Distance-time problem?

When you reverse the direction,

uphill becomes downhill, downhill becomes uphill, and flat stays flat.

Suppose when going from x to y we have

u miles of uphill,

f miles of flat

d miles of downhill

(and the reverse coming back).

x to y: u / 56 + f / 63 + d / 72 = 4 hours

y to x: u / 72 + f / 63 + d / 56 = 5 hours

Subtracting top from bottom:

u/72 - u/56 + d/56 - d/72 = 1

1/72 - 1/56 = 1/8*9 - 1/7*8 = -2/7*8*9

-2/7*8*9 u + 2/7*8*9 d = 1

-2u + 2 d = 7*8*9 = 504

2 (d - u) = 504

d - u = 252

And we want

u/56 + f/63 + d/72 = 4

u/72 + f/63 + d/56 = 5

Now suppose u = 0.

Then we get d = 252

x to y takes: 0/56 + f/63 + 252/72 = 3.5 + f/63 hours

f/63 = 0.5 hour (to bring total to 4)

f = 31.5

and total distance = 252 = 283.5

For the return journey we have

0/72 + 0.5 + 252/56 = 0.5 + 4.5 = 5 hours, as required.

Now we can repeat this calculation for any value of u

that we like up to and including 15.75,

which increases d as well, but reduces f.

When u = 15.75, f becomes 0, so that is the maximum value for u.

(and d = 267.75, and we have

15.75 / 56 + 267.75 / 72 = .28125 hours uphill + 3.71875 hours downhill, total 4

and 15.75 / 72 + 267.75 / 72 = .21875 hours downhill + 4.78125 hours uphill, total 5,

on the reverse trip).

But in every case,

u + f + d = 283.5 miles, and that is the answer to the problem.