A man who recently died left a sum of Rs. 3,90,000 to be divided among his wife, five sons and four daughters. He directed that each son should receive three times as much as each daughter receives, and that each daughter should receive twice as much as their mother receives. Based on these conditions, what was the share of the wife in rupees?

Introduction / Context:
This problem is a ratio based distribution question under the aptitude topic of partnership and shares. Instead of business partners, the shares belong to family members, but the method is the same. The total sum is fixed and the relations between individual shares are given. Our job is to convert these relationships into algebraic form and find the exact share of the wife.

Given Data / Assumptions:

• Total amount left by the man is Rs. 3,90,000.
• Beneficiaries are: wife, five sons and four daughters.
• Each son receives three times as much as each daughter.
• Each daughter receives twice as much as the mother (wife).
• All conditions must hold simultaneously.

Concept / Approach:
The key idea is to represent all individual shares in terms of a single variable. Since the wife's share is the smallest, it is convenient to express daughters' and sons' shares in multiples of the wife's share. Once we express every person's share in terms of the wife's share, we can sum all shares, equate the result to the total amount and solve for the wife's share.

Step-by-Step Solution:
Step 1: Let the wife's share be w rupees.Step 2: Each daughter receives twice the mother, so each daughter's share is 2w.Step 3: Each son receives three times a daughter, so each son's share is 3 * 2w = 6w.Step 4: There is one wife, five sons and four daughters.Step 5: Total share in terms of w is wife + sons + daughters = w + 5 * 6w + 4 * 2w.Step 6: Simplify: w + 30w + 8w = 39w.Step 7: This total 39w equals Rs. 3,90,000, so 39w = 3,90,000.Step 8: Solve for w: w = 3,90,000 / 39 = Rs. 10,000.

Verification / Alternative check:
If the wife receives Rs. 10,000, then each daughter receives 2 * 10,000 = Rs. 20,000 and each son receives 6 * 10,000 = Rs. 60,000. Total for five sons is 5 * 60,000 = Rs. 3,00,000. Total for four daughters is 4 * 20,000 = Rs. 80,000. Adding the wife's share, 3,00,000 + 80,000 + 10,000 = Rs. 3,90,000, which matches the given total amount.

Why Other Options Are Wrong:
If the wife received Rs. 14,000, total amount would be 39 * 14,000 = Rs. 5,46,000, which is too high. For Rs. 12,000, the total would be Rs. 4,68,000. For Rs. 9,000, the total would be Rs. 3,51,000. None of these align with the given amount of Rs. 3,90,000, so they are incorrect. 'None of these' is also incorrect because Rs. 10,000 matches the conditions exactly.

Common Pitfalls:
Candidates sometimes misinterpret which quantity is double or triple, leading to wrong expressions. Another frequent error is incorrectly counting the number of sons or daughters. Always verify by recomputing the total from the final assumed shares to ensure it matches the given sum.

Final Answer:
The wife's correct share of the estate is Rs. 10,000.