Question 394346
A swimming pool can be filled by a large pipe in 4 hours and by a small pipe in 6 hours. A third pipe can empty the pool in 3 hours. How long would it take to fill the pool if all three pipes were open at the same time? Can you help me to get to the starting equation?
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Make this chart:

              number of        number of       rate in
            pools filled         hours        pools/hour
Large pipe                      
Small pipe        
Third pipe     
All three         

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Let x = the time it would take to fill the pool if all three were open. So
put x for the number of hours for all three, and since we want to know how
long it would take to fill 1 pool, we put 1 for the number of pools filled.


              number of        number of       rate in
            pools filled         hours        pools/hour
Large pipe        
Small pipe        
Third pipe       
All three         1                x             

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>>...A swimming pool can be filled by a large pipe in 4 hours...<<

Since it would take the large pipe 4 hours to fill 1 pool, put 1 for
the large pipe's number of pools and 4 for its number of hours.

              number of        number of       rate in
            pools filled         hours        pools/hour
Large pipe        1                4            
Small pipe        
Third pipe       
All three         1                x             

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>>...and by a small pipe in 6 hours...<<

Since it would take the small pipe 6 hours to fill 1 pool, put 1 for
the small pipe's number of pools and 6 for its number of hours.

              number of        number of       rate in
            pools filled         hours        pools/hour
Large pipe        1                4             
Small pipe        1                6             
Third pipe                  
All three         1                x             

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>>...A third pipe can empty the pool in 3 hours...<<

Since it would take the third pipe 3 hours to EMPTY 1 pool, put -1 for
the small pipe's number of pools, because that is a LOSS of one pool! 
and put 3 for its number of hours to "fill negative one pool" which means
to EMPTY 1 pool.


              number of        number of       rate in
            pools filled         hours        pools/hour
Large pipe        1                4            
Small pipe        1                6             
Third pipe       -1                3            
All three         1                x             

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Next we put in the rates in pools/hour by dividing the number of
pools by the number of hours:


              number of        number of       rate in
            pools filled         hours        pools/hour
Large pipe        1                4             1/4
Small pipe        1                6             1/6
Third pipe       -1                3            -1/3
All three         1                x             1/x

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Now the equation comes from:

      The sum of the three rates = their combined rate:
          
                {{{1/4+1/6-1/3=1/x}}}

Solve that and get x = 12 hours.  Hint: multiply through by LCD of 12x
to clear of fractions.

Edwin</pre>