Uniform Motion, Work, and Problem Solving

Question

Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?

Answer 1

r is the slower time

t is the faster time

12 = (R+2)*t <--- asphalt 2 mph faster

12 = R*(t+1) <--- gravel took 1 hour longer

(r+2)*t = r * (t+1)

(r+2)/r = (t+1)/t

1 + 2/r = 1 + 1/t

2/r = 1/t

2t = r

12 = (r+2)*t

12 = (2t+2)*t

12 = 2t^2 + 2t

12 = 2(t^2 + 24)

6 = t^2 + t

0 = t^2 + t - 6

0 = (t + 3)( t - 2)

t+3 results in negative measures

t-2 = 0 ---> t = 2

2 hours on asphalt . that's 12/2 = 6 mph

3 hours on gravel. Thats 12/3 = 4 mph.

Notice the difference in the speed is 2 mph

as given in the problem, so it checks.

6 mph on asphalt for 2 hours

4 mph on gravel for 3 hours