Question

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?

Answer 1

my guess is 3 miles but i could be wrong

Answer 2

yes i believe 3 miles is the right answer

Answer 3

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

Answer 4

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 \(D=3\)

Answer 5

cute question though isn't it?

Answer 6

According to the textbook, it is 3 miles.

Answer 7

I'm not so sure about cute, though

Answer 8

yes that is what i got as well

Answer 9

point being that he averages 6 mph no matter what portion is up, down or flat

Answer 10

we can probably prove that without much difficulty

Answer 11

Just one thing. Why wouldn't you add the work rate of 6mph in? :
\[\frac{ D }{ 4 }+\frac{ D }{ 6 }+\frac{ D }{ 12 }=1\]

Answer 12

what does D represent in your equation? in mine it is the distance form one house to the other

Answer 13

if it is all flat, it is a constant rate of 6 mile per hour, so the distance is 3 miles,

Answer 14

I thought of it as a work problem. So the rates would equal the work rates and the D would be time

Answer 15

oh wait.... D is the distance and the time is the 1

Answer 16

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

Answer 17

lets prove that no matter what part is flat, up or down, he travels at an average rate of 6 mph

Answer 18

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 \(\frac{2x}{\frac{x}{3}}=6\)

Answer 19

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

Answer 20

I guess.

Answer 21

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

Answer 22

lol.

Answer 23

Thanks.

Answer 24

yw