*Tuesday, June 12th, 2018*

A pipe is connected to a tank or cistern. It is used to fill or empty the tank; accordingly, it is called an inlet or an outlet.

**Inlet:**Â A pipe which is connected to fill a tank is known as an inlet.

**Outlet:**Â A pipe which is connected to empty a tank is known as an outlet.

Problems on pipes and cisterns are similar to problems on time and work. In pipes and cistern problems, the amount of work done is the part of the tank of filled or emptied. And, the time taken to do a piece of work is the time take to fill or empty a tank completely or to a desired level.

1) If an inlet connected to a tank fills it in X hours, part of the tank filled in one hour is = 1/X

2) If an outlet connected to a tank empties it in Y hours, part of the tank emptied in one hour is = 1/Y

3) An inlet can fill a tank in X hours and an outlet can empty the same tank in Y hours. If both the pipes are opened at the same time and Y > X, the net part of the tank filled in one hour is given by;

= (1/X – 1/Y)

Therefore, when both the pipes are open the time taken to fill the whole tank is given by;

= (XY/Y-X) Hours

If X is greater than Y, more water is flowing out of the tank than flowing into the tank. And, the net part of the tank emptied in one hour is given by;

= (1/Y – 1/X)

Therefore, when both the pipes are open the time taken to empty the full tank is given by;

= (YX/X-Y) Hours

4) An inlet can fill a tank in X hours and another inlet can fill the same tank in Y hours. If both the inlets are opened at the same time, the net part of the tank filled in one hour is given by;

= (1/X + 1/Y)

Therefore, the time taken to fill the whole tank is given by;

= (XY/Y+X) Hours

In a similar way, If an outlet can empty a tank in X hours and another outlet can empty the same tank in Y hours, the part of the tank emptied in one hour when both the pipes start working together is given by;

= (1/X + 1/Y)

**Example 1:** A drain pipe can drain a tank in 12 hours, and a fill pipe can fill the same tank in 6 hours. A total of n pipes â€“ which include a few fill pipes and the remaining drain pipes â€“ can fill the entire tank in 2

hours. How many of the following values could n take?

a) Â Â 24Â Â Â Â Â Â Â Â Â Â Â Â Â Â b) Â Â 16Â Â Â Â Â Â Â Â Â Â Â Â Â Â c) Â Â 33Â Â Â Â Â Â Â Â Â Â Â Â Â Â d) Â Â 13Â Â Â Â Â Â Â Â Â Â Â Â Â Â e) Â Â 9Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â f) Â Â 8

A.) 3

B.) 4

C.) 2

D.) 1

**Solution:** Two drain pipes can drain the same volume that one fill pipe fills. This means that a D-D-F combination has to have a net volume effect of 0.

In spite of this, the tank still gets filled. Only the fill pipes can manage to fill the tank. In addition to all the net zero effect pipes, we need three more fill pipes in order to fill the tank in 2 hours.

So, we can have as many D-D-Fs as we want, but we need one F-F-F at the end to ensure that the tank gets filled in 2 hours.

So the number of pipes will be â†’ (D â€“ D – F)…….(D â€“ D – F) + (F â€“ F – F).

The number of pipes has to be a multiple of 3. Only options (a), (c) and (e) fit the description. Choice (A).

**Correct Answer: 3**

**Example 2:** Pipe A, B and C are kept open and together fill a tank in t minutes. Pipe A is kept open throughout, pipe B is kept open for the first 10 minutes and then closed. Two minutes after pipe B is closed, pipe C is opened and is kept open till the tank is full. Each pipe fills an equal share of the tank. Furthermore, it is known that if pipe A and B are kept open continuously, the tank would be filled completely in t minutes. How long will C alone take to fill the tank?

A.) 18Â Â Â Â Â Â Â Â Â Â Â Â Â B.) 36Â Â Â Â Â Â Â Â Â Â C.) 27Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â D.) 24

**Solution:** A is kept open for all t minutes and fills one-third the tank. Or, A should be able to fill the entire tank in ‘3t’ minutes.

A and B together can fill the tank completely in t minutes. A alone can fill it in 3t minutes.

A and B together can fillÂ 1/tÂ of the tank in a minute. A alone can fillÂ 1/3tÂ of the tank in a minute. So, in a minute, B can fillÂ 1tâˆ’13t=2/3t. Or, B takesÂ 3t/2Â minutes to fill an entire tank.

To fill one-third the tank, B will takeÂ t/2Â minutes. B is kept open for t – 10 minutes.

t/2Â = t – 10, t = 20 minutes.

A takes 60 minutes to fill the entire tank, B takes 30 minutes to fill the entire tank. A is kept open for all 20 minutes. B is kept open for 10 minutes.

C, which is kept open for 8 minutes also fills one-third the tank. Or, c alone can fill the tank in 24 minutes. Choice (D)

**Correct Answer: 24 minutes**

**Example 3:** Â Pipe A fills a tank at the rate of 100lit/min, Pipe B fills at the rate of 25 lit/min, pipe C drains at the rate of 50 lit/min. The three pipes are kept open for one minute each, one after the other. If the capacity of the tank is 7000 liters, how long will it take to fill the tank if

i.Â Â A is kept open first, followed by B and then C.

ii.Â B first, followed by A, and then C.

iii. B first, followed by C, and then A.

