This scenario is best illustrated, much easier to undertstand and to follow, using a diagram.

When the planes pass each other they're 80 miles apart
Twenty minutes after passing each other, the Northbound covered: , while the Southbound 
covered: 

Twenty minutes () after passimg each other, a right triangle is formed by the planes, with the shorter leg being
80 miles and the longer leg, . 
The distance between them (hypotenuse of the right triangle, so formed) is: 


Now, after passing each other and the planes are 300 miles apart, another right triangle is formed, with the shorter leg being 80 miles 
and the hypotenuse, 300 miles. The longer leg is then: 

However, the longer leg consists of the following segments:
1) 150T miles (the distance the Southbound travelled to get to 300 miles of the Northbound, with T being the 
                 time the Southbound took to travel from the point where they were 20 minutes after passimg each other, to the point
                 where they were 300 miles apart)
2) 50 miles    (the distance covered by the Southbound, 20 minutes after they passed each other)
3)  miles (the distance covered by the Northbound, 20 minutes after they passed each other)
4) 200T miles (the distance the Northbound travelled to get to 300 miles of the Southbound, with T being the time the Northbound took to 
                 travel from the point where they were 20 minutes after passimg each other, to the point where they were 300 miles apart)

This gives us: 
Finally, time they each took to travel from the point where they were 20 minutes after passimg each other, to the point where they 
were 300 miles apart, or 
OR
Time they each took to travel from the point where they passed each other, to the point where they were 300 miles
apart: 20 + 29.56686 = 49.56686 minutes