Divide ₹ 2,708 between A and B, so that A's share at the end of 6 years may equal B's share at the end of 8 years, compound interest being 8% p.a., then find the share of A.

Options

A. ₹ 1458
B. ₹1448
C. ₹ 1438
D. ₹1468

Answer

To solve the problem of dividing ₹ 2,708 between A and B such that A's share at the end of 6 years equals B's share at the end of 8 years with a compound interest rate of 8% per annum, we can follow these steps:

### Step 1: Set up the equation for A's and B's shares
Let A's share be ₹ x and B's share be ₹ (2708 - x). According to the problem, the amount A receives after 6 years should equal the amount B receives after 8 years.

The formula for the amount A receives after 6 years with compound interest is:
\[ A_{6} = x \left(1 + \frac{8}{100}\right)^{6} \]

The formula for the amount B receives after 8 years is:
\[ B_{8} = (2708 - x) \left(1 + \frac{8}{100}\right)^{8} \]

### Step 2: Set the two amounts equal
Since A's amount after 6 years equals B's amount after 8 years, we can set up the equation:
\[ x \left(1 + \frac{8}{100}\right)^{6} = (2708 - x) \left(1 + \frac{8}{100}\right)^{8} \]

### Step 3: Simplify the equation
We can simplify the equation by dividing both sides by \( \left(1 + \frac{8}{100}\right)^{6} \):
\[ x = (2708 - x) \left(1 + \frac{8}{100}\right)^{2} \]

### Step 4: Calculate \( \left(1 + \frac{8}{100}\right)^{2} \)
Calculating \( \left(1 + \frac{8}{100}\right)^{2} \):
\[ \left(1 + \frac{8}{100}\right) = 1.08 \]
\[ \left(1.08\right)^{2} = 1.1664 \]

### Step 5: Substitute back into the equation
Now substitute back into the equation:
\[ x = (2708 - x) \cdot 1.1664 \]

### Step 6: Expand and rearrange the equation
Expanding the right side gives:
\[ x = 2708 \cdot 1.1664 - x \cdot 1.1664 \]
\[ x + x \cdot 1.1664 = 2708 \cdot 1.1664 \]
\[ x(1 + 1.1664) = 2708 \cdot 1.1664 \]
\[ x \cdot 2.1664 = 3156.6592 \]

### Step 7: Solve for x
Now, solve for x:
\[ x = \frac{3156.6592}{2.1664} \]
\[ x \approx 1458 \]

### Step 8: Find B's share
To find B's share:
\[ B = 2708 - x \]
\[ B = 2708 - 1458 \]
\[ B \approx 1250 \]

### Conclusion
Therefore, A's share is approximately ₹ 1458.

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