Three men, four women and six children can complete a work in seven days. A woman does double the work a man does and a child does half the work a man does. How many women alone can complete this work in 7 days ?

### Concept & Formula When different types of workers have varying efficiencies, the best approach is to convert all worker units into a single common unit (e.g., expressing men and children in terms of women) based on their relative efficiencies. $$ \text{Total Work} = (\text{Total Equivalent Units}) \times \text{Days} $$ ### Step-by-Step Solution * **Efficiency Ratios:** Let the 1-day work of a man be $M$, a woman be $W$, and a child be $C$. * Given: A woman does double the work of a man: $W = 2M \Rightarrow M = W/2$. * Given: A child does half the work of a man: $C = M/2$. Substituting $M = W/2$ into the child's equation: $C = (W/2) / 2 = W/4$. * **Convert to Women Equivalent:** The workforce is $3$ men, $4$ women, and $6$ children ($3M + 4W + 6C$). * Substitute $M$ and $C$ in terms of $W$: $3(W/2) + 4W + 6(W/4) = 1.5W + 4W + 1.5W = 7W$. * This means the combined effort of $3$ men, $4$ women, and $6$ children is exactly equal to the effort of $7$ women. * **Calculate Time:** The mixed group ($7W$ equivalent) takes $7$ days. Therefore, $7$ women alone will take exactly the same amount of time: $7$ days. ### Exam Strategy & Shortcut Directly substitute everything into the unit of "women" since the question asks for women alone. $1$ man = $0.5$ women $1$ child = $0.5$ man = $0.25$ women Workforce = $3(0.5) + 4(1) + 6(0.25) = 1.5 + 4 + 1.5 = 7$ women. Since $7$ equivalent women complete the work in $7$ days, it means $7$ women alone complete the work in $7$ days. ### Common Pitfall Getting confused with the efficiency ratios (e.g., assuming a man does double the work of a woman instead of the other way around as stated in the problem). Carefully reading the relative performance is key. ### Final Answer Therefore, the correct answer is **7**.