Mae travels to a house & she fell asleep 1/2 of the way there. She awakes & still has 1/2 the distance she slept to get there.What part...

Okay, the first thing to think about is this: at the point
that she goes to sleep, how much of the journey is left.  The answer of course is
1/2.

So now we look at the remaining half of the journey. 
Let us divide it up into the part she sleeps and the part that still remains.  We should
divide that into three parts because the part she sleeps is twice as much as the part
that still remains.  So of those three equal parts of the second half, she sleeps 2 and
has one remaining.

This means that she sleeps 2/3 of the
last 1/2 of the journey.  If you multiply 2/3 by 1/2, you get 1/3 of the entire
journey.

To do this in numbers rather than
words:

Let T = time awake.

We
know that 2t is time asleep because she slept twice as long as she was
awake.

We know that 2t + t = 1/2 of the whole
journey.

So now 3t =
1/2

Divide both sides by 3
and

t = 1/6

Now we know that
2t was how long she slept.  If t = 1/6, 2t = 2/6, which is
1/3.

So the time she slept, which we are calling 2t is 1/3
of the whole journey.