A principal sum of money is lent out at compound interest compounded annually at the rate of 20% per annum for 2 years. It would give ₹ 2410 more if the interest is compounded half yearly. Find the principal sum.

Options

A. ₹ `120000`
B. ₹ `125000`
C. ₹ `100000`
D. ₹ `110000`

Answer

To solve the problem step by step, we will calculate the principal sum of money (P) based on the given conditions regarding compound interest.

### Step 1: Understanding the Problem
We need to find the principal sum (P) that, when compounded annually at a rate of 20% for 2 years, results in an amount that is ₹2410 less than the amount when compounded half-yearly at the same rate for the same time period.

### Step 2: Calculate Amount when Compounded Annually
The formula for the amount (A) when compounded annually is:
\[ A = P \left(1 + \frac{R}{100}\right)^N \]
Where:
- \( R = 20\% \)
- \( N = 2 \) years

Substituting the values:
\[ A_1 = P \left(1 + \frac{20}{100}\right)^2 \]
\[ A_1 = P \left(1 + 0.2\right)^2 \]
\[ A_1 = P \left(1.2\right)^2 \]
\[ A_1 = P \times 1.44 \]

### Step 3: Calculate Amount when Compounded Half-Yearly
When compounded half-yearly, the rate is halved and the number of compounding periods is doubled. Thus:
- Rate for half-year = \( \frac{20}{2} = 10\% \)
- Number of compounding periods = \( 2 \times 2 = 4 \)

Using the formula for compound interest:
\[ A_2 = P \left(1 + \frac{R/2}{100}\right)^{2N} \]
Substituting the values:
\[ A_2 = P \left(1 + \frac{10}{100}\right)^4 \]
\[ A_2 = P \left(1 + 0.1\right)^4 \]
\[ A_2 = P \left(1.1\right)^4 \]
Calculating \( (1.1)^4 \):
\[ (1.1)^4 = 1.4641 \]
Thus:
\[ A_2 = P \times 1.4641 \]

### Step 4: Set Up the Equation
According to the problem, the difference between the two amounts is ₹2410:
\[ A_2 = A_1 + 2410 \]
Substituting the expressions for \( A_1 \) and \( A_2 \):
\[ P \times 1.4641 = P \times 1.44 + 2410 \]

### Step 5: Simplify the Equation
Rearranging the equation:
\[ P \times 1.4641 - P \times 1.44 = 2410 \]
Factoring out \( P \):
\[ P (1.4641 - 1.44) = 2410 \]
Calculating \( 1.4641 - 1.44 = 0.0241 \):
\[ P \times 0.0241 = 2410 \]

### Step 6: Solve for P
Now, divide both sides by 0.0241:
\[ P = \frac{2410}{0.0241} \]
Calculating the right side:
\[ P = 100,000 \]

### Final Answer
The principal sum is ₹100,000.

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