If some amount becomes 3200 in 2 years and 8000 in 8 years at compound interest then find the Principle amount?

Options

A. ₹ 1180
B. ₹ 1380
C. ₹ 1480
D. ₹ 1280

Answer

To find the principal amount given that it becomes 3200 in 2 years and 8000 in 8 years at compound interest, we can use the formula for compound interest:

\[ A = P \left(1 + \frac{R}{100}\right)^T \]

Where:
- \( A \) = Amount after time \( T \)
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step-by-Step Solution:

1. **Set up the equations**:
- For the first scenario (after 2 years):
\[ A_1 = P \left(1 + \frac{R}{100}\right)^2 = 3200 \quad \text{(Equation 1)} \]
- For the second scenario (after 8 years):
\[ A_2 = P \left(1 + \frac{R}{100}\right)^8 = 8000 \quad \text{(Equation 2)} \]

2. **Divide the equations**:
- Dividing Equation 1 by Equation 2:
\[ \frac{3200}{8000} = \frac{P \left(1 + \frac{R}{100}\right)^2}{P \left(1 + \frac{R}{100}\right)^8} \]
- This simplifies to:
\[ \frac{2}{5} = \frac{\left(1 + \frac{R}{100}\right)^2}{\left(1 + \frac{R}{100}\right)^8} \]

3. **Simplify the equation**:
- This can be rewritten as:
\[ \frac{2}{5} = \left(1 + \frac{R}{100}\right)^{2 - 8} = \left(1 + \frac{R}{100}\right)^{-6} \]
- Taking the reciprocal gives:
\[ \frac{5}{2} = \left(1 + \frac{R}{100}\right)^6 \]

4. **Solve for \( 1 + \frac{R}{100} \)**:
- Taking the sixth root:
\[ 1 + \frac{R}{100} = \left(\frac{5}{2}\right)^{\frac{1}{6}} \]

5. **Calculate \( \frac{R}{100} \)**:
- Subtract 1 from both sides:
\[ \frac{R}{100} = \left(\frac{5}{2}\right)^{\frac{1}{6}} - 1 \]

6. **Substitute back to find \( P \)**:
- Substitute \( R \) back into either Equation 1 or Equation 2 to find \( P \). Using Equation 1:
\[ 3200 = P \left(1 + \frac{R}{100}\right)^2 \]
- Rearranging gives:
\[ P = \frac{3200}{\left(1 + \frac{R}{100}\right)^2} \]

7. **Final Calculation**:
- Substitute \( \left(1 + \frac{R}{100}\right) \) from step 4 into this equation to find \( P \).