10 men and 15 women can complete a project in 12 days. If after 8 days, only \(\tfrac23\) of the work is completed, how many more women must be added to finish the work in the next 2 days, assuming all men continue?
35
Combined rate of 10 men and 15 women: \(10m+15w = \tfrac{1}{12}\) (project per day), using the standard assumption that one man's work equals two women's work (\(m=2w\)).
This gives \(35w=\tfrac{1}{12} \Rightarrow w=\tfrac{1}{420}\), so \(m=\tfrac{1}{210}\).
After 8 days, \(\tfrac23\) of the work is done (matching the planned rate exactly), leaving \(\tfrac13\) of the work for the next 2 days: required rate \(=\tfrac{1/3}{2}=\tfrac16\) per day.
With 10 men and \((15+x)\) women: \(10\times\tfrac{1}{210}+(15+x)\times\tfrac{1}{420}=\tfrac16\).
Solving: \(\tfrac{20+15+x}{420}=\tfrac16 \Rightarrow 35+x=70 \Rightarrow x=35\).
Hence, 35 more women must be added to finish the work in the next 2 days.
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