A man has 3 sons , 2 daughters and a wife. They divided a sum of Rs. 19000 among themselves such that each daughter got 1.5 times the amount received by each son and his wife received Rs. 600 less than each son. What is the total amount ( in Rs.) received by the three sons together ?

Options

A. 2800
B. 3600
C. 5600
D. 8400

Answer

To solve the problem, we need to find out how much money the three sons received together from a total of Rs. 19,000, given the conditions about the distribution of money among the family members.

### Step-by-step Solution:

1. **Define Variables**:
- Let the amount received by each son be \( x \).
- Since there are 3 sons, the total amount received by the sons will be \( 3x \).

2. **Determine the Amount for Daughters**:
- Each daughter receives 1.5 times the amount received by each son.
- Therefore, the amount received by each daughter is \( 1.5x \).
- Since there are 2 daughters, the total amount received by the daughters will be \( 2 \times 1.5x = 3x \).

3. **Determine the Amount for the Wife**:
- The wife receives Rs. 600 less than each son.
- Therefore, the amount received by the wife is \( x - 600 \).

4. **Set Up the Equation**:
- Now we can set up the equation for the total amount distributed among the family:
\[ \text{Total Amount} = \text{Amount received by Sons} + \text{Amount received by Daughters} + \text{Amount received by Wife} \]
\[ 19000 = 3x + 3x + (x - 600) \]

5. **Simplify the Equation**:
- Combine like terms:
\[ 19000 = 3x + 3x + x - 600 \]
\[ 19000 = 7x - 600 \]

6. **Solve for \( x \)**:
- Add 600 to both sides:
\[ 19000 + 600 = 7x \]
\[ 19600 = 7x \]
- Now divide both sides by 7:
\[ x = \frac{19600}{7} = 2800 \]

7. **Calculate the Total Amount Received by the Sons**:
- The total amount received by the three sons together is:
\[ 3x = 3 \times 2800 = 8400 \]

### Final Answer:
The total amount received by the three sons together is **Rs. 8400**.