2440.13 – Driving to Nirvana

We were driving to Nirvana and I fell asleep. No, I was not driving, just eager to get there. So, when we were halfway there I fell asleep. When I woke up, the distance we still had to go was half as far as we went while I was asleep. For what fraction of the trip was I asleep?

Solution

Let $y$ be the whole trip and $x$ be the part that I slept through. Then the portion after I woke up is $\dfrac{x}{2}$ . The half of the trip after I went to sleep is $x + \dfrac{x}{2} = \dfrac{3x}{2}$ . Therefore the whole trip, being twice that half, is $3x$ .

Thus, $y = 3x$ so $x = \dfrac{y}{3}$ and it seems I was asleep for a third of the trip. Did I ever get to Nirvana? That is another story and will be told another time.