A person invested for 3 years and 5 years @ 5% per annum compounded annually, after dividing 2,523 in two parts on the name of his sons respectively. If they got the same amount, how much money invested on each of his son?

Options

A. ₹1,100,₹1,423
B. ₹ 1,523,₹ 1,000
C. ₹ 1,323,₹ 1,200
D. ₹ 1,223,11,300

Answer

To solve the problem, we need to determine how much money was invested for each son, given that the total amount is 2,523 and that one son’s investment is for 3 years while the other’s is for 5 years at a compound interest rate of 5% per annum.

### Step-by-Step Solution:

1. **Define Variables**:
Let the amount invested for the son with the 3-year investment be \( x \). Therefore, the amount invested for the son with the 5-year investment will be \( 2523 - x \).

2. **Write the Compound Interest Formula**:
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where \( A \) is the amount after time \( n \), \( P \) is the principal amount (initial investment), \( r \) is the rate of interest, and \( n \) is the number of years.

3. **Calculate Amounts for Each Son**:
For the first son (3 years):
\[ A_1 = x \left(1 + \frac{5}{100}\right)^3 = x \left(1.05\right)^3 \]
For the second son (5 years):
\[ A_2 = (2523 - x) \left(1 + \frac{5}{100}\right)^5 = (2523 - x) \left(1.05\right)^5 \]

4. **Set the Amounts Equal**:
Since both sons received the same amount:
\[ x \left(1.05\right)^3 = (2523 - x) \left(1.05\right)^5 \]

5. **Simplify the Equation**:
Dividing both sides by \( \left(1.05\right)^3 \):
\[ x = (2523 - x) \left(1.05\right)^2 \]
Expanding this gives:
\[ x = (2523 - x) \cdot 1.1025 \]

6. **Distribute and Rearrange**:
\[ x = 2523 \cdot 1.1025 - x \cdot 1.1025 \]
\[ x + x \cdot 1.1025 = 2523 \cdot 1.1025 \]
\[ x(1 + 1.1025) = 2523 \cdot 1.1025 \]
\[ x \cdot 2.1025 = 2781.2675 \]

7. **Solve for \( x \)**:
\[ x = \frac{2781.2675}{2.1025} \approx 1323 \]

8. **Calculate the Amount for the Second Son**:
The amount invested for the second son is:
\[ 2523 - x = 2523 - 1323 = 1200 \]

### Final Answer:
The amount invested for the first son is **1,323** and for the second son is **1,200**.