A sum of money amounts to Rs. 4500 in 2 years and Rs. 6750 in 4 years Compound Interest. Find the sum.

Answer

To solve the problem step-by-step, we will use the information given about the amounts after 2 years and 4 years to find the principal amount (sum of money).

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Amount after 2 years (A1) = Rs. 4500
- Amount after 4 years (A2) = Rs. 6750

2. **Understand the Relationship Between A1 and A2:**
- The amount after 4 years (A2) can be expressed in terms of the amount after 2 years (A1) using the formula for compound interest:
\[ A2 = A1 \times \left(1 + \frac{R}{100}\right)^T \]
- Here, T = 2 years (the time between A1 and A2).

3. **Set Up the Equation:**
- Substitute the known values into the equation:
\[ 6750 = 4500 \times \left(1 + \frac{R}{100}\right)^2 \]

4. **Simplify the Equation:**
- Divide both sides by 4500:
\[ \frac{6750}{4500} = \left(1 + \frac{R}{100}\right)^2 \]
- This simplifies to:
\[ 1.5 = \left(1 + \frac{R}{100}\right)^2 \]

5. **Take the Square Root:**
- To solve for \(1 + \frac{R}{100}\), take the square root of both sides:
\[ 1 + \frac{R}{100} = \sqrt{1.5} \]

6. **Calculate \(\sqrt{1.5}\):**
- \(\sqrt{1.5} \approx 1.2247\) (you can use a calculator for this).
- Therefore:
\[ 1 + \frac{R}{100} \approx 1.2247 \]

7. **Isolate R:**
- Subtract 1 from both sides:
\[ \frac{R}{100} \approx 0.2247 \]
- Multiply by 100 to find R:
\[ R \approx 22.47\% \]

8. **Find the Principal Amount (P):**
- Use the formula for A1:
\[ A1 = P \times \left(1 + \frac{R}{100}\right)^2 \]
- Substitute A1 and the value of \(1 + \frac{R}{100}\):
\[ 4500 = P \times (1.2247)^2 \]
- Calculate \((1.2247)^2 \approx 1.5\):
\[ 4500 = P \times 1.5 \]

9. **Solve for P:**
- Divide both sides by 1.5:
\[ P = \frac{4500}{1.5} = 3000 \]

### Final Answer:
The principal amount (sum of money) is Rs. 3000.

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