Question 359168
The road from Tedium to Excitement is uphill for 5 miles, level for 4 miles,
 and then downhill for 6 miles.
 John Mayer walks from Excitement to Tedium in 4 hours.
 Later he walks halfway from Tedium to Excitement and back again in 3 hours and 55 minutes.
 Finally he walks all the way to Excitement from Tedium in 3 hours and 52 minutes.
 What are his rates going uphill, downhill, and on level ground assuming that
 these rates remain constant?
:
Let u = uphill rate
Let f = flat rate
Let d = downhill rate
:
Write a time equation for each statement: Time = dist/rate
We are going to do it in minutes, change it to hrs later
:
"John Mayer walks from Excitement to Tedium in 4 hours."
{{{6/u}}} + {{{4/f}}} + {{{5/d}}} = 240
:
"he walks halfway from Tedium to Excitement and back again in 3 hours and 55 min."
{{{5/u}}} + {{{5/f}}} + {{{5/d}}} = 235 (halfway is 7.5 mi)
:
"he walks all the way to Excitement from Tedium in 3 hours and 52 minutes."
{{{5/u}}} + {{{4/f}}} + {{{6/d}}} = 232
:
 What are his rates going uphill, downhill, and on level ground assuming that these rates remain constant?
:
Use elimination on the 1st and 2nd equations
{{{6/u}}} + {{{4/f}}} + {{{5/d}}} = 240
{{{5/u}}} + {{{5/f}}} + {{{5/d}}} = 235
-----------------------------------------Subtraction eliminates d
{{{1/u}}} - {{{1/f}}} = 5
:
Multiply the 1st equation by 6 and the 3rd equation by 5
{{{30/u}}} + {{{30/f}}} + {{{30/d}}} = 1410
{{{25/u}}} + {{{20/f}}} + {{{30/d}}} = 1160
--------------------------------------------------Subtraction eliminates d again
{{{5/u}}} + {{{10/f}}} = 250
:
Multiply the 1st 2 unknown equation by 10, add to the above equation
{{{5/u}}} + {{{10/f}}} = 250
{{{10/u}}} -{{{10/f}}} =  50
------------------addition eliminates f, find u
{{{15/u}}} = 300
u = {{{15/300}}}
u = {{{1/20}}} miles per minute, that's {{{1/20}}} * 60 = 3 mph up hill
:
Use the equation: {{{1/u}}} - {{{1/f}}} = 5 to find f,(were dealing mi/min here)
{{{1/(1/20)}}} - {{{1/f}}} = 5
20 - {{{1/f}}} = 5
-{{{1/f}}} = 5 - 20
{{{-1/f}}} = -15
f = +{{{1/15}}} mi/min, that's {{{1/15}}} * 60 = 4 mph on the flat area
:
We can use the 1st equation, using hrs, to find d
{{{6/3}}} + {{{4/4}}} + {{{5/d}}} = 4
2 + 1 + 5/d = 4
5/d = 4 - 3
5/d = 1
d = 5 mph down hill
:
Summarize here
u = 3 mph uphill
f = 4 mph level
d = 5 mph down hill
:
:
See if this works int he original 3rd equation, using hrs
{{{5/3}}} + {{{4/4}}} + {{{6/5}}} = 3 hr 52 min
1.67 + 1 + 1.2 = 3.87 hrs which is 3 hrs, .87*60 = 52.2 min, close enough