Five men are working to complete a work in 15 days. After five days 10 women are accompanied by them to complete the work in next 5 days. If the work is to be done by women only, then in how many days could the work be over if 10 women have started it ? (Bank Recruitment, 2007)

### Concept & Formula This problem can be solved using the **Man-Days** concept, where the total work is constant and is equal to the product of the number of workers and the time taken. $$ W = M \times D $$ By equating the total work in terms of men to the work done by the mixed group, we can find the ratio of efficiency between a man and a woman. ### Step-by-Step Solution * **Total Work:** Let the 1-day work of a man be $M$ and a woman be $W$. Total work = $5M \times 15 = 75M$. * **Work Done in First 5 Days:** $5$ men work for $5$ days. Work done = $5M \times 5 = 25M$. * **Remaining Work:** $75M - 25M = 50M$. * **Work Done in Next 5 Days:** $5$ men and $10$ women do the remaining work in $5$ days. $ (5M + 10W) \times 5 = 50M $ $ 25M + 50W = 50M $ $ 50W = 25M \Rightarrow 2W = 1M $ This means $1$ man does the work of $2$ women. * **Total Work in Terms of Women:** Total work = $75M = 75(2W) = 150W$. * **Time Taken by 10 Women:** Time = Total Work / (1-day work of $10$ women) = $150W / 10W = 15$ days. ### Exam Strategy & Shortcut Find the efficiency ratio quickly. The $5$ men had $10$ days of work left ($5 \times 10 = 50$ man-days). When $10$ women joined, it was finished in $5$ days. In these $5$ days, the $5$ men did $25$ man-days of work. The remaining $25$ man-days of work was done by $10$ women in $5$ days (which is $50$ woman-days). So, $25M = 50W \Rightarrow 1M = 2W$. Total work is $5 \times 15 = 75M = 150W$. For $10$ women, it takes $15$ days. ### Common Pitfall A common mistake is failing to convert the total work into a single unit (either entirely men or entirely women) before calculating the final required time. ### Final Answer Therefore, the correct answer is **15 days**.