Distance Up and Down!

As we travel from X to Y, we have $a$ miles of uphill travel, $b$ miles of flat ground, and $c$ miles of downhill travel. Thus, the distance from X to Y can be expressed as $a + b + c$ .

With the information in the problem, we can derive two equations. $(1) \frac{a}{56} + \frac{b}{63} + \frac{c}{72} = 4$ $(2) \frac{a}{72} + \frac{b}{63} + \frac{c}{56} = 5$ Multiplying by the lowest common multiple of $56$ , $63$ , and $72$ , i.e. $504$ , we derive two further equations. $(3) 9a + 8b + 7c = 2016$ $(4) 7a + 8b + 9c = 2520$ Now, originally, I had a marvelously complicated method of calculating the solution, but I scrapped all of that when I remembered one simple thing: we were looking for $a + b + c$ . So, I just added equations $(3)$ and $(4)$ together to get $16a + 16b + 16c = 4536$ and divided by 16. $a + b + c = \boxed{283.5}$

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 = 283.5