Two pipes A and B can separately fill a cistern in 60 minutes and 75 minutes respectively. There is a third pipe in the bottom of the cistern to empty it. If all the three pipes are simultaneously opened then the cistern is full in 50 minutes. In how much time, the third pipe alone can empty the cistern?

### Concept & Efficiency Method Instead of working with fractions, we can assume the total capacity of the cistern to be the Least Common Multiple (LCM) of the given times. This gives us whole-number efficiencies (work done per minute) for each pipe, making calculations much simpler. $$ \text{Efficiency} = \frac{\text{Total Work (Capacity)}}{\text{Time}} $$ ### Step-by-Step Solution 1. **Find Total Capacity:** Let the total capacity of the cistern be the LCM of 60, 75, and 50. LCM(60, 75, 50) = 300 units. 2. **Calculate Individual Efficiencies:** * Efficiency of Pipe A = $300 / 60 = +5$ units/min (filling). * Efficiency of Pipe B = $300 / 75 = +4$ units/min (filling). 3. **Calculate Combined Efficiency:** When all three pipes (A, B, and C) are open, the tank fills in 50 minutes. * Net Efficiency (A + B + C) = $300 / 50 = +6$ units/min. 4. **Find Efficiency of Pipe C:** Let the efficiency of the emptying pipe C be $x$. * $A + B + C = 6$ * $5 + 4 + x = 6$ * $9 + x = 6 \Rightarrow x = -3$ units/min. (The negative sign indicates it is emptying the tank). 5. **Calculate Time for Pipe C:** Time taken by pipe C to empty the full tank alone = $\frac{\text{Total Capacity}}{\text{Efficiency of C}}$ * Time = $300 / 3 = 100$ minutes. ### Exam Strategy & Shortcut Use the LCM approach whenever dealing with multiple pipes or workers. It avoids error-prone fraction additions. Once you see 60, 75, and 50, mentally fix 300 as the volume. Then it's just basic arithmetic: $5 + 4 - \text{Leak} = 6$. Leak is 3. $300 / 3 = 100$. ### Common Pitfall A common mistake is assuming the third pipe fills the tank and adding its rate, or setting up the fraction equation incorrectly as $\frac{1}{60} + \frac{1}{75} + \frac{1}{x} = \frac{1}{50}$ without paying attention to the negative sign required for an emptying pipe. ### Final Answer Therefore, the correct answer is **100 min**.