Understanding the Compound Interest Problem

This question asks us to divide a total sum of Rs. 66,300 between two individuals, A and B. The condition for the division is based on their future amounts after receiving compound interest for different time periods. Specifically, the amount A receives after 8 years with compound interest must be equal to the amount B receives after 10 years at the same interest rate.

The interest rate given is 10% per annum, compounded annually.

Applying the Compound Interest Formula

The formula for the amount (\(A\)) after \(n\) years with compound interest is:

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

Where:

• \(P\) is the principal amount (the initial amount).
• \(r\) is the annual interest rate.
• \(n\) is the number of years.

Setting up the Equations for Amount Division

Let the initial amount received by A be \(P_A\) and the initial amount received by B be \(P_B\).

The total amount is Rs. 66,300, so:

$$P_A + P_B = 66300 \quad (*)$$

The interest rate is \(r = 10\%\).

For A, the time period is \(n_A = 8\) years. The amount A receives after 8 years is:

$$A_A = P_A \left(1 + \frac{10}{100}\right)^8 = P_A (1 + 0.1)^8 = P_A (1.1)^8$$

For B, the time period is \(n_B = 10\) years. The amount B receives after 10 years is:

$$A_B = P_B \left(1 + \frac{10}{100}\right)^{10} = P_B (1 + 0.1)^{10} = P_B (1.1)^{10}$$

The problem states that the amount A receives after 8 years is equal to the amount B receives after 10 years. So:

$$A_A = A_B$$

$$P_A (1.1)^8 = P_B (1.1)^{10}$$

Solving for the Initial Amounts (P_A and P_B)

We have the equation \(P_A (1.1)^8 = P_B (1.1)^{10}\). We can rearrange this to find a relationship between \(P_A\) and \(P_B\):

$$\frac{P_A}{P_B} = \frac{(1.1)^{10}}{(1.1)^8}$$

Using the rules of exponents (\(\frac{a^m}{a^n} = a^{m-n}\)):

$$\frac{P_A}{P_B} = (1.1)^{10-8} = (1.1)^2$$

Calculate \((1.1)^2\):

$$(1.1)^2 = 1.1 \times 1.1 = 1.21$$

So, the relationship is:

$$\frac{P_A}{P_B} = 1.21$$

This gives us \(P_A = 1.21 P_B\).

Now substitute this expression for \(P_A\) into the total amount equation \((*)\):

$$P_A + P_B = 66300$$

$$1.21 P_B + P_B = 66300$$

Combine the \(P_B\) terms:

$$(1.21 + 1) P_B = 66300$$

$$2.21 P_B = 66300$$

Now solve for \(P_B\):

$$P_B = \frac{66300}{2.21}$$

To simplify the division, multiply the numerator and denominator by 100 to remove the decimal:

$$P_B = \frac{66300 \times 100}{2.21 \times 100} = \frac{6630000}{221}$$

Let's perform the division. We notice that \(221 \times 3 = 663\). So:

$$P_B = \frac{663 \times 10000}{221} = 3 \times 10000 = 30000$$

So, the amount B receives initially is Rs. 30,000.

Now, find \(P_A\) using the relationship \(P_A = 1.21 P_B\):

$$P_A = 1.21 \times 30000$$

$$P_A = 1.21 \times 3 \times 10000 = 3.63 \times 10000 = 36300$$

So, the amount A receives initially is Rs. 36,300.

Verifying the Solution

The amounts are \(P_A = 36300\) and \(P_B = 30000\). Their sum is \(36300 + 30000 = 66300\), which is the total amount to be divided. This confirms our calculation of the principal amounts is consistent with the total sum.

Let's check the future amounts:

Amount A receives after 8 years: \(A_A = 36300 (1.1)^8\)

Amount B receives after 10 years: \(A_B = 30000 (1.1)^{10}\)

From our calculation \(P_A = 1.21 P_B\), we have \(36300 = 1.21 \times 30000\). Substituting this into the expression for \(A_A\):

$$A_A = (1.21 \times 30000) \times (1.1)^8$$

Since \(1.21 = (1.1)^2\), we have:

$$A_A = (1.1)^2 \times 30000 \times (1.1)^8$$

Using the rule of exponents (\(a^m \times a^n = a^{m+n}\)):

$$A_A = 30000 \times (1.1)^{2+8} = 30000 \times (1.1)^{10}$$

This is exactly the expression for \(A_B = 30000 (1.1)^{10}\).

So, \(A_A = A_B\), confirming that our division satisfies the condition.

The calculated amounts are A = Rs. 36,300 and B = Rs. 30,000.

Comparing with Options

Let's look at the given options:

• Option 1: A = Rs. 36,300, B = Rs. 30,000
• Option 2: A = Rs. 37,000, B = Rs. 29,300
• Option 3: A = Rs. 35,520, B = Rs. 30,810
• Option 4: A = Rs. 35,200, B = Rs. 31,100

Our calculated values, A = Rs. 36,300 and B = Rs. 30,000, match Option 1.

Revision Table: Key Concepts

Additional Information: Ratio Method for Amount Division

Problems like this, where a sum is divided such that the future values are equal, can often be thought of in terms of present values or a ratio. The amount A gets is the present value of a future amount \(X\) received after 8 years, and the amount B gets is the present value of the same future amount \(X\) received after 10 years. The higher the time period, the lower the present value (initial amount) needed to reach the same future amount.

The initial amounts \(P_A\) and \(P_B\) are in inverse proportion to the factors raised to the power of the years.

Specifically, if \(P_A (1+r/100)^{n_A} = P_B (1+r/100)^{n_B}\), then:

$$\frac{P_A}{P_B} = \frac{(1+r/100)^{n_B}}{(1+r/100)^{n_A}} = \left(1 + \frac{r}{100}\right)^{n_B - n_A}$$

In our case, \(r=10\), \(n_A=8\), \(n_B=10\). So:

$$\frac{P_A}{P_B} = \left(1 + \frac{10}{100}\right)^{10 - 8} = (1.1)^2 = 1.21$$

This means \(P_A : P_B = 1.21 : 1\). Or, to work with integers, \(P_A : P_B = 121 : 100\).

The total number of ratio parts is \(121 + 100 = 221\).

The total amount is Rs. 66,300.

Value of one ratio part = \(\frac{66300}{221}\) = Rs. 300.

Amount A receives \(P_A = 121 \times 300 = 36300\).

Amount B receives \(P_B = 100 \times 300 = 30000\).

This ratio method confirms the previous calculation and provides an alternative way to solve such compound interest division problems.

The final answer is A = Rs. 36,300 and B = Rs. 30,000.