A sum of money amounts to Rs. 3200 in 4 years and Rs. 8000 in 8 years on compound interest. Find the sum.

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Understand the Given Information
We know that:
- The amount after 4 years (A1) = Rs. 3200
- The amount after 8 years (A2) = Rs. 8000

### Step 2: Use the Compound Interest Formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \( A \) = Amount after time \( T \)
- \( P \) = Principal amount (initial sum)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 3: Set Up the Equations
From the information given:
1. For 4 years:
\[ A1 = P \left(1 + \frac{R}{100}\right)^4 = 3200 \]

2. For 8 years:
\[ A2 = P \left(1 + \frac{R}{100}\right)^8 = 8000 \]

### Step 4: Relate the Two Equations
We can express \( A2 \) in terms of \( A1 \):
\[ A2 = A1 \cdot \left(1 + \frac{R}{100}\right)^4 \]
Substituting the values:
\[ 8000 = 3200 \cdot \left(1 + \frac{R}{100}\right)^4 \]

### Step 5: Solve for \( \left(1 + \frac{R}{100}\right)^4 \)
Divide both sides by 3200:
\[ \left(1 + \frac{R}{100}\right)^4 = \frac{8000}{3200} \]
\[ \left(1 + \frac{R}{100}\right)^4 = 2.5 \]

### Step 6: Take the Fourth Root
To find \( 1 + \frac{R}{100} \):
\[ 1 + \frac{R}{100} = (2.5)^{\frac{1}{4}} \]

### Step 7: Calculate \( (2.5)^{\frac{1}{4}} \)
Calculating \( (2.5)^{\frac{1}{4}} \):
Using a calculator or approximation:
\[ (2.5)^{\frac{1}{4}} \approx 1.221 \]

### Step 8: Solve for \( R \)
Now we have:
\[ 1 + \frac{R}{100} \approx 1.221 \]
Subtract 1 from both sides:
\[ \frac{R}{100} \approx 0.221 \]
Multiply by 100:
\[ R \approx 22.1 \]

### Step 9: Substitute Back to Find Principal \( P \)
Now substitute \( R \) back into the equation for \( A1 \):
\[ 3200 = P \left(1 + \frac{22.1}{100}\right)^4 \]
\[ 3200 = P \cdot (1.221)^4 \]
Calculating \( (1.221)^4 \):
\[ (1.221)^4 \approx 2.5 \]
So:
\[ 3200 = P \cdot 2.5 \]
Now solve for \( P \):
\[ P = \frac{3200}{2.5} = 1280 \]

### Final Answer
The principal amount (sum) is Rs. 1280.

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