4 men and 6 women can complete a work in 8 days, while 3 men and 7 women can complete it in 10 days. In how many days will 10 women complete it ?

### Concept & Substitution Method First, equate the total work done by the two different groups to find the relationship between the efficiency of a man and a woman. Then, substitute this relationship into one of the original groups to find the total work purely in terms of women's work units. $$\text{Total Work} = \text{Workforce} \times \text{Days}$$ ### Step-by-Step Solution * **Equate total work:** Let the efficiency of a man be $m$ and a woman be $w$. * $8(4m + 6w) = 10(3m + 7w)$. * **Solve for the efficiency ratio:** Expand both sides. * $32m + 48w = 30m + 70w$. * $32m - 30m = 70w - 48w$. * $2m = 22w$. * $1m = 11w$. (One man does the work of 11 women). * **Calculate total work in terms of women:** Substitute $1m = 11w$ into the first group's equation. * First group workforce = $4m + 6w = 4(11w) + 6w = 44w + 6w = 50w$. * This workforce takes 8 days. Total work = $50w \times 8 \text{ days} = 400w$ units of work (or 400 woman-days). * **Find time for target workforce:** We need to find how many days 10 women will take. * Time = Total Work / Target Workforce. * Time = $400w / 10w = 40$ days. ### Exam Strategy & Shortcut Find the man-to-woman equivalence fast: $32m - 30m = 70w - 48w \Rightarrow 2m = 22w \Rightarrow m = 11w$. Convert group 1 to all women: $4(11) + 6 = 50$ women. If 50 women take 8 days, then 10 women (which is $1/5$th the workforce) will take 5 times as long. $8 \times 5 = 40$ days. ### Common Pitfall A frequent error is stopping after finding that $1m = 11w$ and not converting the total work into a common unit (woman-days) before dividing by the target group. ### Final Answer Therefore, the correct answer is **40**.