A bicycle travels up a hill at an average speed of 4m/s and down the same hill through the same distance at an average speed of 8m/s. Express the average speed for the entire journey in terms oftandd.

							5.333 m/s

Average speed is not as we usually do sum of speeds divide by total number of speeds
but Average speed is total number of distances divide by total time take 

let the distance cover by bicycle up a hill be   x 
since distance cover is same in down a hill then Total distance cover during ride is 2x

time taken by travel up a hill = x/4 s
time taken by travel down a hill = x/8 s
total time taken during ride is = x/4 + x/8 =3x/8

Average speed =total distance  / total time taken
                          = 2x / (3x/8)
                          = 16x / 3x = 5.333 sec

Alter method 
  Simple formula for finding Average speed if distances cover is same is 2uv / (u+v)
where u is speed in first ride and v is speed in second ride
Acoording to question 
Average speed = 2 x 4 x 8 / (4+8) = 5.333 sec
						

							5.33 sec
						

							Average speed is (total distances / total time taken) 

let the distance covered up a hill is x then distance covered down a hill is same to be x 
then total distances covered = 2x

time taken to travel up a hill = x/4 sec
time taken to travel down a hill = x/8 sec
total time taken during ride = (x/4) + (x/8) = 3x/8

Average speed = total distance / total time taken
                          = 2x / (3x/8) =  16x / 3x
                          = 5. 33 sec

Alter method
Simplest formula for average speed if distance cover is same is 2uv / (u+v) 
where u is speed in first ride and v is speed in second ride 
According to question 
u= 4 m/s
v= 8 m/s
putting the value of u and v in 2uv / (u+v) we get 2 x 4 x 8 / (4+8) = 5.33 sec