Answer: Choice B) 3 miles
You have the correct answer. Nice work.
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Explanation:
Let's say Phil passes his brother at a sign post, and let's say the sign post is x miles into the 6 mile hike. This means 0 < x < 6.
Once Bill reaches the sign post, he turns around to walk back x miles. His faster brother Phil bikes the remaining 6-x miles to the trail end, before coming back to bike an additional 6 miles to get to the trailhead again. Phil arrives back at the same moment Bill does.
Overall, Phil travels 6-x+6 = 12-x miles in the same amount of time Bill walks back x miles. I recommend drawing out a number line diagram to see why this is the case.
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For Bill, we have this equation
distance = rate*time
x = r*t
r = x/t
Phil has the equation
distance = rate*time
12-x = (3r)*t
Notice that Phil's rate is three times that of Bill. The t is the same for both brothers because they travel in the same timespan.
Plug in r = x/t and solve for x. The t variables divide and cancel.
12-x = 3*(r)*t
12-x = 3*(x/t)*t
12-x = 3x
12 = 3x+x
4x = 12
x = 12/4
x = 3
Phil passed Bill at the 3 mile marker.
This is why the answer is choice B.
