A sum of money is invested at `20%` compound interest (compounded annually). It would fetch `Rs. 723` more in 2 years if interest is compounded half yearly. The sum is

Options

A. `Rs. 15,000`
B. ` Rs.30,000`
C. `Rs.20,000`
D. ` Rs.7.500`

Answer

To solve the problem step by step, we will break down the calculations for both annual and half-yearly compounding and determine the principal amount.

### Step 1: Understanding the Problem
We need to find the principal amount (P) that, when invested at a compound interest rate of 20% per annum, will yield Rs. 723 more when compounded half-yearly compared to when it is compounded annually over 2 years.

### Step 2: Calculate the Amount for Annual Compounding
For annual compounding at 20% for 2 years, the formula for the amount (A) is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- \( P \) = Principal amount
- \( r \) = Rate of interest (20%)
- \( t \) = Time (2 years)

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

### Step 3: Calculate the Amount for Half-Yearly Compounding
For half-yearly compounding, the rate is halved and the time is doubled. Thus:
- Rate = 10% (half of 20%)
- Time = 4 years (double of 2 years)

Using the same formula:
\[ A_{half-yearly} = P \left(1 + \frac{10}{100}\right)^4 = P \left(1.1\right)^4 \]
Calculating \( (1.1)^4 \):
\[ (1.1)^4 = 1.4641 \]
So,
\[ A_{half-yearly} = P \times 1.4641 \]

### Step 4: Set Up the Equation
According to the problem, the difference between the amounts from half-yearly and annual compounding is Rs. 723:
\[ A_{half-yearly} - A_{annual} = 723 \]
Substituting the amounts we calculated:
\[ P \times 1.4641 - P \times 1.44 = 723 \]

### Step 5: Simplify the Equation
Factoring out \( P \):
\[ P (1.4641 - 1.44) = 723 \]
Calculating the difference:
\[ 1.4641 - 1.44 = 0.0241 \]
Thus, we have:
\[ P \times 0.0241 = 723 \]

### Step 6: Solve for P
Now, we can solve for \( P \):
\[ P = \frac{723}{0.0241} \]
Calculating this gives:
\[ P \approx 30000 \]

### Final Answer
The sum of money (principal amount) is Rs. 30,000.

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