Geeta borrowed रु 15,000 for 18 months at a certain rate of interest compounded semi annually. If at the end of six months it amounted to 15,600, calculate :
(i) the rate of interest per annum.
(ii) the total amount of money that Geeta must pay at the end of 18 months inm order to clear the account.

Explanation

To solve the problem step-by-step, we will break it down into two parts as specified in the question.

### Step 1: Calculate the Simple Interest (SI) for the first 6 months

1. **Identify the principal (P)**:
\[ P = 15,000 \text{ rupees} \]

2. **Identify the amount (A) after 6 months**:
\[ A = 15,600 \text{ rupees} \]

3. **Calculate the Simple Interest (SI)**:
\[ SI = A - P = 15,600 - 15,000 = 600 \text{ rupees} \]

### Step 2: Calculate the Rate of Interest (R)

4. **Use the formula for Simple Interest**:
\[ SI = \frac{P \times R \times T}{100} \]
Here, \( T \) is in years. Since 6 months is half a year, \( T = \frac{1}{2} \).

5. **Substitute the known values into the formula**:
\[ 600 = \frac{15,000 \times R \times \frac{1}{2}}{100} \]

6. **Rearranging to find R**:
\[ 600 = \frac{15,000 \times R}{200} \]
\[ 600 \times 200 = 15,000 \times R \]
\[ 120,000 = 15,000 \times R \]
\[ R = \frac{120,000}{15,000} = 8 \text{ percent per annum} \]

### Step 3: Calculate the Total Amount at the end of 18 months

7. **Determine the effective rate for the remaining 12 months**:
Since the interest is compounded semi-annually, the rate for the next 12 months will be half of the annual rate:
\[ \text{Rate for 6 months} = \frac{8}{2} = 4 \text{ percent} \]

8. **Convert 18 months to years**:
\[ T = \frac{18}{12} = 1.5 \text{ years} \]

9. **Calculate the number of compounding periods**:
Since the interest is compounded semi-annually, for 18 months, there will be 3 compounding periods (6 months each):
\[ n = 3 \]

10. **Use the compound interest formula**:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Substituting the values:
\[ A = 15,000 \left(1 + \frac{4}{100}\right)^3 \]
\[ A = 15,000 \left(1 + 0.04\right)^3 \]
\[ A = 15,000 \left(1.04\right)^3 \]

11. **Calculate \( (1.04)^3 \)**:
\[ (1.04)^3 \approx 1.124864 \]

12. **Final calculation of the amount**:
\[ A \approx 15,000 \times 1.124864 \approx 16,872.96 \text{ rupees} \]

### Summary of Results
- **Rate of Interest per annum**: \( 8\% \)
- **Total Amount after 18 months**: \( 16,872.96 \text{ rupees} \)