A doctor wants to divide Rs. 145000 between his son and daughter who are 12 years and 14 years respectively, in such a way that the sum invested at the rate of `12(1)/(2)%` per annum compounded annually will give the same amount to each, when they attain 16 years. How should he divide the sum?

Options

A. Rs. 81000 to son and Rs. 64000 to daughter
B. Rs. 64000 to son and Rs. 81000 to daughter
C. Rs. 45000 to son and Rs. 1000000 to daughter
D. Rs. 1000000 to son and Rs. 45000 to daughter

Explanation

To solve the problem of how to divide Rs. 145,000 between the doctor’s son and daughter, we need to ensure that both receive the same amount when they turn 16 years old, given their current ages and the interest rate. Here’s a step-by-step solution:

### Step 1: Determine the time periods for each child
- The son is currently 12 years old and will be 16 in 4 years.
- The daughter is currently 14 years old and will be 16 in 2 years.

### Step 2: Define the variables
- Let the amount invested for the son be \( x \) rupees.
- Therefore, the amount invested for the daughter will be \( 145000 - x \) rupees.

### Step 3: Calculate the amount for the son after 4 years
Using the formula for compound interest:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where:
- \( P \) is the principal amount (initial investment),
- \( r \) is the rate of interest,
- \( n \) is the number of years.

For the son:
\[ A_{son} = x \left(1 + \frac{12.5}{100}\right)^4 \]
Convert \( 12.5\% \) to a fraction:
\[ 12.5\% = \frac{25}{2}\% = \frac{25}{200} = \frac{1}{8} \]
Thus:
\[ A_{son} = x \left(1 + \frac{1}{8}\right)^4 = x \left(\frac{9}{8}\right)^4 \]

### Step 4: Calculate the amount for the daughter after 2 years
For the daughter:
\[ A_{daughter} = (145000 - x) \left(1 + \frac{12.5}{100}\right)^2 \]
Using the same conversion:
\[ A_{daughter} = (145000 - x) \left(\frac{9}{8}\right)^2 \]

### Step 5: Set the amounts equal
Since both amounts must be equal when they turn 16:
\[ x \left(\frac{9}{8}\right)^4 = (145000 - x) \left(\frac{9}{8}\right)^2 \]

### Step 6: Simplify the equation
Dividing both sides by \( \left(\frac{9}{8}\right)^2 \):
\[ x \left(\frac{9}{8}\right)^2 = 145000 - x \]
This simplifies to:
\[ \frac{81}{64} x = 145000 - x \]

### Step 7: Solve for \( x \)
Rearranging gives:
\[ \frac{81}{64} x + x = 145000 \]
Combining terms:
\[ \left(\frac{81}{64} + \frac{64}{64}\right)x = 145000 \]
\[ \frac{145}{64} x = 145000 \]
Multiplying both sides by \( \frac{64}{145} \):
\[ x = 145000 \times \frac{64}{145} \]
Calculating \( x \):
\[ x = 64000 \]

### Step 8: Find the amount for the daughter
Now, substituting \( x \) back:
\[ Amount_{daughter} = 145000 - 64000 = 81000 \]

### Final Answer
- The share of the son is Rs. 64,000.
- The share of the daughter is Rs. 81,000.