Two Pipes and a Leaking Cistern
A rooftop cistern has two inlet pipes, P and Q, and one outlet drain, R. Pipe P alone can fill the empty cistern in 12 hours. Pipe Q alone can fill it in 18 hours. The drain R, when open, can empty a full cistern in 24 hours. The caretaker, Sumi, opens P and Q together at 6:00 a.m. with the drain closed. At 9:00 a.m. she notices the cistern is only partly full and accidentally opens the drain R as well. She leaves both inlets and the drain open for a further 3 hours, then closes the drain. She wants to know when the cistern will finally be full, and whether opening the drain caused a delay of more than one hour compared with the no-drain plan.
1. How much of the cistern is filled by 9:00 a.m., and what fraction remains empty?
Read the worked answer
Work with per-hour rates in units of cisterns per hour.
P fills 1/12 per hour; Q fills 1/18 per hour.
Combined rate of P and Q = 1/12 + 1/18. LCM of 12 and 18 is 36: 3/36 + 2/36 = 5/36 per hour.
From 6:00 to 9:00 is 3 hours, so fraction filled = 3 × 5/36 = 15/36 = 5/12.
Fraction empty at 9:00 = 1 − 5/12 = 7/12.
2. During 9:00 a.m. to 12:00 noon all three are open. Find the net fraction filled in those 3 hours and the fraction still empty at noon.
Read the worked answer
Net rate with P, Q and drain R all open = 1/12 + 1/18 − 1/24.
Use denominator 72: 6/72 + 4/72 − 3/72 = 7/72 per hour.
In 3 hours (9:00 to noon): 3 × 7/72 = 21/72 = 7/24 filled.
Fraction filled by noon = 5/12 + 7/24 = 10/24 + 7/24 = 17/24.
Fraction empty at noon = 1 − 17/24 = 7/24.
So the drain slowed filling but did not reverse it, because 7/72 is still positive.
3. From noon, only P and Q run. Find the finishing time and compare it with the time the cistern would have finished if the drain had never been opened.
Read the worked answer
With drain opened: At noon, 7/24 remains. P and Q fill at 5/36 per hour.
Time needed = (7/24) ÷ (5/36) = (7/24) × (36/5) = (7 × 36)/(24 × 5) = 252/120 = 2.1 hours = 2 h 6 min.
Finish time = 12:00 noon + 2 h 6 min = 2:06 p.m.
Without drain: Total time for P and Q alone = 1 ÷ (5/36) = 36/5 = 7.2 hours = 7 h 12 min from 6:00 a.m., i.e. 1:12 p.m.
Delay = 2:06 p.m. − 1:12 p.m. = 54 minutes, which is less than one hour. So the answer to Sumi's question is no - the delay was under an hour.