Divide Rs. 1105 between A and B, so that the A’s share at the end of 5 years may equal B’s share at the end of 7 years, compound interest being at 10%.

Answer

To solve the problem of dividing Rs. 1105 between A and B such that A's share at the end of 5 years equals B's share at the end of 7 years with a compound interest rate of 10%, we can follow these steps:

### Step 1: Understand the Problem
We need to find the shares of A and B such that the amount A receives after 5 years is equal to the amount B receives after 7 years.

### Step 2: Use the Compound Interest Formula
The formula for the amount (A) after time (t) with principal (P) and rate (R) is:
\[ A = P \left(1 + \frac{R}{100}\right)^t \]

For A's share after 5 years:
\[ A_A = P_A \left(1 + \frac{10}{100}\right)^5 \]

For B's share after 7 years:
\[ A_B = P_B \left(1 + \frac{10}{100}\right)^7 \]

### Step 3: Set Up the Equation
According to the problem, we need:
\[ P_A \left(1 + \frac{10}{100}\right)^5 = P_B \left(1 + \frac{10}{100}\right)^7 \]

### Step 4: Simplify the Equation
We can express the equation in terms of the ratio of A's and B's shares:
\[ \frac{P_A}{P_B} = \frac{\left(1 + \frac{10}{100}\right)^7}{\left(1 + \frac{10}{100}\right)^5} \]

This simplifies to:
\[ \frac{P_A}{P_B} = \left(1 + \frac{10}{100}\right)^{7-5} = \left(1 + \frac{10}{100}\right)^2 \]

### Step 5: Calculate the Ratio
Calculating the base:
\[ 1 + \frac{10}{100} = 1.1 \]
Thus,
\[ \left(1.1\right)^2 = 1.21 \]

So, we have:
\[ \frac{P_A}{P_B} = \frac{121}{100} \]

### Step 6: Express Total Amount in Terms of Ratio
Let A's share be \( 121x \) and B's share be \( 100x \). The total amount is:
\[ 121x + 100x = 1105 \]
\[ 221x = 1105 \]

### Step 7: Solve for x
To find x:
\[ x = \frac{1105}{221} = 5 \]

### Step 8: Calculate A's and B's Shares
Now we can find A's and B's shares:
- A's share:
\[ P_A = 121x = 121 \times 5 = 605 \]

- B's share:
\[ P_B = 100x = 100 \times 5 = 500 \]

### Final Shares
Thus, A's share is Rs. 605 and B's share is Rs. 500.

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