If compound interest on some amount at the rate of 20% interest for 1.5 years is 9930 then find the principle amount if interest is compounded semi-annually?

Options

A. ₹ 29500
B. ₹ 31000
C. ₹ 35000
D. ₹30000

Answer

To find the principal amount when the compound interest is given, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We are given that the compound interest (CI) for a certain principal amount (P) at a rate of 20% per annum for 1.5 years is 9930, and the interest is compounded semi-annually.

2. **Convert Time Period**:
Since the interest is compounded semi-annually, we need to convert the time period of 1.5 years into half-year periods.
- 1.5 years = 1.5 × 2 = 3 half-year periods.

3. **Identify the Rate**:
The annual interest rate is 20%. Since the interest is compounded semi-annually, we divide this rate by 2.
- Semi-annual rate = 20% / 2 = 10% = 0.10 (in decimal).

4. **Use the Compound Interest Formula**:
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) = the amount after time \( t \)
- \( P \) = principal amount
- \( r \) = annual interest rate (as a decimal)
- \( n \) = number of times interest is compounded per year
- \( t \) = time in years

In our case:
- \( r = 0.20 \)
- \( n = 2 \) (since it's compounded semi-annually)
- \( t = 1.5 \)

5. **Calculate the Amount (A)**:
We know that:
\[ CI = A - P \]
Given \( CI = 9930 \), we can express \( A \) as:
\[ A = P + 9930 \]

Now substituting the values into the compound interest formula:
\[ P + 9930 = P \left(1 + \frac{0.20}{2}\right)^{2 \times 1.5} \]
Simplifying further:
\[ P + 9930 = P \left(1 + 0.10\right)^{3} \]
\[ P + 9930 = P \left(1.10\right)^{3} \]

6. **Calculate \( (1.10)^3 \)**:
\[ (1.10)^3 = 1.331 \]
Therefore:
\[ P + 9930 = P \times 1.331 \]

7. **Rearranging the Equation**:
\[ P \times 1.331 - P = 9930 \]
\[ P(1.331 - 1) = 9930 \]
\[ P(0.331) = 9930 \]

8. **Solve for P**:
\[ P = \frac{9930}{0.331} \approx 30000 \]

### Final Answer:
The principal amount \( P \) is **30,000**.