A total of Rs 18,750 is invested by a man in the bank account of his two sons whose ages are 12 years and 14 years in such a way that they will get equal amount at the age of 18 years at a rate of 5% per annum simple interest. What is the share of the younger child?

Options

A. A)Rs 7500
B. B)Rs 9000
C. C)Rs 8000
D. D)Rs 6500

Answer

To solve the problem step by step, we need to determine how much money should be invested for each son so that they both receive the same amount when they turn 18 years old.

### Step-by-Step Solution:

1. **Identify the Ages and Time Periods:**
- The younger son is currently 12 years old and will be 18 in 6 years (18 - 12 = 6 years).
- The elder son is currently 14 years old and will be 18 in 4 years (18 - 14 = 4 years).

2. **Let the Principal Amounts:**
- Let the principal amount invested for the younger son be \( K \).
- Therefore, the principal amount for the elder son will be \( 18750 - K \).

3. **Calculate the Amount for Each Son:**
- The formula for the amount \( A \) after a certain time with simple interest is:
\[ A = P \left(1 + \frac{R \cdot T}{100}\right) \]
- For the younger son:
\[ A_1 = K \left(1 + \frac{5 \cdot 6}{100}\right) = K \left(1 + 0.30\right) = 1.3K \]
- For the elder son:
\[ A_2 = (18750 - K) \left(1 + \frac{5 \cdot 4}{100}\right) = (18750 - K) \left(1 + 0.20\right) = (18750 - K) \cdot 1.2 \]

4. **Set the Amounts Equal:**
- Since both sons will receive the same amount at age 18, we set \( A_1 = A_2 \):
\[ 1.3K = (18750 - K) \cdot 1.2 \]

5. **Expand and Rearrange the Equation:**
- Expanding the right side:
\[ 1.3K = 22500 - 1.2K \]
- Rearranging gives:
\[ 1.3K + 1.2K = 22500 \]
\[ 2.5K = 22500 \]

6. **Solve for \( K \):**
- Dividing both sides by 2.5:
\[ K = \frac{22500}{2.5} = 9000 \]

7. **Conclusion:**
- The share of the younger child is Rs 9,000.