A man gave `50%` of his savings of `RS. 84,100` to his wife and divided the remaining sum among his two sons A and B of 15 and 13 years of age respectively. He divided it in such a way that each of his sons, when they attain the age of 18 years, would receive the same amount at `5%` compound interest per annum. The share of B was

Options

A. `Rs. 20,000`
B. `Rs. 20,050 `
C. `Rs. 22,000 `
D. `Rs. 22,050`

Answer

To solve the problem step by step, we will follow the outlined logic in the video transcript.

### Step 1: Calculate the Wife's Share
The total savings of the man is Rs. 84,100. He gives 50% of this amount to his wife.

\[ \text{Wife's Share} = 50\% \text{ of } 84,100 = \frac{50}{100} \times 84,100 = 42,050 \]

### Step 2: Calculate the Remaining Amount for Sons A and B
The remaining amount after giving the wife her share is also 50% of Rs. 84,100.

\[ \text{Remaining Amount} = 84,100 - 42,050 = 42,050 \]

### Step 3: Determine the Ages and Time Until Maturity
- Son A is 15 years old and will receive the amount in 3 years (when he turns 18).
- Son B is 13 years old and will receive the amount in 5 years (when he turns 18).

### Step 4: Let the Share of Son A be X
Let the share of Son A be \( X \). Therefore, the share of Son B will be:

\[ \text{Share of Son B} = 42,050 - X \]

### Step 5: Calculate the Future Value for Both Sons
Using the formula for compound interest, the future value for Son A after 3 years at 5% interest is:

\[ \text{Future Value of A} = X \times (1 + \frac{5}{100})^3 = X \times (1.05)^3 \]

The future value for Son B after 5 years at 5% interest is:

\[ \text{Future Value of B} = (42,050 - X) \times (1 + \frac{5}{100})^5 = (42,050 - X) \times (1.05)^5 \]

### Step 6: Set the Future Values Equal
Since both sons will receive the same amount when they turn 18, we set the future values equal:

\[ X \times (1.05)^3 = (42,050 - X) \times (1.05)^5 \]

### Step 7: Simplify the Equation
Now we can simplify the equation:

\[ X \times 1.157625 = (42,050 - X) \times 1.2762815625 \]

Expanding both sides:

\[ 1.157625X = 42,050 \times 1.2762815625 - 1.2762815625X \]

Combining like terms:

\[ 1.157625X + 1.2762815625X = 42,050 \times 1.2762815625 \]

\[ (1.157625 + 1.2762815625)X = 42,050 \times 1.2762815625 \]

Calculating the left side:

\[ 2.4339065625X = 42,050 \times 1.2762815625 \]

Calculating the right side:

\[ 2.4339065625X = 53,694.50 \]

### Step 8: Solve for X
Now, divide both sides by 2.4339065625 to find \( X \):

\[ X = \frac{53,694.50}{2.4339065625} \approx 22,080 \]

### Step 9: Calculate Share of Son B
Now we can find the share of Son B:

\[ \text{Share of Son B} = 42,050 - X = 42,050 - 22,080 = 19,970 \]

### Final Answer
The share of Son B is Rs. 19,970.

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