An airline passenger feel asleep halfway to her destination. When she awoke, the distance remaining was half the distance traveled while she slept. How much of the entire trip was she asleep?
Forget the halfway for a minute.
From the point she fell asleep, she slept twice as long as the rest of the flight after waking. Does that smell like 2/3 + 1/3 to you?
Back to the halfway bit...
She slept 2/3 of the second half of the flight.
She slept 2/3 of half of the flight.
She slept 2/3 * 1/2 of the flight.
To solve this problem, let's break it down into smaller steps:
Step 1: Define the variables.
Let's represent the total distance traveled as 'd'.
Let's represent the distance remaining when she woke up as 'r'.
Let's represent the distance traveled while she was asleep as 'x'.
Step 2: Understand the given information.
According to the problem, when the passenger woke up, the distance remaining was half the distance traveled while she slept. In other words, 'r' is half of 'x'.
Step 3: Formulate the equation.
Since 'r' is half of 'x', we can write this relationship as:
r = (1/2)x
Step 4: Relate the variables to solve the problem.
The total distance traveled 'd' can be expressed as the sum of the distance she traveled while asleep 'x' and the distance remaining when she woke up 'r':
d = x + r
Step 5: Substitute the value of 'r' from the relationship in step 3 into the equation in step 4.
Substituting r = (1/2)x into the equation d = x + r, we get:
d = x + (1/2)x
Step 6: Simplify the equation.
Combining like terms, we have:
d = (3/2)x
Step 7: Solve for 'x'.
To find the distance she traveled while asleep 'x', we can rearrange the equation in step 6 to solve for 'x':
x = (2/3)d
Step 8: Calculate the percentage of the trip asleep.
To determine how much of the entire trip she was asleep, we need to find the ratio of 'x' (the distance traveled while asleep) to 'd' (the total distance traveled):
Percentage of trip asleep = (x / d) * 100
Step 9: Substitute the value of 'x' into the equation in step 8 and calculate.
Substituting x = (2/3)d into the equation, we get:
Percentage of trip asleep = ((2/3)d / d) * 100
= (2/3) * 100
= 66.67%
Therefore, the passenger was asleep for 66.67% of the entire trip.