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 of the journey did she sleep? The answer is 1/3. How are they getting 1/3?
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.
Let the distance Mae has to travel be equal to D. Now she fell asleep when she reached half of the way to the destination. When she woke up she has half the distance that she remained asleep left. Let's say she fell asleep for a distance D. Now when she woke up she has travelled a distance N/2 + D. The distance left for her to reach is D/2. So N/2 + D + D/2 = N.
=> D+ D/2 = N - N/2
=> (3/2) D = N/2
=> D= N/3
So she slept 1/3 of the way.
When she started sleeping it was exactly half the distance.
The remainig distance is exactly another half.
During the the duration of her sleep she traavelled a distance 2x which is twice the remaing distance x when she awoke.
Travelled distance in sleep +distance after she awoke = 2x+x = 1/2 the jounet distance.
3x = 1/2.
x = (1/2)/3 = 1/6.
Therefore the distance travelled in sleep = 2x = 2*(1/6) = 1/3 of the total journey distance.