Equivalent work rates: If 12 men or 18 women can finish a job in 14 days, how long will 8 men and 16 women together take to finish it?

Introduction / Context:
From the two equivalences, deduce individual rates for a man and a woman relative to the whole job. Add the rates for the mixed team (8 men + 16 women) and invert to get the time to finish the complete job under constant productivity assumptions.

Given Data / Assumptions:

• 12 men finish in 14 days ⇒ 12*14 man-days = job ⇒ 1 man = 1/168 job/day.
• 18 women finish in 14 days ⇒ 18*14 woman-days = job ⇒ 1 woman = 1/252 job/day.
• Team: 8 men + 16 women.

Concept / Approach:
Team rate = 8*(1/168) + 16*(1/252). Compute carefully using common denominators to avoid arithmetic slips, then time = 1 / (team rate).

Step-by-Step Solution:

Verification / Alternative check:
Relative rate: 1 man = 1.5 women (since 12 men ≡ 18 women). Then 8 men + 16 women ≡ 8 men + (16/1.5) men ≈ 8 + 10.666… = 18.666… men. Since 12 men take 14 days, 1 man/day rate = 1/168; thus time ≈ (job)/(18.666…/168) = 168/18.666… = 9 days. Confirms result.

Why Other Options Are Wrong:
10, 12, 14, 8 days contradict the exact combined rate 1/9 job/day.

Common Pitfalls:
Adding times instead of rates, or mixing up the man-to-woman equivalence (1 man = 1.5 women).

Final Answer:
9 days