Find the amount and the compound interest on Rs. 32,000 for 1 year at 20% per annum compounded half-yearly.

Answer

To solve the problem of finding the amount and the compound interest on Rs. 32,000 for 1 year at 20% per annum compounded half-yearly, we will follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs. 32,000
- Rate of interest (R) = 20% per annum
- Time (T) = 1 year

### Step 2: Adjust the rate and time for half-yearly compounding
Since the interest is compounded half-yearly:
- The effective rate per half-year = R / 2 = 20% / 2 = 10%
- The number of compounding periods (n) = T * 2 = 1 * 2 = 2

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

### Step 4: Substitute the values into the formula
Substituting the values we have:
\[ A = 32000 \left(1 + \frac{10}{100}\right)^2 \]

### Step 5: Simplify the equation
Calculating inside the parentheses:
\[ A = 32000 \left(1 + 0.1\right)^2 \]
\[ A = 32000 \left(1.1\right)^2 \]
Calculating \( (1.1)^2 \):
\[ (1.1)^2 = 1.21 \]
Now substituting back:
\[ A = 32000 \times 1.21 \]

### Step 6: Calculate the amount
\[ A = 32000 \times 1.21 = 38720 \]
So, the amount after 1 year is Rs. 38,720.

### Step 7: Calculate the compound interest
Compound Interest (CI) can be calculated as:
\[ CI = A - P \]
Substituting the values:
\[ CI = 38720 - 32000 \]
\[ CI = 6720 \]

### Final Results
- Amount after 1 year: Rs. 38,720
- Compound Interest: Rs. 6,720

---