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# Take A Hike

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A man goes out hiking.......on the first part of the trip over level ground, his rate is 4 miles per hour.......then he hikes uphill at 3 mph.

On the return trip......the "uphill" part is now the "downhill"  part and he is able to hike at 6 miles per hour.  Then, finishing up on the level part, he once again is able to maintain the 4 miles per hour pace, as before.

The question is.........how far did he hike if the entire trip took 5 hours......?????   Dec 12, 2015

#1
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Let x = miles on flat land, y = miles on the hill.  He goes out and back, so total miles = 2 (x+y).

The times spent on each segment is the distance travelled divided by the speed:

x/4mph + y/3mph + x/4mph + y/6mph = 5 hours

Multiply both sides by 12 mph to clear the denominator of the fractions; the numbers are now all in miles:

3x + 4y + 3x + 2y = 6x + 6y = 6(x+y) = 12*5 = 60.

So (x+y) = 60/6 = 10 miles ... so the total distance is 20 miles.

Dec 12, 2015

#1
+5

Let x = miles on flat land, y = miles on the hill.  He goes out and back, so total miles = 2 (x+y).

The times spent on each segment is the distance travelled divided by the speed:

x/4mph + y/3mph + x/4mph + y/6mph = 5 hours

Multiply both sides by 12 mph to clear the denominator of the fractions; the numbers are now all in miles:

3x + 4y + 3x + 2y = 6x + 6y = 6(x+y) = 12*5 = 60.

So (x+y) = 60/6 = 10 miles ... so the total distance is 20 miles.

Guest Dec 12, 2015
#2
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Very nice, Guest......I know two other ways of solving this [ there may be more].......but......I like the way you did it, maybe best of alll.....!!!!

I'll post my method in a while......it's a little different   Dec 12, 2015
edited by CPhill  Dec 12, 2015
#3
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Since he hikes on the level part of his trip at an average of 4mph, therefore it takes:

1/4 hour to hike 1 mile. And,

Since he hikes on the steep part of his trip at an average of 3mph, therefore it take:

1/3 hour to hike 1 mile. Then we have:

1/4+1/3=7/12 hours-combined average of the the two rates.

Since the total hike took 5 hours or 300 mintutes, therefore we have:

7/12 X 300=175 minutes-total time for the outgoing hike.But we have:

3/(3+4) X 175=75 minutes that would take him to hike the level part of trail. And, since he hikes on this part of trail at 4mph, therefore he must hike:

75/60 X 4=5 miles-the level part of the hike. Similar calculation will give:

100/60 X 3=5 miles-the steep part of his trail. So, naturally the total outging distance is:

5+5=10 miles X 2=20 miles the total hike distance back and forth. To summarize then:

On the outgoing hike he walks for:

1.25 hours at 4mph, and 1 2/3 hours at 3mph=10 miles in 2 hours and 55 minutes.

On the returning hike he walks for:

50 minutes or 5/6 hours at 6mph and 1.25 hours at 4mph=10 miles in 2 hours and 5 minutes.

This accounts for every part of his hike and the time and distance it took him to walk the whole trail.

Dec 13, 2015