A sum of Rs. 25000 invested at 8% per annum compounded semi-annually amounts to Rs. 28121.60, Then, the time period is

Options

A. 12 yr
B. 1 yr
C. `1 (1)/(2)` yr
D. None of these

Answer

To find the time period for the investment of Rs. 25000 at an interest rate of 8% per annum compounded semi-annually, we can use the formula for compound interest.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Principal (P) = Rs. 25000
- Amount (A) = Rs. 28121.60
- Rate of Interest (R) = 8% per annum
- Compounding Frequency (N) = 2 (since it is compounded semi-annually)

2. **Use the Compound Interest Formula:**
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{R}{N}\right)^{NT} \]
Substituting the known values into the formula:
\[ 28121.60 = 25000 \left(1 + \frac{8}{2 \times 100}\right)^{2T} \]

3. **Simplify the Equation:**
Calculate the interest rate per compounding period:
\[ \frac{8}{2 \times 100} = \frac{8}{200} = 0.04 \]
So, the equation becomes:
\[ 28121.60 = 25000 \left(1 + 0.04\right)^{2T} \]
Which simplifies to:
\[ 28121.60 = 25000 \left(1.04\right)^{2T} \]

4. **Divide Both Sides by 25000:**
\[ \frac{28121.60}{25000} = (1.04)^{2T} \]
Calculate the left side:
\[ 1.124864 = (1.04)^{2T} \]

5. **Take Logarithm on Both Sides:**
Taking logarithm (base 10) on both sides:
\[ \log(1.124864) = \log((1.04)^{2T}) \]
Using the property of logarithms:
\[ \log(1.124864) = 2T \cdot \log(1.04) \]

6. **Solve for T:**
Rearranging gives:
\[ T = \frac{\log(1.124864)}{2 \cdot \log(1.04)} \]
Now, calculate the logarithms:
- \(\log(1.124864) \approx 0.0524\)
- \(\log(1.04) \approx 0.0170\)

Substituting these values:
\[ T = \frac{0.0524}{2 \cdot 0.0170} \approx \frac{0.0524}{0.0340} \approx 1.54 \text{ years} \]

### Final Answer:
The time period (T) is approximately **1.54 years**.