Help with distance word problem?

Question

"You commute 56 miles one way to work . The trip to work takes 10 minutes longer than the trip home. Your average speed on the trip home is 8mph faster. What is your average speed on the trip home?"

I'm having trouble figuring this one. Could anyone help me figure out how to do this?

Thanks. I really appreciate the help. I never thought of the 1/6 of an hour.

Answer 1

distance to work = 56
rate to work = x
time to work = 56/x

distance home = 56
rate home = x + 8
time home = 56/(x+8)

10 minutes = 10/60 hours

56/x = 56/(x + 8) + 10/60
56/x = 56/(x + 8) + 1/6
56(x + 8)(6) = 56(x)(6) + 1(x)(x + 8)
336(x + 8) = 336x + x(x + 8)
336x + 2688 = 336x + x^2 + 8x
0 = x^2 + 8x - 2688
0 = (x + 56)(x - 48)
x + 56 = 0 ..or.. x - 48 = 0
x = -56 (which makes no sense) ..or.. x = 48 (which does)

average rate on trip home = x + 8 = 48 + 8 = 56 mph

Answer 2

let x be the speed you travel on they way to work. And t be the time it takes to ho to work

The average speed on the way home is 8 mi/hr faster.
x + 8 is the speed you travel home.

10min = 1/6hr
Since we know it takes t hrs to go to work and the time 1/6hr longer than the trip home, then we t - 1/6 is the it takes you to go home

distance = speed * time

on the way to work:
56 = xt

on the way home
56 = (x + 8) (t - 1/6)

solve for t
t = 56/x

56 = (x + 8) (56/x - 1/6)

56 = 56 - x/6 + 448/x - 4/3

0 = -x/6 + 448/x - 4/3

0 = -x/6 + 448/x - 4/3

0 = -x^2/6 - 4x/3 + 448

use quadratic formula and you'll get x = 48

so the speed on the way home is 48 + 8 = 56 km/hr <= answer

Answer 3

_______________Out_____Back
Distance (miles)___56_____56
Speed (mph)______x______x + 8
Time (hours)______7t/6____t

x = 56 / (7t/6)
x = 48/t

t = 56/ (x + 8)
t = 56 / (48/t + 8)
t = 56t / (48 + 8t)
8t² + 48t = 56t
8t² - 8 t = 0
8t (t - 1) = 0
t = 0 , t = 1

x + 8 = 56 / 1
x = 48

Average speed on way home = 48 mph