SOLUTION: there are three pipes that can fill a pool. pipe A can fill the pool in 8 hours. pipes A and C can fill the pool in 6 hours. pipes B and C can fill the pool in 10 hours, how long w

Therefore pipe A can fill 1 pool in 8 hours.  Therefore
A's pool-filling rate is 1 pool per 8 hours, which can be
written as  or .

Suppose pipe B can fill the pool in x hours.

Then pipe B can fill 1 pool in x hours.  Therefore
B's pool-filling rate is 1 pool per x hours, which can be
written as  or .

Suppose pipe C can fill the pool in y hours.

Then pipe C can fill 1 pool in y hours.  Therefore
C's pool-filling rate is 1 pool per y hours, which can be
written as  or . 

So A and C's combined pool-filling rate is 1 pool per 6 hours, 
which can be written as  or .

Now we use the fact that A and C's combined pool-filling rate
is the sum of their individual pool-filling rates to get this
equation:



So



Above we have C's pool-filling rate as ,
so we solve that for 

     
     
     

so C's pool-filling rate is ,

So B and C's combined pool-filling rate is 1 pool per 10 hours, 
which can be written as  or .

Now we use the fact that B and C's combined pool-filling rate
is the sum of their individual pool-filling rates to get this
equation:



So



Above we have B's pool-filling rate as ,
so we solve that for 



     
     
      

so B's pool-filling rate is .

Suppose all three pipes can fill the pool in z hours.

So the combined pool-filling rate of all three is 1 pool per z hours, 
which can be written as  or .  

Then the equation comes from





Multiply through by LCD = 120z

15z + 7z + 5z = 120
           27z = 120
             z = 
             z =  or  hour

or 4 hours, 26 minutes, 40 seconds.

Edwin