So you have defined two different functions. You have

and

For these functions to be compatable they must agree on where you define the top and the bottom.

If you defince the top to 1 and the bottom to be 0 then you must have these values

and

So you get the two functions

Now if you define their difference it will only exist on the intersection of their domains!

But notice that

Since f is continous the intermediate value theorem says there exists a such that

So we get that

So the hiker was at the same place at the same time.

Note: Since the function is linear would could solve for the value of t explicitly, but the power of the IVT is we don't have to be able to find it.