Question 644105
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How to do it in your head:

The LCM of 7 and 3 is 21.  If they worked together for 21 hours, 
Sam would finish 3 projects while Peter would finish 7 projects. 
So in 21 hours they would finish 10 projects.  So together they 
could finish just 1 project in just 1/10th of that 21 hours or 
2.1 hours, or 2 hours, 6 minutes.

How to do it with algebra:

Let x be the answer.

Make this chart

                  Projects     No. of hours       rate in 
                  finished      required        projects/hour         
Sam alone
Peter alone
Both together

We put the hours required in each case and 1 for the number of
projects completed, which is 1 in all three cases.

                  Projects     No. of hours       rate in 
                  finished      required        projects/hour         
Sam alone            1              7     
Peter alone          1              3
Both together        1              x

Fill in the rates by dividing projects finished by hours:

                  Projects     No. of hours       rate in 
                  finished      required        projects/hour         
Sam alone            1              7               1/7   
Peter alone          1              3               1/3 
Both together        1              x               1/x

The equation comes from 

                     {{{(matrix(4,1,

                     "Sam's", rate, working, alone))}}} + {{{(matrix(4,1,

                     "Peter's", rate, working, alone))}}} = {{{(matrix(5,1,

                     their,combined, rate, working, together))}}}

                                 {{{1/7}}} + {{{1/3}}} = {{{1/x}}}

Multilply through by LCD = 21x

                                3x + 7x = 21
                                    10x = 21
                                      x = 2.1 hours, or 2 hours 6 minutes  


Edwin</pre>