx 5302 \]

4. **Calculate Compound Interest (CI1):**
\[ CI1 = A - P = 5302 - 5000 = 302 \]

### Step 2: Calculate Compound Interest when Compounded Semi-Annually

1. **Adjust the rate and time for semi-annual compounding:**
- Semi-annual rate = \( \frac{4}{2} = 2\% \)
- Time in half years = \(1.5 \times 2 = 3\) half-year periods.

2. **Use the same formula for compound interest:**
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
- Here, \(n = 3\) (number of compounding periods).

3. **Calculate the amount (A):**
\[ A = 5000 \left(1 + \frac{2}{100}\right)^3 = 5000 \left(1 + 0.02\right)^3 = 5000 \left(1.02\right)^3 \]
- Calculate \(1.02^3\):
\[ 1.02^3 \approx 1.061208 \]
- Now, calculate \(A\):
\[ A \approx 5000 \times 1.061208 \approx 5306.04 \]

4. **Calculate Compound Interest (CI2):**
\[ CI2 = A - P = 5306.04 - 5000 = 306.04 \]

### Step 3: Find the Difference Between the Two Compound Interests

1. **Calculate the difference:**
\[ \text{Difference} = CI2 - CI1 = 306.04 - 302 = 4.04 \]

### Final Answer
The difference between the compound interest when compounded semi-annually and annually is approximately **₹4.04**.

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