Men–women equivalence and joint workforce: Either 12 men or 18 women can complete a job in 14 days. How many days will 8 men and 16 women working together take to finish the job?

Introduction / Context:
When different worker types have different efficiencies, convert everyone to a common unit (e.g., “man-equivalents” or “woman-equivalents”). Use total work constancy (work = rate * time) to find the duration for a mixed team.

Given Data / Assumptions:

• 12 men finish in 14 days ⇒ total work W = 12 * 14 = 168 man-days.
• 18 women finish in 14 days ⇒ W = 18 * 14 = 252 woman-days.
• Thus 168 man-days = 252 woman-days ⇒ 1 man = 252/168 = 1.5 women.
• Team: 8 men + 16 women.

Concept / Approach:
Convert everyone to man-equivalents. Then compute team’s daily capacity and divide total man-days by this capacity to get total days.

Step-by-Step Solution:
Total work W = 168 man-days.16 women = 16 / 1.5 = 10.666… man-equivalents.Team capacity = 8 + 10.666… = 18.666… man-equivalents per day.Time = W / capacity = 168 / 18.666… = 9 days.

Verification / Alternative check:
Using woman-equivalents instead: 1 woman = 2/3 man. 8 men = 12 women; plus 16 women = 28 women/day. Total W = 252 woman-days. 252/28 = 9 days. Same result confirms robustness.

Why Other Options Are Wrong:

• 10, 12, 13 days do not match the exact equivalence calculations.

Common Pitfalls:

• Mistaking that 12 men ≡ 18 women implies 1 man = 1.5 women (not the inverse).

Final Answer:
9 days