Three women and eighteen children together complete a work in 2 days. If six women alone can complete the same work in 3 days, then how many days will nine children alone take to finish the work?

Introduction / Context:
Use the women-only completion time to define the total job, then determine a child’s rate by balancing the mixed team against the known total. Finally, scale to nine children.

Given Data / Assumptions:

• 6 women → 3 days → total work W = 18w (where w = woman/day).
• (3w + 18c) * 2 days = W.

Concept / Approach:
From (3w + 18c)*2 = 18w, solve for c in terms of w. Then compute time for 9 children = W / (9c).

Step-by-Step Solution:

Verification / Alternative check:
Back-substitute c = w/3 into the mixed team to confirm the total matches W.

Why Other Options Are Wrong:
They conflict with the derived equivalence c = w/3.

Common Pitfalls:
Forgetting to multiply by the number of days when forming equations; dropping the factor 2.

Final Answer:
6