Question 498679
Pipe A can fill a tank in 6 hours, and pipe B can fill it in 2 hours less time than it take drain pipe C to empty the tank. With all three pipes open it takes 3 hours 20 minutes to fill the tank. How long would it take pipe C to empty it
<pre>
Make this chart:

               Number of     Time to      Rate in
              tanks filled    fill       tanks/hours
Pipe A
Pipe B
Drain C
All three

The number of tanks filled is 1 for each of those except Drain C, and
we put -1 for the number of tanks it filled. We can fill in the times
for C and All three from the problem, and we get this:



               Number of     Time to      Rate in
              tanks filled    fill       tanks/hours
Pipe A            1            6  
Pipe B            1           
Drain C          -1            
All three         1           3 1/3


Let the time for C to drain be x.  Then B's time to fill is x-2.
Fill those in:

               Number of     Time to      Rate in
              tanks filled    fill       tanks/hours
Pipe A            1            6           
Pipe B            1           x-2         
Drain C          -1            x           
All three         1          3 1/3


Fill in the rates in tanks/hour by dividing the number of
tanks filled by the number of hours:
         

               Number of     Time to      Rate in
              tanks filled    fill       tanks/hours
Pipe A            1            6            1/6
Pipe B            1           x-2          1/(x-2)
Drain C          -1            x           -1/x
All three         1          3 1/3          {{{1/(3&1/3)}}}


{{{(matrix(4,1,
rate,
for,
pipe,
A))+(matrix(4,1,
rate,
for,
pipe,
B))+(matrix(4,1,
rate,
for,
drain,
C))=(matrix(5,1,
rate,
for,
all,
three,
together)) }}}

{{{1/6+1/(x-2)-1/x=1/(3&1/3)}}}

First we simplify the last term: {{{1/(3&1/3)=1/(10/3)=1*expr(3/10)=3/10}}}

{{{1/6+1/(x-2)-1/x=3/10)}}}

The LCD is 30x(x-2) so multiply through by that:

{{{5x(x-2)+30x-30(x-2)=9x(x-2)}}}

{{{5x^2-10x+30x-30x+60=9x^2-18x}}}

{{{5x^2-10x+60=9x^2-18x}}}

{{{0=4x^2+8x-60}}}

{{{0=x^2-2x-15}}}

{{{(x-5)(x+3)=0}}}

{{{x-5=0}}},  {{{x+3=0}}}
{{{x=5}}}     {{{x=-3}}}

Discard the negative time.

It would take drain C 5 hours to empty the tank. 

Edwin</pre>