## Grazes 3 years ago 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?

1. satellite73

my guess is 3 miles but i could be wrong

2. satellite73

yes i believe 3 miles is the right answer

3. satellite73

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

4. satellite73

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 $$D=3$$

5. satellite73

cute question though isn't it?

6. Grazes

According to the textbook, it is 3 miles.

7. Grazes

I'm not so sure about cute, though

8. satellite73

yes that is what i got as well

9. satellite73

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

10. satellite73

we can probably prove that without much difficulty

11. Grazes

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

12. satellite73

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

13. satellite73

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

14. Grazes

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

15. Grazes

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

16. satellite73

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

17. satellite73

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

18. satellite73

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 $$\frac{2x}{\frac{x}{3}}=6$$

19. satellite73

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

20. Grazes

I guess.

21. satellite73

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

22. Grazes

lol.

23. Grazes

Thanks.

24. satellite73

yw