SOLUTION: Hi Bob went up a mountain at 5km per hour and down the mountain at 7.5 km per hour. What was the average speed up and down the mountain. Thanks

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Question 1183833: Hi
Bob went up a mountain at 5km per hour and down the mountain at 7.5 km per hour.
What was the average speed up and down the mountain.
Thanks
Found 3 solutions by josgarithmetic, greenestamps, ikleyn:
Answer by josgarithmetic(39613) About Me  (Show Source):
You can put this solution on YOUR website!
Whole trip, d
Up the mountain, d/2
Down the mountain, d/2

Time going up, %28d%2F2%29%281%2F5%29
Time going down, %28d%2F2%29%281%2F7.5%29
Whole time, d%2F10%2Bd%2F15=3d%2F30%2B2d%2F30=5d%2F30=d%2F6

Average speed for whole trip
d%2F%28d%2F6%29
highlight%286%29kilometers per hour

Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


First solution method:

As shown by the other tutor, using d for the distance each way.

Second solution method -- easier for many students, because no variables are involved.

Choose a "nice" number for the distance; then solve in the same way as when using the variable d. Since the two speeds are 5 and 7.5km/hr, let the distance each way be 15km. Then

time up the mountain = 15/5 = 3 hours
time down = 15/7.5 = 2 hours

Total distance 30km; total time 5 hours; average speed = total distance divided by total time = 30/5 = 6km/hr.

Third solution method -- harder to understand; but faster if you understand how to use it.

The ratio of speeds is 5:7.5 = 2:3; since the distances are the same, that means the ratio of times at the two speeds is 3:2.

So 3/5 of the time he is traveling at 5km/hr and 2/5 of the time he is traveling at 7.5km/hr. That makes the average speed

%283%2F5%29%285%29%2B%282%2F5%29%287.5%29+=+15%2F5%2B15%2F5+=+3%2B3+=+6

Answer by ikleyn(52750) About Me  (Show Source):
You can put this solution on YOUR website!
.

When half of the distance is covered with the average speed "u",  

and the other half of the distance is covered with the average speed "v",


then the average speed covering the total distance is 


    w = %282%2Au%2Av%29%2F%28u+%2B+v%29.      (1)


In your case,  u = 5 km/h;  v = 7.5 km/h;  hence, the average speed covering the total distance is  


    w = %282%2A5%2A7.5%29%2F%285%2B7.5%29 = 75%2F12.6 = 6 kilometers per hour.     ANSWER


So, to solve the problem, you can start from the first principles and derive the formula, as other tutors 

did it in their posts.   It is how the beginner students do.


The other way is to apply this ready to use formula (1)  and answer the question quickly.


Good student should know both methods.

Solved.

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    - Calculating an average speed: a train going from A to B and back
    - One more problem on calculating an average speed
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