Hiker's rate is 6 mph, which is 3 11 of the bicycle's rate. how fast do they move towards each other? how fast do they get closer if the bicyclist catching up with the hiker?

Let h and b represent the hiker's rate and bicycle's rate, respectively.
.. h = 3/11*b . . . . . . the hiker's rate d 3/11 of the bicycle's rate
.. 11/3*h = b . . . . . . multiply by the inverse of the coefficient of b
.. 11/3*6 mph = b . . . substitute the given information
.. 22 mph = b . . . . evaluate

a) If the hiker and bicycle are moving toward each other, the rate at which distance is being consumed is
.. h +b = 6 mph +22 mph = 28 mph

b) if the bicycle is catching up to the hiker, the rate at which the distance between them is being consumed is
.. b -h = 22 mph -6 mph = 16 mph