an airline passanger fell asleep halfway to her destination. When she woke up, the distance remaining was half the distance traveled while she was asleep. What portion of the trip was she asleep?
After falling asleep, half the trip was left. She slept 2/3 of the remaning half. 2/3 of 1/2 equals 1/3 of the entire trip spent sleeping.
"an airline passenger fell asleep *halfway* to her destination."
halfway= 1/2
Let's assume that the total distance of the trip is represented by "D."
When the passenger fell asleep, she had traveled half the distance, so the distance traveled before she fell asleep is D/2.
When she woke up, the remaining distance was half the distance traveled while she was asleep, which is (D/2)/2, or D/4.
Since she woke up halfway through the remaining distance, we can conclude that she was asleep for the other half.
Therefore, the portion of the trip that she was asleep for is 1/2, or 50%.
To find the portion of the trip that the passenger was asleep, we need to understand the relationship between the distance traveled while she was asleep and the total distance of the trip.
Let's assign variables to the important information:
Let's call the total distance of the trip "D".
Let's call the distance traveled while the passenger was asleep "A".
Let's call the distance remaining when she woke up "R".
According to the problem, the distance remaining when she woke up was half the distance traveled while she was asleep. This can be expressed as the equation: R = (1/2)A.
We also know that the total distance of the trip is the sum of the distance traveled while she was asleep and the distance remaining: D = A + R.
We can substitute the value of R from the first equation into the second equation:
D = A + (1/2)A
Combining like terms, we get:
D = (3/2)A
To find the portion of the trip the passenger was asleep, we can rearrange this equation to solve for A:
A = (2/3)D
Therefore, the passenger was asleep for (2/3) of the total trip. In other words, she was asleep for two-thirds of the journey.