Find the amount and the compound interest on Rs. 8,000 in `1""1/2` years at 10% per year compounded half-yearly.

Answer

To find the amount and the compound interest on Rs. 8,000 in 1.5 years at 10% per year compounded half-yearly, we can follow these steps:

### Step 1: Determine the values of n and r
- The total time \( t = 1.5 \) years.
- Since the interest is compounded half-yearly, we need to calculate \( n \) (the number of compounding periods).

\[ n = 2 \times t = 2 \times 1.5 = 3 \]

- The annual interest rate \( R = 10\% \). Since it is compounded half-yearly, we need to find the rate per half-year \( r \).

\[ r = \frac{R}{2} = \frac{10}{2} = 5\% \]

### Step 2: Use the compound interest formula
The formula for the amount \( A \) when compounded is given by:

\[ A = P \left(1 + \frac{r}{100}\right)^n \]

Where:
- \( P \) is the principal amount (Rs. 8000),
- \( r \) is the rate per period (5%),
- \( n \) is the number of compounding periods (3).

### Step 3: Substitute the values into the formula
Now, substituting the values into the formula:

\[ A = 8000 \left(1 + \frac{5}{100}\right)^3 \]

This simplifies to:

\[ A = 8000 \left(1 + 0.05\right)^3 = 8000 \left(1.05\right)^3 \]

### Step 4: Calculate \( (1.05)^3 \)
Now we need to calculate \( (1.05)^3 \):

\[ (1.05)^3 = 1.157625 \]

### Step 5: Calculate the amount \( A \)
Now, we can calculate the amount:

\[ A = 8000 \times 1.157625 \approx 9261 \]

### Step 6: Calculate the compound interest
The compound interest (CI) is calculated by subtracting the principal from the amount:

\[ CI = A - P = 9261 - 8000 = 1261 \]

### Final Results
- The amount after 1.5 years is approximately **Rs. 9261**.
- The compound interest earned is **Rs. 1261**.

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