Find the rate percent per annum, if `Rs.2000` amounts to `Rs.2,315.25` In a year and a half, interest being compounded half yearly.

Options

A. `11.5%`
B. `10%`
C. `5%`
D. `20% `

Answer

To find the rate percent per annum when Rs. 2000 amounts to Rs. 2315.25 in a year and a half, with interest being compounded half-yearly, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values**:
- Principal (P) = Rs. 2000
- Amount (A) = Rs. 2315.25
- Time (T) = 1.5 years

2. **Convert the time into half-year periods**:
Since the interest is compounded half-yearly, we need to convert the time into half-year periods:
\[ T = 1.5 \text{ years} = 3 \text{ half-year periods} \]

3. **Use the compound interest formula**:
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^{n} \]
where:
- \( A \) = Amount
- \( P \) = Principal
- \( r \) = Rate of interest per annum
- \( n \) = Number of compounding periods

Substituting the known values:
\[ 2315.25 = 2000 \left(1 + \frac{r}{200}\right)^{3} \]

4. **Rearranging the equation**:
Divide both sides by 2000:
\[ \frac{2315.25}{2000} = \left(1 + \frac{r}{200}\right)^{3} \]
Calculate the left side:
\[ 1.157625 = \left(1 + \frac{r}{200}\right)^{3} \]

5. **Taking the cube root**:
To eliminate the exponent, take the cube root of both sides:
\[ 1 + \frac{r}{200} = \sqrt[3]{1.157625} \]
Calculate the cube root:
\[ 1 + \frac{r}{200} \approx 1.05 \]

6. **Isolate \( r \)**:
Subtract 1 from both sides:
\[ \frac{r}{200} \approx 0.05 \]
Multiply both sides by 200:
\[ r \approx 10 \]

7. **Conclusion**:
The rate of interest per annum is approximately **10%**.