Rs. 12000 invested at 20% per annum compounded semiannually amount to Rs. 13200. The period of investment is

Options

A. `1` yr
B. `1/2` yr
C. `2` yr
D. `2 (1)/(2)` yr

Answer

To solve the problem of finding the period of investment when Rs. 12000 is invested at 20% per annum compounded semiannually to amount to Rs. 13200, we can use the formula for compound interest.

### Step-by-Step Solution:

1. **Identify the given values:**
- Principal (P) = Rs. 12000
- Rate of interest (R) = 20% per annum
- Amount (A) = Rs. 13200
- Compounding frequency (N) = 2 (since it is compounded semiannually)

2. **Convert the annual interest rate to the semiannual rate:**
- Semiannual rate = R / N = 20% / 2 = 10% = 0.10 (as a decimal)

3. **Use the compound interest formula:**
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{N}\right)^{Nt} \]
Substituting the known values into the formula:
\[ 13200 = 12000 \left(1 + 0.10\right)^{2t} \]

4. **Simplify the equation:**
\[ 13200 = 12000 \times (1.1)^{2t} \]
Divide both sides by 12000:
\[ \frac{13200}{12000} = (1.1)^{2t} \]
\[ 1.1 = (1.1)^{2t} \]

5. **Set the exponents equal to each other:**
Since the bases are the same, we can equate the exponents:
\[ 2t = 1 \]

6. **Solve for t:**
\[ t = \frac{1}{2} \text{ years} \]

### Final Answer:
The period of investment is **0.5 years** or **6 months**.