Question 838067
<pre>
First we find their cistern-filling rates in cisterns per minuter.

Let the time required for the faster pipe to fill the cistern be x minutes
So the faster pipe's filling rate is 1 cistern per x minutes or
{{{1_cistern/x_minutes}}} or {{{1/x}}} cisterns per minute

Then the time required for the slower pipe to fill the cistern is x+72
minutes.  So the slower pipe's filling rate is 1 cistern per x+72 minutes 
or 
{{{(1_cistern)/(x+72_minutes)}}} or {{{1/(x+72)}}} cisterns per minute.

Since the faster pipe's rate is 7 times faster than the slower pipe's rate,
we have:

      {{{1/x}}} {{{""=""}}} {{{7}}}{{{""*""}}}{{{1/(x+72)}}}
      {{{1/x}}} {{{""=""}}} {{{7/(x+72)}}}
Cross-multiply:
      x + 72 = 7x
          72 = 6x
          12 = x

So the faster pipe's filling rate is {{{1/x}}} = {{{1/12}}} cistern per minute.
And the slower pipe's filling rate is {{{1/(x+72)}}} = {{{1/84}}} cistern per minute.

Now we have their respective filling rates in cisterns per minute.

Let t = the number of minutes required to fill the pipe when both pipes
are open.  So their combined filling rate is 1 cistern per t minutes 
or {{{1_cistern/t_minutes}}} or {{{1/t}}} cisterns per minute.

The equation comes from:

   {{{(matrix(5,1,

The, faster, "pipe's", filling, rate))}}}{{{""+""}}}{{{(matrix(5,1,

The, slower, "pipe's", filling, rate))}}}{{{""=""}}}{{{(matrix(4,1,

Their, combined, filling, rate))}}}

   {{{1/12}}}{{{""+""}}}{{{1/84}}}{{{""=""}}}{{{1/t}}}

Multiply through by 84t

    7t + t = 84
        8t = 84
         t = {{{84/8}}} = 10.5 minutes.

Edwin</pre>