There are two methods, (1) in your head and (2) by algebra.

(1) In your head.

The LCM of 6 minutes and 8 minutes is 24 minutes.  Working together for 24 minutes, 
washing identical cars, the first guy would wash 4 cars and the second
guy would wash 3 cars.  That's 7 cars in 24 minutes or 1 car in 24/7 or  minutes.

(2)  By algebra.

Make this chart:
                             Number of         No. of      Rate in  
                           cars washed         Minutes     Cars/minute
1st guy washing alone
2nd guy washing alone
Both painting together

Let x be the number of minutes required for both to wash 1 car, so fill in
1 for the number of cars washed by both washing together and x for the
number of minutes.  Also fill in 1 car and 6 and 8 for the numbers of minutes
required for each to wash 1 car separately:

                             Number of         No. of      Rate in  
                           cars washed         Minutes     Cars/minute
1st guy washing alone           1                 6
2nd guy washing alone           1                 8
Both painting together          1                 x

Now fill in the Rates in cars/minute by dividing cars by minutes:

                             Number of         No. of      Rate in  
                           cars washed         Minutes     Cars/minute
1st guy washing alone           1                 6           1/6
2nd guy washing alone           1                 8           1/8
Both painting together          1                 x           1/x


                     The equation comes from

       +  = 

                     +  = 

 Multiply through by 24x
                  
                    4x + 3x = 24
                         7x = 24
                          x =  =  minutes.

Edwin