If some amount becomes 4500 in 2 years and 6750 in 4 years at compound interest then find the Principle amount?

Options

A. ₹ 3200
B. ₹ 2250
C. ₹3000
D. ₹ 2000

Answer

To solve the problem, we need to find the principal amount (P) given that the amount becomes ₹4500 in 2 years and ₹6750 in 4 years at compound interest.

### Step-by-Step Solution:

1. **Identify the Amounts:**
- Amount after 2 years (A2) = ₹4500
- Amount after 4 years (A4) = ₹6750

2. **Calculate the Interest Earned from Year 2 to Year 4:**
- The interest earned from Year 2 to Year 4 can be calculated as:
\[ \text{Interest from Year 2 to Year 4} = A4 - A2 = 6750 - 4500 = 2250 \]

3. **Calculate the Rate of Interest:**
- The formula for the compound interest for the period from Year 2 to Year 4 is:
\[ \text{Interest} = A2 \times r^2 \]
- Where \( r \) is the rate of interest. We can rearrange this to find \( r \):
\[ 2250 = 4500 \times r^2 \]
- Dividing both sides by 4500:
\[ r^2 = \frac{2250}{4500} = \frac{1}{2} \]
- Taking the square root:
\[ r = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \approx 0.7071 \text{ or } 70.71\% \]

4. **Calculate the Principal Amount (P):**
- We know that:
\[ A2 = P(1 + r)^2 \]
- Substituting the known values:
\[ 4500 = P(1 + 0.7071)^2 \]
- Calculate \( (1 + 0.7071)^2 \):
\[ (1 + 0.7071)^2 = (1.7071)^2 \approx 2.9129 \]
- Now, substituting back:
\[ 4500 = P \times 2.9129 \]
- Solving for \( P \):
\[ P = \frac{4500}{2.9129} \approx 1545.45 \]

5. **Final Answer:**
- The principal amount is approximately ₹1545.45.