Find, to the nearest rupee, the amount and the compound interest on 9000 for `1 (1)/(2)` years at 8% per annum, the interest being compounded half-yearly.

Answer

To solve the problem of finding the amount and compound interest on ₹9000 for 1.5 years at an 8% annual interest rate compounded half-yearly, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = ₹9000
- Time (T) = 1.5 years = \( \frac{3}{2} \) years
- Rate (R) = 8% per annum
- Since the interest is compounded half-yearly, we need to adjust the time and rate accordingly.

### Step 2: Adjust the time and rate for half-yearly compounding
- The number of compounding periods per year = 2 (since it is compounded half-yearly).
- Therefore, the total number of compounding periods (n) = \( 2 \times \frac{3}{2} = 3 \).
- The effective rate per half-year = \( \frac{8}{2} = 4\% \).

### Step 3: Use the compound interest formula
The formula for the amount (A) when interest is compounded is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( P \) = Principal
- \( r \) = Rate of interest per period
- \( n \) = Number of compounding periods

Substituting the values:
\[ A = 9000 \left(1 + \frac{4}{100}\right)^3 \]

### Step 4: Calculate the amount
First, calculate \( 1 + \frac{4}{100} = 1.04 \).
Now substitute this back into the formula:
\[ A = 9000 \times (1.04)^3 \]
Calculating \( (1.04)^3 \):
\[ (1.04)^3 = 1.124864 \]
Now calculate the amount:
\[ A = 9000 \times 1.124864 \approx 10123.776 \]

### Step 5: Round the amount to the nearest rupee
Rounding \( 10123.776 \) gives us:
\[ A \approx ₹10124 \]

### Step 6: Calculate the compound interest
Compound Interest (CI) can be calculated using the formula:
\[ CI = A - P \]
Substituting the values:
\[ CI = 10123.776 - 9000 \approx 1123.776 \]
Rounding \( 1123.776 \) gives us:
\[ CI \approx ₹1124 \]

### Final Results
- Amount (A) = ₹10124 (to the nearest rupee)
- Compound Interest (CI) = ₹1124 (to the nearest rupee)