Alan works 6miles away from the apartment and rollerblades from his work place to the apartment at a constant rate of 7miles per hour . Soren works 8 miles away from the apartment and bikes from his work place at a constant rate of 12miles per hour . How much time in hours do Alan and Soren have to travel to be the same distance from their apartment ?

Let the distance remaining after time t be x miles.

Distance traveled by Alan = (6-x) miles
Distance traveled by Soren = (8-x) miles

Time takes by the two is the same. Therefore,
(8-x)/12 = (6-x)/7 => 7(8-x) = 12(6-x) => 56-7x = 72-12x => 12x-7x = 72 -56 => 5x= 16 => x = 16/5 = 3.2 miles.

And then, time taken (t) = (8-3.2)/12 = (6-3.2)/7 = 0.4 hours