When they run in one direction, they will meet for the first time when the faster runner will cover 
the distance in one circumference longer that the slower runner.


So, let assume that "u" is the rate running of Albert and "v" is the rate of Bob,
and let assume that Albert is faster than Bob.


Then the criterion :first time meet after start" is this equation


    u*t - v*t = 600  meters,  or

    (u - v)*12 = 600,   which implies

    u - v = 600/12 = 50 meter per minute.



When they run in opposite direction, the criterion to meet is


    u*t + v*t = 600 meters,  or

    (u + v)*4 = 600,  which implies

     u + v = 600/4 = 150  meters per second.


Thus you have these two equations to find unknowns "u" and "v"

    u + v = 150     (1)

    u - v =  50     (2)


To solve the system, add the equations. You will get  

    2u = 150 + 50 = 200,  u = 200/2 = 100 m/sec.


Then from equation (1) you find  v = 150 - u = 150 - 100 = 50 m/sec.


Thus the rates "u" and "v" are just found.


Then you make the last step to get the answer:


    time for Albert = 600/100 = 6 minuter per lap,  and

    time for Bob    = 600/50 = 12 minuter per lap.