A.) 279.25 mins, 280 mins and 280 mins

B.) 280 mins, 280 mins and 279.25 mins

C.) 279 mins, 280 mins and 279.25 mins

D.) 277 mins, 277 mins and 45 secs and 279 mins

**Solution:**

i.)Â A is kept open first, followed by B and then C

Each cycle of 3 minutes, 75 liters get filled. 100 + 25 – 50. So, after 3 minutes the tank would have 75 liters

6 mins – 150 liters

9 mins – 225 liters

30 mins – 750 liters

270 mins – 6750 liters

273 mins – 6825 liters

276 mins – 6900 litres

During the 277th min, Pipe A is Opened and it fills at the rate of 100 litres/min. Therefore, at the end of 277 mins, the tank is filled.

The most important thing in these type of questions is to think in terms of cycles till we reach close to the required target and then think in simple steps.

ii.)Â Â B first, followed by A, and then C

Each cycle of 3 minutes, 75 liters get filled. 25 + 100 – 50. So, after 3 minutes the tank would have 75 liters.

6 mins – 150 liters

9 mins – 225 liters

30 mins – 750 liters

270 mins – 6750 liters

273 mins – 6825 liters

276 mins – 6900 litres

during the 277th min, Pipe B is Opened and it fills at the rate of 25 litres/min. At the end of 277 mins, 6925 litres. Then, Pipe A is opened and it fills at the rate of 100 litres/min. Therefore, it takes 3/4 th of a minute to fill the remaining 75 litres. Time taken = 277 + 3/4 th minute = 277 mins and 45 secs .

iii.)Â Â B first, followed by C, and then A

In this case also, Each cycle of 3 minutes, 75 litres get filled. 25 -50 + 100. But there is a small catch here. In the first set of 3 minutes, we would fill up to about 100 litres. After 1 minute, we would be at 25 liters, after 2 minutes, we would be at 0 liters and in the third minute, the tank would be 100 litres full.

6 mins – 175 liters

9 mins – 250 liters

30 mins – 775 liters

270 mins – 6775 liters

273 mins – 6850 liters

276 mins – 6925 liters

279 minutes – 7000 liters

So, it would take 279 mins to fill the tank.

The most important thing in these type of questions is to think in terms of cycles till we reach close to the required target and then think in simple steps.

**Answer Choice (D).**

**Â **

**Example 4: **A tank is fitted with 8 pipes, some of which that fill the tank and others that empty the tank. Each of the pipes that fills the tank fills it in 8 hours, while each of those that empty the tank empties it in 6 hours. If all the pipes are kept open when the tank is full, it will take 6 hours to drain the tank. How many of these are fill pipes?

- 2 fill pipes
- 4 fill pipes
- 6 fill pipes
- 5 fill pipes

**Solution:** Let the number of fill pipes be ‘n’

Therefore, there will be (8 – n) waste pipes.

Each of the fill pipes can fill the tank in 8 hours.

Therefore, each of the fill pipes will fillÂ 1/8th of the tank in an hour.

Hence, n fill pipes will fillÂ n/8th of the tank in an hour.

Similarly, each of the waste pipes will drain the full tank in 6 hours.

âˆ´ each of the waste pipes will drainÂ 1/6th of the tank in an hour.

(8 – n) waste pipes will drainÂ (8-n)/6th of the tank in an hour.

Between the fill pipes and the waste pipes, they drain the tank in 6 hours.

That is, when all 8 of them are opened,Â 1/6th of the tank gets drained in an hour.

(Amount of water filled by fill pipes in 1 hour – Amount of water drained by waste pipes 1 hour) =Â (1/6^{th} ) of the tank

Therefore,

(n/8) â€“ ((8âˆ’n)/)6Â =Â -1/6

N**ote**: The right hand side has a negative sign because the tank gets drained.

Cross multiplying and solving the equations, 14n – 64 = -8

or 14n = 56 or n = 4

**The correct answer is Choice (2).**

**Example 5:****Â **Pipe A usually fills a tank in 2 hours. On account of a leak at the bottom of the tank, it takes pipe A 30 more minutes to fill the tank. How long will the leak take to empty a full tank if pipe A is shut?

- 2 hours 30 minutes
- 5 hours
- 4 hours
- 10 hours

Pipe A fills the tank normally in 2 hours.

Therefore, it will fill 1/2 of the tank in an hour.

Let the leak take x hours to empty a full tank when pipe A is shut.

Therefore, the leak will empty 1/x of the tank in an hour.

The net amount of water that gets filled in the tank in an hour when pipe A is open and when there is a leak = Â Â (1/2 â€“ 1/x) of the tank. —– (1)

Now, when there is a leak, the problem states that it takes two and a half hours to fill the tank. i.e. 5/2hours.

Therefore, in an hour, 2/5th of the tank gets filled. —– (2)

Equating (1) and (2), we get 1/2 â€“ 1/x = 2/5

=> 1/x = 1/2 â€“ 2/5 = 1/10

=> x = 10 hours.

The correct answer is Choice (4).

**Conclusion:**Â Most of the questions asked in CAT or other similar exams are of the same pattern as above from Pipes and Cisterns. If we remember the basic properties and have some practice on the above pattern of questions, these will be easy to handle in exams.

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