Some amount is invested for 2 years at the rate of 20% Compound interest. Had the interested been compounded half yearly, one would have gotten 723 more. Find the amount?

Options

A. ₹28000
B. ₹ 32000
C. ₹30000
D. ₹34000

Answer

To solve the problem step by step, we will first calculate the total amount of interest earned when compounded annually and then when compounded half-yearly. Finally, we will set up an equation based on the difference in interest amounts to find the principal amount.

### Step 1: Calculate the total interest when compounded annually.
- The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
where \(A\) is the amount after time \(t\), \(P\) is the principal amount, \(r\) is the rate of interest, and \(t\) is the time in years.

- Given:
- \(r = 20\%\)
- \(t = 2\) years

- Therefore, the amount after 2 years when compounded annually is:
\[ A = P \left(1 + \frac{20}{100}\right)^2 = P \left(1.2\right)^2 = P \times 1.44 \]

### Step 2: Calculate the total interest when compounded half-yearly.
- When compounded half-yearly, the rate is halved and the time is doubled:
- New rate \(r = \frac{20}{2} = 10\%\)
- New time \(t = 2 \times 2 = 4\) half-years

- The amount after 4 half-years is:
\[ A = P \left(1 + \frac{10}{100}\right)^4 = P \left(1.1\right)^4 \]

- Calculate \( (1.1)^4 \):
\[ (1.1)^4 = 1.4641 \]
- Thus, the amount after 4 half-years is:
\[ A = P \times 1.4641 \]

### Step 3: Set up the equation based on the difference in interest.
- The difference in the amounts when compounded half-yearly and annually is given as 723:
\[ P \times 1.4641 - P \times 1.44 = 723 \]

- Simplifying the equation:
\[ P (1.4641 - 1.44) = 723 \]
\[ P \times 0.0241 = 723 \]

### Step 4: Solve for \(P\).
- Divide both sides by 0.0241:
\[ P = \frac{723}{0.0241} \approx 30000 \]

### Final Answer:
The principal amount \(P\) is approximately **30,000**.

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