Reena, Aastha and Shloka can independently complete a piece of work in 6 hours, 4 hours and 12 hours respectively. If they work together, how much time will they take to complete that piece of work?

### Concept & Multi-Person Combined Efficiency To find the total time required for three or more people working together, sum their individual one-hour rates (efficiencies) to find their combined one-hour rate. The total time is the reciprocal of this combined rate. ### Step-by-Step Solution * **Given:** Reena's time = 6 hours. Aastha's time = 4 hours. Shloka's time = 12 hours. * **Calculation:** Reena's 1-hour work = $\frac{1}{6}$ Aastha's 1-hour work = $\frac{1}{4}$ Shloka's 1-hour work = $\frac{1}{12}$ * Combined 1-hour work of all three together: $= \frac{1}{6} + \frac{1}{4} + \frac{1}{12}$ To add these fractions, find the LCM of 6, 4, and 12, which is 12. $= \frac{2}{12} + \frac{3}{12} + \frac{1}{12}$ $= \frac{6}{12}$ $= \frac{1}{2}$ * The combined efficiency is $\frac{1}{2}$ of the work per hour. * Total time taken = Reciprocal of $\frac{1}{2} = 2$ hours. ### Exam Strategy & Shortcut Use the Total Work (LCM) method. Let Total Work = LCM(6, 4, 12) = 12 units. Reena's efficiency = 12 / 6 = 2 units/hr. Aastha's efficiency = 12 / 4 = 3 units/hr. Shloka's efficiency = 12 / 12 = 1 unit/hr. Total combined efficiency = 2 + 3 + 1 = 6 units/hr. Total Time = Total Work / Combined Efficiency = 12 / 6 = 2 hours. This entirely avoids fractional addition. ### Common Pitfall A common mistake is simply adding the hours together ($6 + 4 + 12 = 22$) or taking an average of the hours. Remember that adding more workers will always decrease the total time compared to the fastest single worker. ### Final Answer Therefore, the correct answer is **2 hours**.