An amount was lent for two years at the rate of `20%` per annum compounded annually. Had the compounding been done half yearly, the interest would have Increased by `Rs. 241`. What was the amount (in Rs.) lent?

Options

A. 10000
B. 12000
C. 20000
D. 24000

Answer

To solve the problem step by step, we will first calculate the compound interest for both annual and half-yearly compounding methods and then find the principal amount lent.

### Step 1: Calculate Compound Interest (CI) for Annual Compounding
Given:
- Principal = P (unknown)
- Rate = 20% per annum
- Time = 2 years

The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- \( A \) = Amount after time \( t \)
- \( r \) = Rate of interest
- \( t \) = Time in years

For annual compounding:
\[ A = P \left(1 + \frac{20}{100}\right)^2 \]
\[ A = P \left(1 + 0.2\right)^2 \]
\[ A = P \left(1.2\right)^2 \]
\[ A = P \times 1.44 \]

The compound interest (CI) for annual compounding is:
\[ CI_{annual} = A - P = P \times 1.44 - P = P(1.44 - 1) = P \times 0.44 \]

### Step 2: Calculate Compound Interest (CI) for Half-Yearly Compounding
For half-yearly compounding:
- Rate per half-year = \( \frac{20}{2} = 10\% \)
- Time = 2 years = 4 half-years

Using the same formula:
\[ A = P \left(1 + \frac{10}{100}\right)^4 \]
\[ A = P \left(1 + 0.1\right)^4 \]
\[ A = P \left(1.1\right)^4 \]

Calculating \( (1.1)^4 \):
\[ (1.1)^4 = 1.4641 \]
So,
\[ A = P \times 1.4641 \]

The compound interest (CI) for half-yearly compounding is:
\[ CI_{half-yearly} = A - P = P \times 1.4641 - P = P(1.4641 - 1) = P \times 0.4641 \]

### Step 3: Set Up the Equation Based on Given Information
According to the problem, the difference in interest between half-yearly and annual compounding is Rs. 241:
\[ CI_{half-yearly} - CI_{annual} = 241 \]
Substituting the values we calculated:
\[ P \times 0.4641 - P \times 0.44 = 241 \]
\[ P(0.4641 - 0.44) = 241 \]
\[ P(0.0241) = 241 \]

### Step 4: Solve for Principal Amount (P)
Now, we can solve for \( P \):
\[ P = \frac{241}{0.0241} \]
Calculating this gives:
\[ P \approx 10000 \]

### Final Answer
The amount lent was Rs. 10,000.

---