What sum of money will amount to Rs 27,783 in one and a half years at 10% per annum compounded half yearly ?

Answer

To find the sum of money (Principal, P) that will amount to Rs 27,783 in one and a half years at an interest rate of 10% per annum compounded half yearly, we can follow these steps:

### Step 1: Understand the Compounding Terms
Since the interest is compounded half yearly, we need to adjust the rate and time accordingly:
- Rate (r) = 10% per annum, so for half yearly, it will be:
\[ r = \frac{10}{2} = 5\% \text{ per half year} \]
- Time (t) = 1.5 years, which means:
\[ t = 1.5 \times 2 = 3 \text{ half years} \]

### Step 2: Use the Compound Interest Formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- A = Amount after time t
- P = Principal amount (the sum we want to find)
- r = Rate of interest per period
- t = Number of periods

### Step 3: Substitute the Known Values
We know:
- A = Rs 27,783
- r = 5%
- t = 3

Substituting these values into the formula:
\[ 27,783 = P \left(1 + \frac{5}{100}\right)^3 \]
This simplifies to:
\[ 27,783 = P \left(1 + 0.05\right)^3 \]
\[ 27,783 = P \left(1.05\right)^3 \]

### Step 4: Calculate \( (1.05)^3 \)
Now we calculate \( (1.05)^3 \):
\[ (1.05)^3 = 1.157625 \]
So, we can rewrite the equation:
\[ 27,783 = P \times 1.157625 \]

### Step 5: Solve for P
To find P, we rearrange the equation:
\[ P = \frac{27,783}{1.157625} \]
Calculating this gives:
\[ P \approx 24,000 \]

### Final Answer
The sum of money that will amount to Rs 27,783 in one and a half years at 10% per annum compounded half yearly is approximately Rs 24,000.

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