A man had two sons. To the elder, he left 5/11 of his property, to the younger 5/11 of the remainder, the rest to the widow. Find the share of the sons if the widow gets ₹ 3600.

## Concept & Logic This is a successive distribution problem. When a portion is given from the "remainder", the new fraction must be multiplied by the remaining fraction of the total, not simply subtracted from the total. $$ Remaining\ Fraction = 1 - Fraction\ Given $$ ## Step-by-Step Solution * **Given:** Elder son gets $\frac{5}{11}$ of the total property. Younger gets $\frac{5}{11}$ of the remainder. Widow gets ₹ 3600. Let total property be $P$. * **Calculation:** Elder son's share = $\frac{5}{11}P$. * **Deduction:** The remainder of the property after the elder son's share is $P - \frac{5}{11}P = \frac{6}{11}P$. * **Calculation:** Younger son's share is $\frac{5}{11}$ of this remainder: Younger = $\frac{5}{11} \times \frac{6}{11}P = \frac{30}{121}P$. * **Deduction:** The widow gets whatever is left from the remainder after the younger son is paid: Widow's fraction = $\frac{6}{11}P - \frac{30}{121}P = \frac{66P - 30P}{121} = \frac{36}{121}P$. * **Calculation:** Equate the widow's fractional share to her actual amount: $\frac{36}{121}P = 3600$ $P = \frac{3600 \times 121}{36} = 12100$. * **Calculation:** Find the exact share of both sons: Elder son = $\frac{5}{11} \times 12100 = 5500$. Younger son = $\frac{30}{121} \times 12100 = 3000$. * **Conclusion:** The shares are ₹ 5500 and ₹ 3000, which do not match options (a), (b), or (c). ## Exam Strategy & Shortcut Work with ratios. The fractions involve 11, so let the total be a multiple of $11 \times 11 = 121$ units. Total = 121 units. Elder = $\frac{5}{11}$ of 121 = 55 units. Remainder = 66 units. Younger = $\frac{5}{11}$ of 66 = 30 units. Remainder (Widow) = 36 units. We know 36 units = ₹ 3600, so 1 unit = ₹ 100. Thus, Elder = 5500, Younger = 3000. These are not in the options, confirming "None of these". ## Common Pitfall Treating "$\frac{5}{11}$ of the remainder" as simply $\frac{5}{11}$ of the total. Students who do this incorrectly calculate the widow's share as $1 - \frac{5}{11} - \frac{5}{11} = \frac{1}{11}$, leading to completely wrong valuations for the property. ## Final Answer Therefore, the correct answer is **None of these**.