# Pipes and Cisterns

This is similar to Time and Work problems. The only difference is that here rates of filling or emptying a tank are given whereas in the previous section rates of work of persons are given. The methodology remains same.

**Illustration 12**:

Two pipes A and B fill a tank in 20 minutes and 40 minutes respectively. A pipe C at the bottom can empty the tank in 60 minutes. If all three pipes were open simultaneously, how long does it take to fill the empty tank?

Pipe A fills 1/20th of the tank in a minute, Pipe B fills 1/40th of the tank in a minute, Pipe C drains 1/60th of the tank in a minute.

Therefore, if all three are open the net effect = (1/20 + 1/40 – 1/60)^{th} of the tank will be filled in a minute i.e. 7/120th of the tank will be filled in a minute.

Therefore, the tank will be filled in 120/7 minutes.

**Illustration 13**:

Two pipes A and B can fill a cistern in 20 and 24 minutes respectively. Both pipes being opened, find when the first pipe must be turned off, so that the cistern may be filled in 12 minutes?

Since the cistern is to be filled in 12 minutes, Second pipe can fill only 12/24 = 1⁄2 of the cistern in total time. This means the other half must be filled by the first pipe. The first pipe can fill the whole tank in 20 minutes, so half of the tank it can fill in half of the 20 minutes i.e. 10 minutes. Now the first pipe is opened from the beginning, it should be turned off after 10 minutes.

**Illustration 14**:

Pipes A and B fill a tank in 30 minutes and 15 minutes. Pipe C drains 12 litres of water in a minute. If all of them are kept open when the tank is full, the tank empties in 30 minutes. How much water can the tank hold?

Pipe A fills 1/30 th of the tank and pipe B fills 1/15th of the tank in 1 minute. Pipe C drains 12 litres in a minute ⇒ in 30 minutes it drains 12 × 30 = 360 litres. So, if Q is the total capacity, then:

Full tank + input in 30 min = output in 30 min

- Q(Q/30 + Q/15 ) * 30 = 360
- Q = 90 litres.

**Illustration 15:**

If two taps can fill a tank in 40 & 30 min respectively and a third pipe can empty it in 20 minutes. If the three are opened together the tank will fill in:

1. 60 min 2. 120 min 3. 150 min. 4. 45 min.

As explained, the basic concepts are the same as time and work. In this case, all three taps together result in 1/40 + 1/30 – 1/20 = 1/120 Hence it would take 120 min if all three pipes were turned on together.

**Illustration 16**:

A cistern is filled in 9 hours and it takes 10 hours when there is a leak in its bottom. If the cistern is full, in what time shall the leak empty it?

Cistern filled in 1 hour by the filling pipe = 1/9. Cistern filled by the leak and the filling pipe in 1 hour = 1/10. Cistern emptied by the leak in one hour = 1/9 -1/10 = 1/90. Hence the leak can empty the tank in 90 hours.