A sum of ₹ 1200 is invested at compound interest (compounded half yearly). If the rate of interest is `10%` per annum, then what will be the amount after 18 months?

Options

A. ₹ 1389.15
B. ₹ 1185.45
C. ₹ 1563.25
D. ₹ 1295.35

Answer

To solve the problem step by step, we will use the formula for compound interest. The formula for the amount \( A \) after \( t \) years is given by:

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

Where:
- \( A \) = Amount after time \( t \)
- \( P \) = Principal amount (initial investment)
- \( r \) = Rate of interest per annum
- \( n \) = Number of compounding periods

### Step 1: Identify the given values
- Principal \( P = ₹1200 \)
- Rate of interest \( r = 10\% \) per annum
- Time \( t = 18 \) months

### Step 2: Convert the time into years and find the number of compounding periods
Since the interest is compounded half-yearly, we need to convert the time from months to years:
\[ t = \frac{18}{12} = 1.5 \text{ years} \]
Since it is compounded half-yearly, we need to find the number of half-year periods:
\[ n = 1.5 \times 2 = 3 \text{ half-years} \]

### Step 3: Adjust the rate of interest for half-yearly compounding
Since the rate is given per annum, we need to divide it by 2 to find the rate for half a year:
\[ r = \frac{10}{2} = 5\% \]

### Step 4: Substitute the values into the formula
Now we can substitute the values into the compound interest formula:
\[ A = 1200 \left(1 + \frac{5}{100}\right)^3 \]

### Step 5: Simplify the expression
Calculate \( 1 + \frac{5}{100} \):
\[ 1 + \frac{5}{100} = 1 + 0.05 = 1.05 \]
Now raise it to the power of 3:
\[ A = 1200 \times (1.05)^3 \]

### Step 6: Calculate \( (1.05)^3 \)
Calculating \( (1.05)^3 \):
\[ (1.05)^3 = 1.157625 \]

### Step 7: Calculate the final amount
Now, multiply by the principal:
\[ A = 1200 \times 1.157625 = 1389.15 \]

### Final Answer
The amount after 18 months will be:
\[ A = ₹1389.15 \]