"The Hikers" Algebraic Problem

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mtberger1

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Oct 9, 2006
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Problem: Some hikers take a walk leaving at 9:00AM and returning at 3:00PM. They follow the same route out and back. If their walking speed is 3mph uphill, 6mph downhill, and 4 mph on flat land, HOW FAR did they walk?

When I solved this problem, I found that the hikers walked 26 miles. I got this answer by using the formula, Distance = Rate x Time. First, I found that the hikers walked 6 hours total as they were hiking from 9:00am to 3:00pm. Next, I averaged the walking speeds of 3mph uphill, 6mph downhill, and 4mph on flat land (4 1/3 avg.), for I wasn't given nor could calculate the amount of each type of terrain on the hike, and thus had to assume that it was equally divided among the three types of terrain. I then plugged my data in the D= R x T formula by using 6 hours as the time and4 1/3 as the rate. I solved for D (Distance) and found that the hikers walked 26 miles. IS THIS CORRECT?

Thanks for your help! :D
 
No, not correct.

Distance d travelled at x mph, then at 2x mph, does not result
in an average speed of 3x/2; average speed is 4x/3:
try 3mph over 18 miles, then 6mph over 18 miles: average = 4mph, not 4.5 mph.

So with your problem, pretty simple: average speed = 4mph, for 6 hours. OK?
 
thanks guys. Denis, could you please clarify how you calculated the average speed of the hikers? It would help me a great deal. Thanks. :)
 
I would just assume that it's all flat. It doesn't clarify the incline.
4*6 hrs=24 miles.
 
mtberger1 said:
...could you please clarify how you calculated the average speed of the hikers?
Click to expand...

A.......d miles, @ 6mph........B.......d miles, @ 3mph.......C

distance AB = distance BC = d

AB is travelled at 6mph, BC is travelled at 3mph;
["looks" like average speed is simple: (6 + 3) / 2 = 4.5mph]

BUT:

time A to B = d / 6
time B to C = d / 3
so:
time A to C = d / 6 + d / 3 = d / 2

So we have total distance 2d miles covered in d / 2 hours;
since speed = distance / time, then:
speed = 2d / (d / 2)
speed = 4d / d
speed = 4

Above could represent the downhill (6mph) going, then the uphill (3mph) returning;
SAME results achieved when doing the uphill going, downhill returning, since the
relative speeds are the same and the distance don't matter...avg. speed still 4mph.
 
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