Seven men can complete a work in 12 days. They started the work and after 5 days, two men left. In how many days will the work be completed by the remaining men ?

### Concept & Formula The core concept is to equate the total work to the sum of work done in different phases using the formula: $$ \text{Total Work} = M_1D_1 + M_2D_2 $$ where $M$ represents the number of men and $D$ represents the days worked. ### Step-by-Step Solution * **Total Work:** Total work = $7 \text{ men} \times 12 \text{ days} = 84$ man-days. * **Work Done initially:** $7$ men worked for $5$ days. Work done = $7 \times 5 = 35$ man-days. * **Remaining Work:** $84 - 35 = 49$ man-days. * **Remaining Workforce:** $2$ men left, so there are $7 - 2 = 5$ men remaining. * **Time to Complete Remaining Work:** Time = Remaining Work / Remaining Men * Time = $49 / 5 = 9.8$ days (or $9 \frac{4}{5}$ days). * Checking the given options: (a) 5, (b) 6, (c) 7, (d) 8. None of these match $9.8$ days. ### Exam Strategy & Shortcut Use the remaining work perspective directly: After $5$ days, the $7$ men have $7$ days of work left ($12 - 5 = 7$). So, remaining work is $7 \times 7 = 49$ man-days. Now, this $49$ man-days of work is to be done by $5$ men. Time = $49 / 5 = 9.8$ days. ### Common Pitfall Students might incorrectly calculate the remaining men or misread the question as asking for the *total* number of days to complete the work rather than the days taken by the remaining men. ### Final Answer Therefore, the correct answer is **None of these**.