Three pipes P, Q and R can fill a cistern in 40 minutes, 80 minutes and 120 minutes, respectively. Initially, all the pipes are opened. After how much time (in minutes) should the pipes Q and R be turned off so that the cistern will be completely filled in just half an hour?

Question

Accepted Answer

Let the total capacity of the cistern be 120 units (LCM of 40, 80, 120 for easy calculation).

Pipe P fills in 40 min ⇒ 120/40 = 3 units/min
Pipe Q fills in 80 min ⇒ 120/80 = 1.5 units/min
Pipe R fills in 120 min ⇒ 120/120 = 1 unit/min

All pipes together fill per minute = 3 + 1.5 + 1 = 5.5 units/min

Let Q and R be turned off after x minutes. So, for x minutes, all three pipes work. For the remaining (30 - x) minutes, only P works.

Total water filled = (all three pipes for x min) + (only P for 30 - x min)

So,
5.5x + 3(30 - x) = 120
5.5x + 90 - 3x = 120
(5.5x - 3x) = 120 - 90
2.5x = 30
x = 12

Therefore, Q and R should be turned off after 12 minutes to fill the cistern in exactly 30 minutes.

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