Variation on Combined Work

Date: 06/13/2002 at 01:45:33
From: Vibhuti Dhand
Subject: word problems in linear equations in two variables

It takes 12 hours to fill a swimming pool using two pipes. If the
larger pipe is used for 4 hours and the smaller for 9 hours, only half
of the pool is filled. 

How long would it take for each pipe alone to fill the pool?

Date: 06/13/2002 at 13:18:07
From: Doctor Ian
Subject: Re: word problems in linear equations in two variables

Hi Vibhuti,

It sounds to me like you're saying:

  A large pipe can fill 1/2 a pool in 4 hours. 

  A small pipe can fill 1/2 the same pool in 9 hours.

  Together, the pipes can fill the pool in 12 hours. 

But this can't be right, because then the answers are trivial:  
The large pipe can fill the pool in 8 hours, and the small one 
can fill it in 18.  

Or are you saying:

  If you run a large pipe for 4 hours, and run a small pipe
  for 9 hours, you can fill 1/2 a pool.  

  If you run both pipes together for 12 hours, you can 
  fill the whole pool. 

Is that correct? 

If so, let's say that the pool holds G gallons, that the large pipe
contributes R gallons per hour, and the small pipe contributes r
gallons per hour.  Then

     4R + 9r = G/2           (First condition)

    12(R + r) = G            (Second condition)

Doubling the first equation gives us 

     8R + 18r = G

    12(R + r) = G

Two things that are equal to the same thing are equal to each 
other, so 

  8R + 18r = 12(R + r)

So you can use this to find out the ratio of the rates for the 
two pipes, 

  R = kr
  
Then you can substitute that back into the equation

  12(kr + r) = G

to find out how long the smaller pipe needs to fill the whole 
pool; which will tell you how long it takes the larger pipe to 
fill the whole pool. 

Can you take it from here? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/