Find the amount and the C.I. on Rs. 8,000 in `1""1/2` years at 20% per year compounded half-yearly.

Answer

To solve the problem of finding the amount and the compound interest (C.I.) on Rs. 8,000 in 1.5 years at a rate of 20% per annum compounded half-yearly, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Principal (P) = Rs. 8,000
- Rate of interest (R) = 20% per annum
- Time (T) = 1.5 years

2. **Convert the Rate and Time for Half-Yearly Compounding:**
- Since the interest is compounded half-yearly, we need to divide the annual rate by 2:
\[ r = \frac{R}{2} = \frac{20}{2} = 10\% \]
- The time period in half-years (n) is:
\[ n = 2 \times T = 2 \times 1.5 = 3 \text{ half-years} \]

3. **Apply the Compound Interest Formula:**
- The formula for the amount (A) when compounded is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
- Substitute the values into the formula:
\[ A = 8000 \left(1 + \frac{10}{100}\right)^3 \]
- Simplifying inside the brackets:
\[ A = 8000 \left(1 + 0.1\right)^3 = 8000 \left(1.1\right)^3 \]

4. **Calculate \( (1.1)^3 \):**
- Calculate \( (1.1)^3 \):
\[ (1.1)^3 = 1.1 \times 1.1 \times 1.1 = 1.331 \]
- Now substitute this back into the amount formula:
\[ A = 8000 \times 1.331 = 10648 \]

5. **Find the Compound Interest (C.I.):**
- The formula for compound interest is:
\[ \text{C.I.} = A - P \]
- Substitute the values:
\[ \text{C.I.} = 10648 - 8000 = 2648 \]

### Final Results:
- Amount (A) = Rs. 10,648
- Compound Interest (C.I.) = Rs. 2,648