Amit and Sumit can plough a field in 4 days. Sumit alone can plough the field in 6 days. In how many days will Amit alone plough the field?

### Concept & Time and Work When the combined work time of two individuals and the individual time of one are given, the time taken by the second individual can be found by subtracting the known individual's 1-day work from their combined 1-day work. ### Step-by-Step Solution * **Given:** Time taken by Amit and Sumit together = 4 days. Time taken by Sumit alone = 6 days. * **Calculation:** Combined 1-day work of (Amit + Sumit) = $\frac{1}{4}$ Sumit's 1-day work = $\frac{1}{6}$ * Amit's 1-day work = (Combined 1-day work) - (Sumit's 1-day work) Amit's 1-day work = $\frac{1}{4} - \frac{1}{6}$ Find the LCM of 4 and 6, which is 12. Amit's 1-day work = $\frac{3}{12} - \frac{2}{12} = \frac{1}{12}$ * Since Amit does $\frac{1}{12}$ of the work in 1 day, he will take 12 days to complete the whole work alone. ### Exam Strategy & Shortcut For two people where combined time ($T$) and one person's time ($y$) is known, the time taken by the other person ($x$) is given by the formula: $$x = \frac{y \times T}{y - T}$$ Substitute $T = 4$ and $y = 6$: $x = \frac{6 \times 4}{6 - 4} = \frac{24}{2} = 12$ days. ### Common Pitfall A frequent error is subtracting the days directly ($6 - 4 = 2$ days). Work is inversely proportional to time, so you must always subtract the rates of work, not the time taken. ### Final Answer Therefore, the correct answer is **12 days**.