An amount of money appreciates to Rs. 7000 after 4 years and to Rs. 10000 after 8 years at a certain compound interest compounded annually. The initial amount for money was :

Answer

To solve the problem step by step, we will use the formula for compound interest and the information given in the question.

### Step 1: Understand the Problem
We are given that an amount of money appreciates to Rs. 7000 after 4 years and to Rs. 10000 after 8 years. We need to find the initial amount (P).

### Step 2: Write the Compound Interest Formula
The formula for the amount (A) after time (T) at a certain rate (R) compounded annually is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]

### Step 3: Set Up the Equations
From the information provided:
1. After 4 years, the amount is Rs. 7000:
\[ 7000 = P \left(1 + \frac{R}{100}\right)^4 \quad \text{(Equation 1)} \]
2. After 8 years, the amount is Rs. 10000:
\[ 10000 = P \left(1 + \frac{R}{100}\right)^8 \quad \text{(Equation 2)} \]

### Step 4: Divide Equation 2 by Equation 1
To eliminate P, we divide Equation 2 by Equation 1:
\[ \frac{10000}{7000} = \frac{P \left(1 + \frac{R}{100}\right)^8}{P \left(1 + \frac{R}{100}\right)^4} \]
This simplifies to:
\[ \frac{10}{7} = \left(1 + \frac{R}{100}\right)^{8-4} \]
\[ \frac{10}{7} = \left(1 + \frac{R}{100}\right)^4 \quad \text{(Equation 3)} \]

### Step 5: Solve for \(1 + \frac{R}{100}\)
Taking the fourth root of both sides:
\[ 1 + \frac{R}{100} = \left(\frac{10}{7}\right)^{\frac{1}{4}} \]

### Step 6: Substitute Back into Equation 1
Now we substitute this value back into Equation 1 to find P:
\[ 7000 = P \left(\left(\frac{10}{7}\right)^{\frac{1}{4}}\right)^4 \]
This simplifies to:
\[ 7000 = P \cdot \frac{10}{7} \]

### Step 7: Solve for P
Now, we can solve for P:
\[ P = 7000 \cdot \frac{7}{10} = 7000 \cdot 0.7 = 4900 \]

### Final Answer
The initial amount of money was Rs. 4900.

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