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Joe rides his bike to his friend Jon's house and returns home by the same route. Joe rides his bike at constant speeds of 6 , on level ground, 4 mph when going uphill, and 12 mph when going downhill. If his total time riding was 1 hour, how far is it to Jon's house?

Mathematics
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my guess is 3 miles but i could be wrong
yes i believe 3 miles is the right answer
first off, since you are not told how many miles are uphill, downhill or flat, it cannot matter since it doesn't matter we can assume that it is all flat, and that it took him 1 hour going 6 mph to get there and back, so his total trip was 6 miles and half of it is 3

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Other answers:

suppose on the other hand that it is uphill all the way there. then it is downhill all the way back his travel time there is \(\frac{D}{4}\) and back it will be \(\frac{D}{12}\) and the total is 1 hour. solve \[\frac{D}{4}+\frac{D}{12}=1\] and you still get \(D=3\)
cute question though isn't it?
According to the textbook, it is 3 miles.
I'm not so sure about cute, though
yes that is what i got as well
point being that he averages 6 mph no matter what portion is up, down or flat
we can probably prove that without much difficulty
Just one thing. Why wouldn't you add the work rate of 6mph in? : \[\frac{ D }{ 4 }+\frac{ D }{ 6 }+\frac{ D }{ 12 }=1\]
what does D represent in your equation? in mine it is the distance form one house to the other
if it is all flat, it is a constant rate of 6 mile per hour, so the distance is 3 miles,
I thought of it as a work problem. So the rates would equal the work rates and the D would be time
oh wait.... D is the distance and the time is the 1
you lost me there this problem is about distance, rate and time i would put D = distance between the houses, because that is what you need to solve for
lets prove that no matter what part is flat, up or down, he travels at an average rate of 6 mph
clearly he averages 6 mph on the flat part, because that is what is says now suppose \(x\) miles are uphill going then returning those same \(x\) miles are downhill the rate uphill is \(4\) so the time uphill is \(\frac{x}{4}\) and the rate downhill is \(12\) so the time downhill is \(\frac{x}{12}\) the total time is for those parts are therefore \(\frac{x}{4}+\frac{x}{12}=\frac{x}{3}\) and the total distance travelled is \(2x\) therefor the average speed is \(\frac{2x}{\frac{x}{3}}=6\)
meaning no matter what portion is up, down or flat, his average rate is 6 mph but all this is really extra work since you are not told what portion is up down or flat, it makes no difference if it makes no difference, assume that it is all flat and you get 3 right away
I guess.
for sure, good trick to remember if you are taking a standardized test and have to answer quickly if you are not told a number you think you need make up one and work with that since we were not told what portion of the trip was flat, i made it all flat
lol.
Thanks.
yw

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