On an up hill hike ted climbs at 3 mph. going back down, he runs at 5 mph. if it takes him forty minutes longer to climb up than run down, then what is the length of the hike?

The main formula is:

Distance traveled = (average)Speed*Time.

i) when Ted Climbs up:

let D be the distance, T the time it takes Ted to climb, S_u=3mph is the speed as Ted climbs up.

then:   D=3T

ii) when Ted goes down:

D is the distance, T-40 the time it takes Ted go down, S_d=5mph is the speed as Ted climbs up.

then:   D=5(T-40)

equalizing the equations in i and ii:

(D=)       3T=5(T-40)
              3T=5T-200
              200=2T
              T=100     (minutes)

the total length of the hike is T+T-40=100+100-40=160 (minutes)

160 minutes

Equatorial triangle and obtuce angle