Pipes & Cistern – (Some Selective Questions)
Short methods for answering some of the questions:
Let there be three pipes A, B and C. Let us assume that:
Pipe A takes ‘x’ minutes to fill
Pipe B takes ‘y’ minutes to fill/drain
Pipe C takes ‘z’ minutes to fill/drain.
If A and B are opened, the time taken to fill is given by —- xy/(x + y)
If A, B and C are opened the time taken to fill is given by — xyz/ (xz + yz + xy)
If A fills and B drains the time taken to fill is given by ——- xy/ (y-x)
If A and B fills and C drains the time taken to fill is given by — xyz/ (xz + yz – xy)
1. Two pipes A and B can fill a tank in 36 hours and 45 hours respectively. If both the pipes are opened simultaneously, then, in how much time the tank will be full?
Ans: 20 hours. ( Use the formula — xy/x+y)
2. A pipe can fill the tank in 15 hours. But due to a leak at the bottom it takes 20 hours to fill.If the tank is full then in how much time the leak will empty the tank?
Ans: 60 hours. (Use the formula — xy/y-x. Treat the leak as another pipe that drains the tank)
3. Pipe A and pipe B can fill a tank in 20 hours and 30 hours respectively while pipe C can drain a full tank in 40 hours. If all the three pipes are opened in how much time the tank will become full?
Ans: 17.14 hours (approx.) (use the formula — xyz/ (xz+yz-xy)
4. Pipe A fills a tank of capacity 700 liters at the rate of 40 liters per minute. Pipe B fills the same tank at the rate of 30 liters per minute. Pipe C at the bottom of the tank drains at the rate of 20 liters per minute. If pipe A is kept open for a minute and closed, followed by pipe B also kept open for a minute and closed and finally pipe C kept open for a minute and closed. If these successive operations continue in how much time the tank will be full?
Ans: 42 minutes.
Pipe A in one minute fills —– 40 liters
Pipe B in one minute fills —– 30 liters
Pipe C in one minute drains – 20 liters.
Thus in 3 minutes the tank gets filled ——– 40 + 30 – 20 = 50 liters. (This is one cycle of operation)
The tank capacity is 700 liters and to fill completely will require
700/50 = 14 cycles. Since each cycle of operation is 3 minutes the
total time required to fill the tank is —– 14 x 3 = 42 minutes.
5. Nine large pipes will drain a pond in eight hours and 6 small pipes will drain the same pond in sixteen hours. How long will it take 3 large pipes and 5 small pipes to drain the pond?
Ans: 10 hours 40 minutes or 10 2/3 hours.
9 large pipes take 8 hours to drain the tank.
In one hour 9 large pipes will drain 1/8 tank and
One large pipe in one hour will drain —-1/8 x 1/9 = 1/72 tank.
3 large pipes in one hour will drain — 3 x 1/72 -> 3/72 -> 1/24 tank. ——— (i)
6 small pipes take 16 hours to drain the tank.
In one hour 6 small pipes will drain 1/16 tank and
One small pipe in one hour will drain —- 1/16 x 1/6 = 1/96 tank.
5 small pipes in one hour will drain — 5 x 1/96 -> 5/96 tank. ——————–(ii)
Thus in one hour
3 large pipes and 5 small pipes will drain (i) + (ii) —— 1/24 + 5/96 -> 27/288 -> 3/32 tank.
Hence to drain the entire tank the time taken will be — 32/3 -> 10 2/3 hours.
6. Two pipes A and B can fill a cistern in 12 minutes and 15 minutes respectively, but a third pipe C can empty the full cistern in 6 minutes. A and B are kept open for 5 minutes after which C also is opened. In how much more time the cistern will be empty?
Ans: 45 minutes.
Pipe A in one minute fills — 1/12 . In 5 minutes it will fill —- 5/12 cistern.—— (i)
Pipe B in one minute fills — 1/15 . In 5 minutes it will fill —5/15 -> 1/3 cistern — (ii)
Thus in 5 minutes A and B would have filled —– (i) + (ii) — 5/12 + 1/3 -> (5+4)/12 = ¾ cistern
Hence after 5 minutes the cistern is full up to ¾ level.
Now pipe C also is opened.
In one minute all the three pipes would fill/drain —
1/12 + 1/15 – 1/6 -> (5 + 4 – 10)/60 = -1/60. (minus)
So in each minute the cistern gets drained up 1/60 level.
The cistern was ¾ full when C was opened.
So the time to drain this quantity ——– (3/4) / (1/60) = 45 minutes.
7. 12 buckets of water with each bucket capacity 13.5 liters will fill a tank. How many buckets of capacity 9 liters will be required to fill the tank?
The capacity of the tank is —– 13.5 x 12 = 162 liters.
The number of buckets of capacity 9 liters required to fill this quantity — 162/9 = 18 buckets.
8. A leak in the bottom of a tank can empty the full tank in 6 hours. Let this leak be called B.
An inlet pipe A can fill the tank at the rate of 4 liters per minute. When the tank is full the inlet pipe is opened and due to the leak it now takes 8 hours to drain. What is the capacity of the tank in liters?
Ans: 5760 liters.
Let the capacity be ‘x’ . Then A takes —– x/4×60 -> x/240 minutes to fill. —– (i)
B takes ——- x/ 6 x60 -> x/360 minutes to drain.
When these two are in operation, the tank gets filled —- x/240 – x/360 = x/720 liters.
Now it is given the time to drain a full tank now is – 8 hours.
Thus we have — x/720 = 8 hours — or ‘x’ = 5760 liters.