12 men can complete a piece of work in 4 days, while 15 women can complete the same work in 4 days. 6 men start working on the job and after working for 2 days, all of them stopped working. How many women should be put on the job to complete the remaining work, if it is to be completed in 3 days?

### Concept & Efficiency Ratio When different groups can complete the same work in the exact same amount of time, their total workforce numbers represent equivalent efficiencies. We can use this to quickly convert between the groups. $$W = M \times D$$ ### Step-by-Step Solution * **Determine equivalence:** Since 12 men complete the work in 4 days and 15 women also complete it in 4 days, their total group efficiencies are equal. * Therefore, $12 \text{ men} = 15 \text{ women}$. * Simplifying by dividing by 3: $4 \text{ men} = 5 \text{ women}$. * **Calculate total work:** Let's measure the total work in "man-days". * Total work = $12 \text{ men} \times 4 \text{ days} = 48 \text{ man-days}$. * **Calculate work done:** 6 men work for 2 days. * Work completed = $6 \times 2 = 12 \text{ man-days}$. * **Calculate remaining work:** $48 - 12 = 36 \text{ man-days}$. * **Determine required workforce:** This remaining work (36 man-days) must be finished in 3 days. * Required men = $36 \text{ man-days} / 3 \text{ days} = 12 \text{ men}$. * **Convert to women:** The problem asks for the number of women needed. * Using our ratio $4 \text{ men} = 5 \text{ women}$, we multiply both sides by 3 to get $12 \text{ men}$. * $12 \text{ men} = 3 \times 5 \text{ women} = 15 \text{ women}$. ### Exam Strategy & Shortcut Since both groups initially take 4 days, you instantly know that 12m = 15w. Instead of finding total work, you can see 6 men working for 2 days is exactly one-quarter of the total initial work ($12m \times 4d$). Thus, $3/4$ of the work remains. $3/4$ of the women's total effort ($15w \times 4d$) is $15w \times 3d$. Since they have 3 days to do it, they need exactly 15 women. ### Common Pitfall A common error is converting the work done by men into women's work too early, which can introduce messy fractions. Keep everything in one unit (men) until the very last step. ### Final Answer Therefore, the correct answer is **15**.