Divide Rs. 10230 into two parts such that the first part after 10 years is equal to the second part after 7 years, compound inter est being 20% per annum com pounded yearly.

Options

A. Rs. 4150 , Rs. 56080
B. Rs. 3950, Rs. 6280
C. Rs. 3750 , Rs. 6480
D. Rs. 3550 , Rs. 6680

Answer

To solve the problem of dividing Rs. 10,230 into two parts such that the first part after 10 years is equal to the second part after 7 years, with a compound interest rate of 20% per annum, we can follow these steps:

### Step 1: Define the Parts
Let the first part be \( P \) and the second part be \( Q \).

### Step 2: Write the Amounts After Interest
According to the problem, the amount of the first part after 10 years is equal to the amount of the second part after 7 years.

The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) is the amount after time \( n \)
- \( P \) is the principal amount
- \( r \) is the rate of interest
- \( n \) is the number of years

For the first part \( P \) after 10 years:
\[ A_P = P \left(1 + \frac{20}{100}\right)^{10} = P \left(1.2\right)^{10} \]

For the second part \( Q \) after 7 years:
\[ A_Q = Q \left(1 + \frac{20}{100}\right)^{7} = Q \left(1.2\right)^{7} \]

### Step 3: Set the Amounts Equal
Since \( A_P = A_Q \), we can write:
\[ P \left(1.2\right)^{10} = Q \left(1.2\right)^{7} \]

### Step 4: Simplify the Equation
Dividing both sides by \( (1.2)^7 \):
\[ P \left(1.2\right)^{3} = Q \]
This can be rewritten as:
\[ P = Q \left(\frac{1}{(1.2)^3}\right) \]

### Step 5: Calculate \( (1.2)^3 \)
Calculating \( (1.2)^3 \):
\[ (1.2)^3 = 1.728 \]
Thus:
\[ P = \frac{Q}{1.728} \]

### Step 6: Express \( P \) in Terms of \( Q \)
Rearranging gives:
\[ P = \frac{125}{216} Q \]

### Step 7: Use the Total Amount
We know that:
\[ P + Q = 10230 \]
Substituting \( P \):
\[ \frac{125}{216} Q + Q = 10230 \]

### Step 8: Combine Terms
Combining the terms:
\[ \left(\frac{125 + 216}{216}\right) Q = 10230 \]
\[ \frac{341}{216} Q = 10230 \]

### Step 9: Solve for \( Q \)
Multiplying both sides by \( \frac{216}{341} \):
\[ Q = 10230 \times \frac{216}{341} \]
Calculating \( Q \):
\[ Q = 6480 \]

### Step 10: Find \( P \)
Using \( Q \) to find \( P \):
\[ P = 10230 - Q = 10230 - 6480 = 3750 \]

### Final Answer
Thus, the two parts are:
- \( P = 3750 \)
- \( Q = 6480 \)