A machine P can print one lakh books in 8 hours, machine Q can print the same number of books in 10 hours while machine R can print them in 12 hours. All the machines are started at 9 a.m. while machine P is closed at 11 a.m. and the remaining two machines complete the work. Approximately at what time will the work be finished ?

### Concept & Total Work and Rate This problem relies on assigning a "Total Work" unit value using the Least Common Multiple (LCM) of the given times to find individual machine printing rates, then tracking the work completed in distinct time phases. ### Step-by-Step Solution * **Given:** P prints in 8 hrs, Q in 10 hrs, R in 12 hrs. They run from 9 a.m. to 11 a.m. (2 hours). Then P closes, and Q + R finish the rest. * **Calculation:** Let Total Work = LCM(8, 10, 12) = 120 units. * Rate of Machine P = $120 / 8 = 15$ units/hr. * Rate of Machine Q = $120 / 10 = 12$ units/hr. * Rate of Machine R = $120 / 12 = 10$ units/hr. * For the first 2 hours (9 a.m. to 11 a.m.), P, Q, and R work together. * Combined rate of (P + Q + R) = $15 + 12 + 10 = 37$ units/hr. * Work done in 2 hours = $37 \times 2 = 74$ units. * Remaining work = Total Work - Work Done = $120 - 74 = 46$ units. * After 11 a.m., P is closed. Q and R continue working. * Combined rate of (Q + R) = $12 + 10 = 22$ units/hr. * Time taken by Q and R to finish the remaining work = Remaining Work / (Q + R rate) = $46 / 22 = 23 / 11$ hours. * $23 / 11 \approx 2.09$ hours, which is approximately 2 hours and 5 minutes. * Finish time = 11:00 a.m. + 2 hours 5 minutes = 1:05 p.m. * The closest option is approximately 1 p.m. ### Exam Strategy & Shortcut Mentally find the LCM (120) and rates (15, 12, 10). Sum the rates for the first phase: 37. Multiply by 2 hours = 74. Subtract from 120 to get 46. Sum the rates for the second phase: 22. Divide 46 by 22, which is slightly more than 2. Thus, the completion time is roughly 2 hours after 11 a.m., placing it closest to 1 p.m. ### Common Pitfall A common mistake is forgetting to add the remaining time to the time the initial phase ended (11 a.m.), or struggling to convert fractional hours into standard time, which isn't strictly necessary here since "approximately" allows for selecting the nearest whole hour. ### Final Answer Therefore, the correct answer is **1 p.m.**