What sum of money would amount to Rs 9261 in `1 (1)/(2)` yr at 10% per annum, interest being compounded half-yearly?

Options

A. Rs 8500
B. Rs 8000
C. Rs 8200
D. Rs 8400

Answer

To find the sum of money (principal) that would amount to Rs 9261 in \(1 \frac{1}{2}\) years at 10% per annum, compounded half-yearly, we can use the formula for compound interest:

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

Where:
- \(A\) = the amount of money accumulated after n years, including interest.
- \(P\) = the principal amount (the initial sum of money).
- \(r\) = the annual interest rate (decimal).
- \(n\) = the number of times that interest is compounded per year.
- \(t\) = the time the money is invested for in years.

### Step 1: Identify the values
- \(A = 9261\)
- \(r = 10\% = 0.10\)
- \(n = 2\) (since the interest is compounded half-yearly)
- \(t = 1 \frac{1}{2} = 1.5\) years

### Step 2: Substitute the values into the formula
We need to rearrange the formula to solve for \(P\):

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

Substituting the known values:

\[ P = \frac{9261}{\left(1 + \frac{0.10}{2}\right)^{2 \times 1.5}} \]

### Step 3: Calculate the interest rate per period
Calculate \(\frac{r}{n}\):

\[ \frac{0.10}{2} = 0.05 \]

### Step 4: Calculate the exponent
Calculate \(nt\):

\[ nt = 2 \times 1.5 = 3 \]

### Step 5: Calculate the expression inside the parentheses
Now substitute back into the equation:

\[ P = \frac{9261}{\left(1 + 0.05\right)^{3}} = \frac{9261}{\left(1.05\right)^{3}} \]

### Step 6: Calculate \((1.05)^{3}\)
Calculate \(1.05^3\):

\[ 1.05^3 = 1.157625 \]

### Step 7: Calculate \(P\)
Now substitute this value back into the equation for \(P\):

\[ P = \frac{9261}{1.157625} \approx 8000 \]

### Final Answer
The sum of money that would amount to Rs 9261 in \(1 \frac{1}{2}\) years at 10% per annum, compounded half-yearly, is approximately **Rs 8000**.

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