On an uphill hike, Ted climbs at a rate of 3 miles an hour. Going down, he runs at a rate of 5 miles an hour. If it takes him 40 minutes longer to climb up than run down, what is the total length of Ted's hike?

Let the total length of Ted's hike be L and the total time spent be T.

The time spent going down is t, and the time spent going up is t+40 (if t is measured in minutes) or t + 2 hr/3 (if t is measured in hours).  Note that t + t + 2/3 must equal T, the total hiking time, with all measurerments in hours.

Distance uphill:  L = (3 mph)(t+2/3) = Distance downhill:  (5 mph)(t)

We need only find t, the am't of time req'd for Ted to go up to down.

3t + 2 = 5t, or 2 = 2t, or t = 1 (hour)

It will take Ted 1 hour to descend the hill and 1 2/3 hour to ascend the hill.

The total length of Ted's hike was then 2 2/3 hours, or 2 hours 40 minutes.

By the way, the distance in each direction is (5 mph)(1 hr) = 5 miles.