Calculate the compound interest on Rs 4,000 in `1(1)/(2)` years at 10 % per annum compounded half-yearly.

Answer

To calculate the compound interest on Rs 4,000 in 1.5 years at 10% per annum compounded half-yearly, follow these steps:

### Step 1: Identify the Principal, Rate, and Time
- **Principal (P)** = Rs 4,000
- **Rate (R)** = 10% per annum
- **Time (T)** = 1.5 years

### Step 2: Adjust the Rate and Time for Half-Yearly Compounding
Since the interest is compounded half-yearly:
- The effective rate for half-yearly compounding is \( R/2 = 10\% / 2 = 5\% \).
- The time in half-year periods is \( T \times 2 = 1.5 \times 2 = 3 \) half-year periods.

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

Substituting the values:
\[ A = 4000 \left(1 + \frac{5}{100}\right)^3 \]

### Step 4: Calculate the Amount
First, calculate \( 1 + \frac{5}{100} \):
\[ 1 + \frac{5}{100} = 1 + 0.05 = 1.05 \]
Now raise it to the power of 3:
\[ A = 4000 \times (1.05)^3 \]
Calculating \( (1.05)^3 \):
\[ (1.05)^3 = 1.157625 \]
Now, calculate the amount:
\[ A = 4000 \times 1.157625 = 4630.5 \]

### Step 5: Calculate the Compound Interest
The formula for compound interest (CI) is:
\[ CI = A - P \]
Substituting the values:
\[ CI = 4630.5 - 4000 = 630.5 \]

### Final Answer
The compound interest on Rs 4,000 in 1.5 years at 10% per annum compounded half-yearly is **Rs 630.5**.

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