A sum of money was lent at 10% per annum, compounded annually, for 2 years. If the interest was compounded half-yearly, he would have received Rs. 220.25 more. Find the sum.

Options

A. Rs. 40000
B. Rs. 45000
C. Rs. 48000
D. Rs. 50000

Answer

To solve the problem step by step, we need to find the principal amount (sum of money) that was lent at 10% per annum, compounded annually, and compare it to the amount received when compounded half-yearly.

### Step 1: Understand the Interest Rates and Time Periods
- The annual interest rate is 10%.
- The time period for both cases is 2 years when compounded annually.
- When compounded half-yearly, the effective interest rate becomes half of the annual rate, which is 5% (10% / 2).

### Step 2: Calculate the Amount for Annual Compounding
Using the formula for compound interest:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) is the amount after time \( n \),
- \( P \) is the principal,
- \( r \) is the rate of interest,
- \( n \) is the number of years.

For annual compounding:
- \( r = 10\% \)
- \( n = 2 \)

Thus, the amount after 2 years is:
\[ A = P \left(1 + \frac{10}{100}\right)^2 = P \left(1.1\right)^2 = P \times 1.21 \]

### Step 3: Calculate the Amount for Half-Yearly Compounding
For half-yearly compounding:
- The rate per half-year is 5% (10% / 2).
- The number of compounding periods in 2 years is 4 (2 years × 2).

Thus, the amount after 2 years is:
\[ A = P \left(1 + \frac{5}{100}\right)^4 = P \left(1.05\right)^4 \]

Calculating \( (1.05)^4 \):
\[ (1.05)^4 = 1.21550625 \]

So, the amount after 2 years with half-yearly compounding is:
\[ A = P \times 1.21550625 \]

### Step 4: Set Up the Equation for the Difference
According to the problem, the difference between the amounts received from half-yearly and annual compounding is Rs. 220.25:
\[ P \times 1.21550625 - P \times 1.21 = 220.25 \]

Factoring out \( P \):
\[ P \left(1.21550625 - 1.21\right) = 220.25 \]

Calculating the difference:
\[ 1.21550625 - 1.21 = 0.00550625 \]

### Step 5: Solve for Principal \( P \)
Now we can solve for \( P \):
\[ P \times 0.00550625 = 220.25 \]
\[ P = \frac{220.25}{0.00550625} \]

Calculating \( P \):
\[ P \approx 40000 \]

### Conclusion
The sum of money lent is Rs. 40,000.

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