Divide Rs. 15494 between A and B so that A's share at the end of 9 years may be equal to B's share at the end of 11 years, compound interest being 20% per annum. Then A's share is :

Options

A. (a) rs 9000
B. (b) rs 9244
C. (c) rs 9144
D. (d) rs 10000

Answer

To solve the problem of dividing Rs. 15494 between A and B such that A's share at the end of 9 years is equal to B's share at the end of 11 years with a compound interest rate of 20% per annum, we can follow these steps:

### Step 1: Define Variables
Let A's share be \( x \). Then, B's share will be \( 15494 - x \).

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

### Step 3: Set Up the Equation
According to the problem, A's amount after 9 years will be equal to B's amount after 11 years:
\[ x \left(1 + \frac{20}{100}\right)^9 = (15494 - x) \left(1 + \frac{20}{100}\right)^{11} \]

### Step 4: Simplify the Equation
Substituting \( 1 + \frac{20}{100} = 1.2 \):
\[ x \cdot (1.2)^9 = (15494 - x) \cdot (1.2)^{11} \]

### Step 5: Divide Both Sides by \( (1.2)^9 \)
This simplifies to:
\[ x = (15494 - x) \cdot (1.2^2) \]

### Step 6: Calculate \( (1.2)^2 \)
Calculating \( (1.2)^2 \):
\[ (1.2)^2 = 1.44 \]
Thus, the equation becomes:
\[ x = (15494 - x) \cdot 1.44 \]

### Step 7: Expand and Rearrange the Equation
Expanding the right side:
\[ x = 15494 \cdot 1.44 - 1.44x \]
Rearranging gives:
\[ x + 1.44x = 15494 \cdot 1.44 \]
\[ 2.44x = 15494 \cdot 1.44 \]

### Step 8: Calculate \( 15494 \cdot 1.44 \)
Calculating \( 15494 \cdot 1.44 \):
\[ 15494 \cdot 1.44 = 22212.16 \]

### Step 9: Solve for \( x \)
Now, we solve for \( x \):
\[ x = \frac{22212.16}{2.44} \approx 9100 \]

### Step 10: Final Answer
Thus, A's share is approximately:
\[ \boxed{9100} \]