<PRE><B>Question 296406</B>
Joe runs up a mountain trail 4 miles long and then returns on the same trail downhill.
 Uphill he goes one mile per hour faster than half his downhill rate.
 If the round trip is one hour and forty minutes, then what is his downhill rate?
:
Change 1 hr 40 min to: {{{5/3}}} hrs
:
Let r = his downhill rate
then
(.5r+1) = his uphill rated
:
Write a time equation: Time = distance/rate
:
Time up + time down = 1{{{2/3}}} hrs
{{{4/((.5r+1))}}} + {{{4/r}}} = {{{5/3}}}
multiply equation by 3r(.5r+1), results
4(3r) + 4*3(.5r+1) = 5r(.5r+1)
:
12r + 12(.5r+1) = 2.5r^2 + 5r
:
12r + 6r + 12 = 2.5r^2 + 5r
:
18r + 12 = 2.5r^2 + 5r
:
Arrange as a quadratic equation
0 = 2.5r^2 + 5r - 18r - 12 
2.5r^2 - 13r - 12 = 0
:
multiply by 2 to get rid of the decimal
5r^2 - 26r - 24 = 0
:
Factors to
(5r + 4)(r - 6) = 0
:
Positive solution
r = 6 mph is the downhill rate
:
:
Check the solution: .5(6) + 1 = 4 mph is the uphill rate
Times
4/4 + 4/6 = 10/6 = 5/3 hr</PRE>