Inferring relative productivity from completion times: 2 men and 1 woman can complete a work in 14 days, while 4 women and 2 men can do the same work in 8 days. If a man earns ₹ 90 per day under pay proportional to productivity, what is a woman’s daily wage?

Introduction / Context:This question links work rates to wages. If wages are proportional to productivity, then the ratio of a man’s wage to a woman’s wage must match the ratio of their work rates. We use the two completion scenarios to deduce the man-to-woman productivity ratio.

Given Data / Assumptions:

• (2m + 1w) completes in 14 days ⇒ rate = (2M + W)
• (2m + 4w) completes in 8 days ⇒ rate = (2M + 4W)
• Let total work = 1 unit; M and W denote man and woman daily work rates
• Man’s daily wage = ₹ 90; wages ∝ rates

Concept / Approach:Write rates as 1/14 = 2M + W and 1/8 = 2M + 4W after normalizing by total work. Eliminate M to find the ratio M : W. Then use wage proportionality to scale from man’s wage ₹ 90 to woman’s wage.

Step-by-Step Solution:2M + W = 1/142M + 4W = 1/8Subtract first from second: 3W = 1/8 − 1/14 = (7 − 4)/56 = 3/56 ⇒ W = 1/56Then 2M + W = 1/14 ⇒ 2M = 1/14 − 1/56 = (4 − 1)/56 = 3/56 ⇒ M = 3/112Ratio M : W = (3/112) : (1/56) = (3/112) : (2/112) = 3 : 2Wage ratio man : woman = 3 : 2; man = ₹ 90 ⇒ woman = (2/3)*90 = ₹ 60

Verification / Alternative check:Productivity proportionality is consistent with many wage problems: if you double the workers of one type, the team rate adjusts linearly, which aligns with our equations.

Why Other Options Are Wrong:₹ 48 or ₹ 72 imply different productivity ratios; ₹ 135 exceeds the man’s wage, contradicting the 3 : 2 ratio.

Common Pitfalls:Using days (inverse) instead of rates directly; miscomputing 1/8 − 1/14; assuming equal wages despite different productivities.

Final Answer:₹ 60