A sum of Rs. 1682 is to be divided between A and B who are respectively 20 years and 22 years old. They invest their shares at 5% per annum, compounded annually. At the age of 25 years both receive equal amounts. Find the share of each.

Options

A. Rs. 730, Rs. 952
B. Rs. 750 , Rs. 932
C. Rs. 700 , Rs. 982
D. Rs. 800 , Rs. 882

Answer

To solve the problem step by step, we will find the shares of A and B based on the given conditions.

### Step 1: Understand the problem
A sum of Rs. 1682 is to be divided between A and B, who are 20 years and 22 years old, respectively. They invest their shares at 5% per annum, compounded annually. After 5 years, when A is 25 and B is 27, both receive equal amounts.

### Step 2: Define the variables
Let A's share be \( x \) and B's share be \( 1682 - x \).

### Step 3: Calculate the amounts received after 5 years
A invests his share for 5 years, while B invests his share for 3 years. The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Where:
- \( A \) is the amount received after \( n \) years,
- \( P \) is the principal amount (initial investment),
- \( R \) is the rate of interest (5%),
- \( n \) is the number of years.

For A:
\[ A_A = x \left(1 + \frac{5}{100}\right)^5 = x \left(1.05\right)^5 \]

For B:
\[ A_B = (1682 - x) \left(1 + \frac{5}{100}\right)^3 = (1682 - x) \left(1.05\right)^3 \]

### Step 4: Set the amounts equal
Since both receive equal amounts after 5 years:
\[ x \left(1.05\right)^5 = (1682 - x) \left(1.05\right)^3 \]

### Step 5: Simplify the equation
Dividing both sides by \( \left(1.05\right)^3 \):
\[ x \left(1.05\right)^2 = 1682 - x \]
\[ x \cdot 1.1025 = 1682 - x \]

### Step 6: Rearranging the equation
\[ x \cdot 1.1025 + x = 1682 \]
\[ x(1.1025 + 1) = 1682 \]
\[ x(2.1025) = 1682 \]

### Step 7: Solve for x
\[ x = \frac{1682}{2.1025} \approx 800 \]

### Step 8: Find B's share
\[ B's \, share = 1682 - x = 1682 - 800 = 882 \]

### Conclusion
A's share is Rs. 800 and B's share is Rs. 882.