Divide Rs. 1301 between A and B, so that the amount of A after 7 years is equal to the amount of B after 9 years, the interest being compounded at 4% per annum. Find the part A.

Answer

To solve the problem of dividing Rs. 1301 between A and B such that the amount of A after 7 years is equal to the amount of B after 9 years, we will follow these steps:

### Step 1: Define Variables
Let the amount that A receives be Rs. x. Then, the amount that B receives will be Rs. (1301 - x).

### Step 2: Write the Formula for Compound Interest
The formula for the amount \( A \) after \( n \) years with principal \( P \) and rate \( r \) compounded annually is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]

### Step 3: Calculate the Amount for A
For A, who receives Rs. x after 7 years at an interest rate of 4%:
\[ A_A = x \left(1 + \frac{4}{100}\right)^7 = x \left(1.04\right)^7 \]

### Step 4: Calculate the Amount for B
For B, who receives Rs. (1301 - x) after 9 years at the same interest rate:
\[ A_B = (1301 - x) \left(1 + \frac{4}{100}\right)^9 = (1301 - x) \left(1.04\right)^9 \]

### Step 5: Set the Amounts Equal
According to the problem, the amounts after the respective years are equal:
\[ x \left(1.04\right)^7 = (1301 - x) \left(1.04\right)^9 \]

### Step 6: Simplify the Equation
We can simplify this equation by dividing both sides by \( (1.04)^7 \):
\[ x = (1301 - x) \left(1.04\right)^2 \]

### Step 7: Expand and Rearrange
Expanding the right side:
\[ x = (1301 - x) \cdot 1.0816 \]
\[ x = 1301 \cdot 1.0816 - x \cdot 1.0816 \]
\[ x + x \cdot 1.0816 = 1301 \cdot 1.0816 \]
\[ x(1 + 1.0816) = 1301 \cdot 1.0816 \]
\[ x \cdot 2.0816 = 1400.0816 \]

### Step 8: Solve for x
Now, divide both sides by 2.0816:
\[ x = \frac{1400.0816}{2.0816} \approx 672.00 \]

### Step 9: Find the Amount for B
Now, we can find the amount for B:
\[ B = 1301 - x = 1301 - 672 = 629 \]

### Conclusion
Thus, the part of A is approximately Rs. 672.

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