Divide Rs. 3903 between A and B, so that A’s share at the end of 7 years may equal to B’s share at the end of 9 years, compound interest being at 4 percent.

Options

A. Rs.2028, Rs.1875
B. Rs.2018, Rs.1885
C. Rs.2008, Rs.1895
D. Rs.2038, Rs.1865

Answer

To solve the problem of dividing Rs. 3903 between A and B such that A’s share at the end of 7 years equals B’s share at the end of 9 years with a compound interest rate of 4%, we can follow these steps:

### Step 1: Understand the Compound Interest Formula
The formula for calculating the amount (A) after a certain time (t) with compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
where:
- \( A \) = Amount after time \( t \)
- \( P \) = Principal amount (initial investment)
- \( r \) = Rate of interest per annum
- \( t \) = Time in years

### Step 2: Set Up the Equations
Let \( P_1 \) be A's share and \( P_2 \) be B's share. According to the problem:
- A's amount after 7 years:
\[ A_1 = P_1 \left(1 + \frac{4}{100}\right)^7 \]
- B's amount after 9 years:
\[ A_2 = P_2 \left(1 + \frac{4}{100}\right)^9 \]

Since \( A_1 = A_2 \), we can equate the two:
\[ P_1 \left(1 + \frac{4}{100}\right)^7 = P_2 \left(1 + \frac{4}{100}\right)^9 \]

### Step 3: Simplify the Equation
Dividing both sides by \( P_2 \) and rearranging gives:
\[ \frac{P_1}{P_2} = \frac{\left(1 + \frac{4}{100}\right)^9}{\left(1 + \frac{4}{100}\right)^7} \]
This simplifies to:
\[ \frac{P_1}{P_2} = \left(1 + \frac{4}{100}\right)^{9-7} = \left(1 + \frac{4}{100}\right)^2 \]

### Step 4: Calculate the Value of \( \left(1 + \frac{4}{100}\right)^2 \)
Calculating \( \left(1 + \frac{4}{100}\right)^2 \):
\[ \left(1 + 0.04\right)^2 = (1.04)^2 = 1.0816 \]

### Step 5: Set Up the Ratio
From the previous step, we have:
\[ \frac{P_1}{P_2} = 1.0816 \]
This can be expressed as:
\[ P_1 = 1.0816 P_2 \]

### Step 6: Use the Total Amount
We know that:
\[ P_1 + P_2 = 3903 \]
Substituting \( P_1 \) from the ratio:
\[ 1.0816 P_2 + P_2 = 3903 \]
\[ 2.0816 P_2 = 3903 \]

### Step 7: Solve for \( P_2 \)
Now, we can solve for \( P_2 \):
\[ P_2 = \frac{3903}{2.0816} \approx 1875 \]

### Step 8: Solve for \( P_1 \)
Now, substituting \( P_2 \) back to find \( P_1 \):
\[ P_1 = 3903 - P_2 = 3903 - 1875 = 2028 \]

### Final Result
Thus, the amounts are:
- A's share \( P_1 = 2028 \)
- B's share \( P_2 = 1875 \)