Understanding the Compound Interest Division Problem

The question asks us to divide a total sum of Rs. 2,602 between two individuals, A and B. The condition for this division is based on their respective amounts after a certain period, earning a compound interest of 4% per annum. Specifically, the amount A receives after 6 years should be equal to the amount B receives after 8 years.

Compound Interest Formula

We use the formula for compound interest to calculate the future amount:

$$A = P \left(1 + \frac{r}{100}\right)^t$$

Where:

• A is the future amount
• P is the principal amount (the initial share)
• r is the annual interest rate
• t is the time period in years

Applying the Condition

Let the principal amount given to A be $P_A$ and the principal amount given to B be $P_B$. The total amount is Rs. 2,602, so:

$$P_A + P_B = 2602 \quad \quad (1)$$

The interest rate r is 4% per annum.

The amount A gets after 6 years is:

$$A_A = P_A \left(1 + \frac{4}{100}\right)^6 = P_A (1.04)^6$$

The amount B gets after 8 years is:

$$A_B = P_B \left(1 + \frac{4}{100}\right)^8 = P_B (1.04)^8$$

According to the problem statement, these amounts are equal:

$$A_A = A_B$$

$$P_A (1.04)^6 = P_B (1.04)^8 \quad \quad (2)$$

Calculating the Ratio of Shares ($P_A : P_B$)

To find the ratio of their shares, we rearrange equation (2):

$$\frac{P_A}{P_B} = \frac{(1.04)^8}{(1.04)^6}$$

Using the rule of exponents $\frac{a^m}{a^n} = a^{m-n}$:

$$\frac{P_A}{P_B} = (1.04)^{8-6} = (1.04)^2$$

Now, we calculate $(1.04)^2$:

$$1.04 \times 1.04 = 1.0816$$

So, the ratio of the shares is:

$$\frac{P_A}{P_B} = 1.0816$$

This means that for every Rs. 1 that B gets, A gets Rs. 1.0816.

Dividing the Total Amount

We can express the ratio as $P_A : P_B = 1.0816 : 1$.

To make calculations easier, we can convert the ratio to integers or simpler fractions if needed, but here we can work with the decimal.

The sum of the ratio parts is $1.0816 + 1 = 2.0816$.

Now, we can find the share of A:

$$P_A = \text{Total Amount} \times \frac{\text{A's ratio part}}{\text{Sum of ratio parts}}$$

$$P_A = 2602 \times \frac{1.0816}{2.0816}$$

$$P_A = 2602 \times \frac{1.0816}{2.0816} \approx 2602 \times 0.51956 \approx 1352$$

And the share of B:

$$P_B = \text{Total Amount} \times \frac{\text{B's ratio part}}{\text{Sum of ratio parts}}$$

$$P_B = 2602 \times \frac{1}{2.0816}$$

$$P_B = 2602 \times \frac{1}{2.0816} \approx 2602 \times 0.48044 \approx 1250$$

Alternatively, once $P_A$ is calculated, $P_B$ can be found using equation (1):

$$P_B = 2602 - P_A = 2602 - 1352 = 1250$$

Conclusion

The total amount of Rs. 2,602 is divided between A and B such that A receives Rs. 1,352 and B receives Rs. 1,250. This division ensures that the amount of A after 6 years equals the amount of B after 8 years at a 4% compound interest rate.