Associated Topics || Dr. Math Home || Search Dr. Math

### Two Pipes and a Drain

```
Date: 01/07/98 at 09:15:17
From: andrei
Subject: 2 pipes

A swimming pool has 2 inlet pipes. One fills the pool in 4 hours, the
other in 6 hours. The outlet pipe empties the pool in 5 hours.

Once the outlet pipe was left open when the pool was being filled. In
how many hours was the pool full?
```

```
Date: 01/07/98 at 12:48:48
From: Doctor Anthony
Subject: Re: 2 pipes

In 1 hour the first inlet pipe fills 1/4 of the pool.

In 1 hour second inlet pipe fills 1/6 of the pool.

In 1 hour the drain empties 1/5 of the pool.

If all the inlets and the drain are open, then in 1 hour the pool
fills by

1/4 + 1/6 - 1/5  = 13/60  of its volume.

Time to fill the pool =  60/13 hours =  4 hours 36 minutes 55 secs.

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 01/07/98 at 09:17:16
From: andrei
Subject: 2 pipes

A hot water faucet fills a tub in 30 minutes, and the cold water
faucet in 20 minutes. The tub can be drained in 15 minutes.

If both faucets are open while the drain is open, how soon will the
tub be full?
```

```
Date: 01/09/98 at 09:25:19
From: Doctor Ezick
Subject: Re: 2 pipes

Good Question!

The trick to this question is to take advantage of the fact that the
problem leaves the size of tub unspecified. The fact that the size of
the tub is unspecified means one of two things.

1. It doesn't matter - the fill time is the same no matter how large
the tub is.

2. We can't answer the question absolutely - the best we can do is to
give an expression that depends upon the size of the tub.

For the moment take my word for it that we are dealing with case one.

Since the size of the tub does not make any difference, let's say that
it's 60 gallons (60 is the least common multiple of 15, 20 and 30,
which is going to make for some nice divisions in a second).

What does this mean?

If the hot water faucet fills the tub in 30 minutes it must be pumping
in 60 gallons / 30 minutes = 2 gallons per minute.

Likewise, the cold water faucet is pumping in 60 / 20 = 3 gallons per
minute.

This means that 2 + 3 = 5 gallons per minute of water are flowing into
the tub.

Meanwhile, if the drain can drain a full tub in 15 minutes, that means
that 60 / 15 = 4 gallons per minute are being drained from the tub.

The result: Every minute 5 gallons of water go in and 4 go out.
This means that 1 gallon of water per minute is going into the tank.
Since the tank holds 60 gallons, it takes 60 gallons / 1 gallon per
minute = 60 minutes to fill the tank!

The question now is: How do we know that the size of the tank does not
matter?  We could solve this problem the same way letting the size be
unspecified by setting up an equation as follows:

Size
------------------------------------------------ = Time to Fill
(Size/20 mins) + (Size/30 mins) - (Size/15 mins)
(Hot Water)     (Cold Water)       (Drain)

This is the same equation we just solved with 60 gallons in the place
of Size.

Notice that we can factor out Size from the denominator and then
divide it by the Size in the numerator to get 1.

Size                       1
---- x ---------------------------------------  = Time to fill
Size   (1/20 mins) + (1/30 mins) - (1/15 mins)

Using 60 as the common denominator.

1                            1
1x---------------------------------------  =  ------------- = 60 mins
(3/60 mins) + (2/60 mins) - (4/60 mins)     (1/60 mins)

Notice how the value Size got factored out of the problem - this
demonstrates that what I asked you to take my word for is in fact
true: that size doesn't matter in this problem. Any value you choose
to use for Size, from a thimble to the Grand Canyon, produces the same

Hope this helps,

-Doctor Ezick,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Word Problems

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search