15. Three villages are connected by paths as shown. From Downend to Uphill, the detour via Middleton is $1 \mathrm{~km}$ longer than the direct path. From Downend to Middleton, the detour via Uphill is $5 \mathrm{~km}$ longer than the direct path. From Uphill to Middleton, the detour via Downend is $7 \mathrm{~km}$ longer than the direct path. What is the length of the shortest of the three direct paths between the villages?

A $1 \mathrm{~km}$
B $2 \mathrm{~km}$
C $3 \mathrm{~km}$
D $4 \mathrm{~km}$
E $5 \mathrm{~km}$
Solution
C

Let the lengths of the direct paths from Uphill to Middleton, Middleton to Downend and Downend to Uphill be $x \mathrm{~km}, y \mathrm{~km}$ and $z \mathrm{~km}$ respectively. The information in the question tells us that $x+y=z+1, x+z=y+5$ and $y+z=x+7$. When we add these three equations, we obtain $2 x+2 y+2 z=z+y+x+13$ and hence $x+y+z=13$. Therefore $13=2 z+1$, $13=2 y+5$ and $13=2 x+7$, which have solutions $z=6, y=4$ and $x=3$. Hence the length of the shortest of the direct paths is the one from Uphill to Middleton with length $3 \mathrm{~km}$.