Answer: One fourth of the entire trip.
Step-by-step explanation:
The initial distance is D.
" He fell asleep halfway home."
Then he fells asleep when the distance between his actual position and his house was half of D, or:
D/2.
"He didn't wake up until he still had half as far to go as he had already
gone while asleep."
So he wakes up when his actual position is a fourth of the initial distance:
(D/2)/2 = D/4.
Then if the entire trip has a distance D, and he was sleeping between:
D/2 - D/4 = 2D/4 - D/4 = D/4.
in a trip of a distance D, he was asleep a distance of D/4.
Then, returning to the question:
How much of the entire trip home was Samuel asleep?
This is equal to the quotient between the distance that he travels asleep and the total distance:
r = (D/4)/D = 1/4.
Then he was asleep in 1/4 of the entire trip.
