A man has three sons, two daughters and a wife. They divide a sum of Rs 19000 among themselves such that each daughter gets 1.5 times the amount received by each son and the wife receives Rs 600 less than each son. What is the total amount received by the three sons together?

Introduction / Context:
This question deals with distribution of money among family members with given conditions relating the amounts each member receives. The distribution involves multiple dependents and a spouse, and the relationships are expressed as multiples or differences relative to a son share. We are asked to compute the total received by all three sons combined from the total Rs 19000.

Given Data / Assumptions:
The family consists of three sons, two daughters and one wife. Total amount to be divided among them is Rs 19000. Each daughter gets 1.5 times the amount received by each son. The wife receives Rs 600 less than each son. We need to find total amount received by the three sons together.

Concept / Approach:
We start by assuming the amount received by each son is x rupees. Using the conditions, we express each daughter and the wife share in terms of x. Then we sum all these shares for all family members, set this equal to the total Rs 19000 and solve for x. Finally we multiply x by three to get the total amount received by the three sons together. This approach leverages simple algebra with linear equations.

Step-by-Step Solution:
Step 1: Let amount received by each son be x rupees. Step 2: Each daughter receives 1.5 times amount of each son, so each daughter gets 1.5x. Step 3: The wife receives x - 600 rupees. Step 4: There are 3 sons, so total for sons = 3x. Step 5: There are 2 daughters, so total for daughters = 2 * 1.5x = 3x. Step 6: Wife share is x - 600. Step 7: Sum of all shares = amount for sons + daughters + wife = 3x + 3x + (x - 600). Step 8: Combine like terms: 3x + 3x + x - 600 = 7x - 600. Step 9: This total must equal 19000, so 7x - 600 = 19000. Step 10: Add 600 to both sides to get 7x = 19600. Step 11: Therefore x = 19600 / 7 = Rs 2800. Step 12: Total amount received by 3 sons together = 3x = 3 * 2800 = Rs 8400.

Verification / Alternative check:
Using x = 2800, each son gets 2800, each daughter gets 1.5 * 2800 = 4200, and the wife gets 2800 - 600 = 2200. Sons receive 3 * 2800 = 8400, daughters receive 2 * 4200 = 8400, and wife receives 2200. The total is 8400 + 8400 + 2200 = 19000, exactly matching the given total. This confirms that x has been calculated correctly and that total amount for the three sons is indeed Rs 8400.

Why Other Options Are Wrong:
Rs 2800 is the amount received by one son, not by all three sons. Rs 3600 and Rs 5600 do not satisfy the equation formed by the conditions and would change the daughter and wife shares, leading to a total different from Rs 19000. Only Rs 8400 is consistent with all conditions and gives a correct distribution for all family members.

Common Pitfalls:
Some learners incorrectly apply the 1.5 factor to the total for all sons rather than to each son. Others may forget that the wife amount is x minus 600 and treat it as a fixed amount independent of x. Mistakes also occur when counting family members, for example using the wrong number of sons or daughters. Writing each share separately in terms of x and checking the final sum against the total greatly reduces such errors.

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