What is the compound interest on Rs. 38000 at 20% per annum for a period of `(3)/(2)` years compounded semi-annually ?

Answer

To find the compound interest on Rs. 38,000 at a rate of 20% per annum for a period of \( \frac{3}{2} \) years compounded semi-annually, we will follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs. 38,000
- Rate of interest (R) = 20% per annum
- Time (T) = \( \frac{3}{2} \) years

### Step 2: Adjust the rate and time for semi-annual compounding
Since the interest is compounded semi-annually:
- The rate per half-year = \( \frac{20\%}{2} = 10\% \)
- The total number of compounding periods (n) = \( 2 \times \frac{3}{2} = 3 \) (since there are 2 compounding periods in a year)

### Step 3: Use the compound interest formula
The formula for the amount (A) when compounded is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Substituting the values we have:
\[ A = 38000 \left(1 + \frac{10}{100}\right)^3 \]
\[ A = 38000 \left(1 + 0.1\right)^3 \]
\[ A = 38000 \left(1.1\right)^3 \]

### Step 4: Calculate \( (1.1)^3 \)
Calculating \( (1.1)^3 \):
\[ (1.1)^3 = 1.1 \times 1.1 \times 1.1 = 1.331 \]

### Step 5: Calculate the total amount (A)
Now substituting back into the amount formula:
\[ A = 38000 \times 1.331 \]
\[ A \approx 50578 \]

### Step 6: Calculate the compound interest (CI)
The compound interest is given by:
\[ CI = A - P \]
\[ CI = 50578 - 38000 \]
\[ CI = 12578 \]

### Final Answer
The compound interest on Rs. 38,000 at 20% per annum for \( \frac{3}{2} \) years compounded semi-annually is Rs. 12,578.

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