Rs. 39030 is divide between 'a' and 'b' in such a way that amount given to 'a' on C.I. in 7 years is equal to amount given to 'b' on C.I. in 9 years. Find the part of 'a'. If the rate of interest is 4%.

Answer

To solve the problem step by step, we need to find the amount given to 'A' when Rs. 39,030 is divided between 'A' and 'B' such that the amount received by 'A' after 7 years on compound interest is equal to the amount received by 'B' after 9 years on compound interest, with an interest rate of 4%.

### Step 1: Define the variables
Let the amount given to 'A' be \( P_A \) and the amount given to 'B' be \( P_B \). According to the problem, we have:
\[ P_A + P_B = 39030 \]

### Step 2: Write the formula for compound interest
The formula for the amount \( A \) after \( n \) years on compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( P \) is the principal amount,
- \( r \) is the rate of interest,
- \( n \) is the number of years.

### Step 3: Set up the equations for 'A' and 'B'
For 'A', the amount after 7 years is:
\[ A_A = P_A \left(1 + \frac{4}{100}\right)^7 \]
For 'B', the amount after 9 years is:
\[ A_B = P_B \left(1 + \frac{4}{100}\right)^9 \]

### Step 4: Equate the amounts
Since the amounts are equal, we can write:
\[ P_A \left(1 + \frac{4}{100}\right)^7 = P_B \left(1 + \frac{4}{100}\right)^9 \]

### Step 5: Simplify the equation
Let \( r = 1 + \frac{4}{100} = 1.04 \). The equation becomes:
\[ P_A (1.04)^7 = P_B (1.04)^9 \]
Dividing both sides by \( (1.04)^7 \):
\[ P_A = P_B \cdot \frac{(1.04)^9}{(1.04)^7} \]
This simplifies to:
\[ P_A = P_B \cdot (1.04)^2 \]

### Step 6: Substitute \( P_B \) in terms of \( P_A \)
From the equation \( P_A + P_B = 39030 \), we can express \( P_B \) as:
\[ P_B = 39030 - P_A \]
Substituting this into the equation for \( P_A \):
\[ P_A = (39030 - P_A) \cdot (1.04)^2 \]

### Step 7: Solve for \( P_A \)
Expanding and rearranging gives:
\[ P_A = (39030 - P_A) \cdot 1.0816 \]
\[ P_A = 39030 \cdot 1.0816 - P_A \cdot 1.0816 \]
\[ P_A + P_A \cdot 1.0816 = 39030 \cdot 1.0816 \]
\[ P_A (1 + 1.0816) = 39030 \cdot 1.0816 \]
\[ P_A \cdot 2.0816 = 42161.088 \]
\[ P_A = \frac{42161.088}{2.0816} \]
\[ P_A \approx 20200 \]

### Step 8: Find the part of 'A'
Thus, the part of 'A' is approximately Rs. 20,200.