Men–Women rate comparison: 16 men and 12 women together finish a work in 20 days. Separately, 18 women can finish the same work in 40 days. In how many days will 12 men and 27 women together finish the work?

Introduction / Context:
We first deduce individual man and woman daily rates using the two conditions, then add rates for the requested combination to compute time.

Given Data / Assumptions:

• 16m + 12w completes in 20 days ⇒ 16m + 12w = 1/20.
• 18w completes in 40 days ⇒ 18w = 1/40 ⇒ w = 1/720.

Concept / Approach:
Find m from the first equation using w, then compute (12m + 27w) time.

Step-by-Step Solution:
12w = 12/720 = 1/60.So 16m = 1/20 − 1/60 = 1/30 ⇒ m = 1/480.Desired rate = 12m + 27w = 12/480 + 27/720 = 1/40 + 3/80 = 1/16.Time = 1 / (1/16) = 16 days.

Verification / Alternative check:
Back-substitute m and w into both original conditions to confirm consistency before final calculation.

Why Other Options Are Wrong:
18, 15, 14, and 20 days result from arithmetic slips in converting w or combining rates.

Common Pitfalls:
Using proportional assumptions without solving; mixing up total “work” with rates.

Final Answer:
16 days