Question 895935
It takes you 8 hours to dig a hole.
<pre>
Your rate is 1 hole per 8 hours or 

{{{matrix(1,2,1,hole)/matrix(1,2,8,hours)}}} = {{{matrix(1,2,1/8,hole/hour)}}}
</pre>
if your brother were to dig the hole by himself, how long would it take him?
<pre>
Let the answer be x hours.
Your brother's rate is 1 hole per x hours or 

{{{matrix(1,2,1,hole)/matrix(1,2,x,hours)}}} = {{{matrix(1,2,1/x,hole/hour)}}}
</pre>
it takes you and your brother 5 hours to dig that same hole together.
<pre>

Then your combined rate is 1 hole per 5 hours or 

{{{matrix(1,2,1,hole)/matrix(1,2,5,hours)}}} = {{{matrix(1,2,1/5,hole/hour)}}}

The equation comes from:

{{{(matrix(5,1,Your,hole-digging,rate, in, "hole/hour"))}}}{{{""+""}}}{{{(matrix(6,1,Your, "brother's", hole-digging, rate, in, "hole/hour"))}}} {{{""=""}}} {{{(matrix(6,1,Your,combined, hole-digging, rate, in, "hole/hour"))}}}

So

   {{{matrix(1,2,1/8,hole/hour)}}}{{{""+""}}}{{{matrix(1,2,1/x,hole/hour)}}} {{{""=""}}} {{{matrix(1,2,1/5,hole/hour)}}}

      {{{1/8}}}{{{""+""}}}{{{1/x}}} {{{""=""}}} {{{1/5}}}

Multiply through by LCD = 40x


      {{{5x}}}{{{""+""}}}{{{40}}} {{{""=""}}} {{{8x}}}
             
            {{{-3x}}}{{{""=""}}}{{{-40}}}

             {{{x}}}{{{""=""}}}{{{(-40)/(-3)}}}

             {{{x}}}{{{""=""}}}{{{40/3}}}

             {{{x}}}{{{""=""}}}{{{13&1/3}}}{{{hours}}}

Edwin</pre>