Find the amount of Rs 1500 after `1(1)/(2)` years at the rate of 12% per annum compounded half-yearly.

Options

A. 1786.52
B. 1789
C. 1785.6
D. 1784

Answer

To find the amount of Rs 1500 after \(1 \frac{1}{2}\) years at the rate of 12% per annum compounded half-yearly, we can follow these steps:

### Step 1: Identify the variables
- Principal (P) = Rs 1500
- Rate of interest (R) = 12% per annum
- Time (T) = \(1 \frac{1}{2}\) years = \(1.5\) years

### Step 2: Convert the time into half-year periods
Since the interest is compounded half-yearly, we need to convert the time into half-year periods:
- Total half-year periods (N) = \(1.5 \text{ years} \times 2 = 3 \text{ half-years}\)

### Step 3: Calculate the half-yearly interest rate
The annual interest rate is 12%, so the half-yearly interest rate will be:
- Half-yearly rate (r) = \(\frac{12}{2} = 6\%\)

### Step 4: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^N \]
Where:
- \(A\) = Amount after time \(T\)
- \(P\) = Principal amount
- \(r\) = Rate of interest per period
- \(N\) = Number of periods

### Step 5: Substitute the values into the formula
Substituting the values we have:
\[ A = 1500 \left(1 + \frac{6}{100}\right)^3 \]
\[ A = 1500 \left(1 + 0.06\right)^3 \]
\[ A = 1500 \left(1.06\right)^3 \]

### Step 6: Calculate \( (1.06)^3 \)
Calculating \( (1.06)^3 \):
\[ (1.06)^3 = 1.191016 \]

### Step 7: Calculate the final amount
Now, substituting back to find \(A\):
\[ A = 1500 \times 1.191016 \approx 1786.52 \]

### Final Answer
The amount after \(1 \frac{1}{2}\) years is approximately Rs 1786.52.

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