Two pipes can fill a tank with water in 15 and 12 hours respectively and a third pipe can empty it in 4 hours. If the pipes be opened in order at 8, 9 and 11 a.m. respectively, the tank will be emptied at (S.S.C., 2005)

### Concept & Net Emptying Rate When fill pipes and an emptying pipe operate simultaneously, assign positive efficiencies to filling and negative to emptying. If the net efficiency is negative, the tank will empty. $$ \text{Net Efficiency} = \sum \text{Fill Rates} - \sum \text{Empty Rates} $$ ### Step-by-Step Solution 1. **Given**: Pipe A fills in 15 hrs, Pipe B fills in 12 hrs, Pipe C empties in 4 hrs. 2. **Calculation**: - Total capacity = LCM(15, 12, 4) = 60 units. - Efficiency of A = +60 / 15 = +4 units/hr. - Efficiency of B = +60 / 12 = +5 units/hr. - Efficiency of C = -60 / 4 = -15 units/hr. 3. Calculate the volume filled before the emptying pipe opens at 11 a.m.: - **8 a.m. to 11 a.m.** (3 hrs): Pipe A is open. Work = 3 * 4 = 12 units. - **9 a.m. to 11 a.m.** (2 hrs): Pipe B is open. Work = 2 * 5 = 10 units. - Total water in the tank at 11 a.m. = 12 + 10 = 22 units. 4. Calculate emptying time after 11 a.m.: - After 11 a.m., all pipes are open. - Net efficiency = +4 + 5 - 15 = -6 units/hr (tank is emptying at 6 units/hr). - Time required to empty the 22 units = 22 / 6 = 11/3 hours = 3 hours and 40 minutes. 5. Determine final time: - 11:00 a.m. + 3 hours 40 minutes = 2:40 p.m. ### Exam Strategy & Shortcut Find the accumulated volume quickly: A works 3 hours, B works 2 hours $\rightarrow 3(4) + 2(5) = 22$ units. Net rate is $-6$. Time is $22/6 = 3\frac{2}{3}$ hours. Convert $2/3$ hour to 40 mins and add to 11 a.m. ### Common Pitfall A common mistake is trying to empty the entire capacity (60 units) rather than just the amount of water that actually collected in the tank by 11 a.m. (22 units). ### Final Answer Therefore, the correct answer is **2 : 40 p.m.**