Rs . 20000 is being compounded at 20% p.a . If the rate of interest is changed half yearly. What will be the amount after 2 years ?

Options

A. 28292
B. 27292
C. 29282
D. 22358

Answer

To solve the problem of finding the amount after 2 years when Rs. 20,000 is compounded at 20% per annum with half-yearly compounding, we can follow these steps:

### Step 1: Understand the Compounding Frequency
Since the interest is compounded half-yearly, we need to adjust the annual interest rate and the time period accordingly.

- Annual interest rate (R) = 20%
- Half-yearly interest rate = 20% / 2 = 10%

### Step 2: Determine the Number of Compounding Periods
The time period given is 2 years. Since the interest is compounded half-yearly, we will have:

- Number of compounding periods (N) = 2 years × 2 = 4 periods

### Step 3: Use the Compound Interest Formula
The formula for compound interest is given by:

\[ A = P \left(1 + \frac{R}{100}\right)^N \]

Where:
- \(A\) = Amount after time \(t\)
- \(P\) = Principal amount (initial investment)
- \(R\) = Annual interest rate (as a percentage)
- \(N\) = Number of compounding periods

### Step 4: Substitute the Values into the Formula
Now we can substitute the values into the formula:

- \(P = 20000\)
- \(R = 10\) (half-yearly rate)
- \(N = 4\)

So, we have:

\[ A = 20000 \left(1 + \frac{10}{100}\right)^4 \]

### Step 5: Simplify the Equation
Now, simplify the expression inside the parentheses:

\[ A = 20000 \left(1 + 0.10\right)^4 \]
\[ A = 20000 \left(1.10\right)^4 \]

### Step 6: Calculate \( (1.10)^4 \)
Now, calculate \( (1.10)^4 \):

\[ (1.10)^4 = 1.4641 \quad \text{(approximately)} \]

### Step 7: Calculate the Final Amount
Now substitute back to find \(A\):

\[ A = 20000 \times 1.4641 \]
\[ A \approx 29282 \]

### Final Answer
The amount after 2 years will be approximately **Rs. 29,282**.

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