Divide 28,730 between A and B so that when their shares are lent out at 10 per cent compound interest compounded per year, the amount that A receives in 3 years is the same as what B receives in 5 years.

Answer

To solve the problem of dividing ₹28,730 between A and B such that the amount A receives in 3 years at 10% compound interest is equal to the amount B receives in 5 years, we can follow these steps:

### Step 1: Define Variables
Let the amount A receives be \( X \). Therefore, the amount B receives will be:
\[ 28730 - X \]

### Step 2: Write the Compound Interest Formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \( A \) is the total amount after time \( T \)
- \( P \) is the principal amount (initial amount)
- \( R \) is the rate of interest
- \( T \) is the time in years

### Step 3: Calculate Amount for A
For A, the principal is \( X \), the rate \( R \) is 10%, and the time \( T \) is 3 years. Thus, the amount A receives after 3 years is:
\[ A_A = X \left(1 + \frac{10}{100}\right)^3 = X \left(\frac{11}{10}\right)^3 \]

### Step 4: Calculate Amount for B
For B, the principal is \( 28730 - X \), the rate \( R \) is 10%, and the time \( T \) is 5 years. Thus, the amount B receives after 5 years is:
\[ A_B = (28730 - X) \left(1 + \frac{10}{100}\right)^5 = (28730 - X) \left(\frac{11}{10}\right)^5 \]

### Step 5: Set the Amounts Equal
According to the problem, the amounts received by A and B are equal:
\[ X \left(\frac{11}{10}\right)^3 = (28730 - X) \left(\frac{11}{10}\right)^5 \]

### Step 6: Simplify the Equation
We can simplify the equation by dividing both sides by \( \left(\frac{11}{10}\right)^3 \):
\[ X = (28730 - X) \left(\frac{11}{10}\right)^2 \]

### Step 7: Expand and Rearrange
Expanding the right side gives:
\[ X = (28730 - X) \cdot \frac{121}{100} \]
\[ X = \frac{28730 \cdot 121}{100} - \frac{121X}{100} \]
Now, multiply through by 100 to eliminate the fraction:
\[ 100X = 28730 \cdot 121 - 121X \]

### Step 8: Combine Like Terms
Rearranging gives:
\[ 100X + 121X = 28730 \cdot 121 \]
\[ 221X = 28730 \cdot 121 \]

### Step 9: Solve for X
Now, divide both sides by 221:
\[ X = \frac{28730 \cdot 121}{221} \]

### Step 10: Calculate X
Calculating the numerator:
\[ 28730 \cdot 121 = 3476530 \]
Now divide:
\[ X = \frac{3476530}{221} \approx 15730 \]

### Step 11: Calculate B's Amount
Now, find B's amount:
\[ B = 28730 - X = 28730 - 15730 = 13000 \]

### Final Answer
Thus, the amounts are:
- Amount A receives: ₹15,730
- Amount B receives: ₹13,000