Need help with racing problem.

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Two runners are running on a track of length d. X runs the whole track with an average speed of s.
Y runs at a speed of s/2 for half the track and then runs at a speed of 2s for the rest of the way. Who will win and where is one when the other wins?

Let's let s = 50mph and let half of d = 50 miles
X is going 50mph and Y is going 25mph for half the track.
I can use the formula rate*time = distance.

50t=50, t=1 So it would take X 1 hour to reach 50 miles and so 2 hours to reach 100 miles.
25t=50, t=2, So Y did the first half in 2 hours then 100t=50, t=1/2, So it then took him a half hour to finish. So total time to finish for Y is 2 and a half hours. Which means X wins.

Does that make sense? I don't know how to answer second part of question.

 

[HallsofIvy]

Wow! Running at 50 mph is pretty good! Of course, that 100 mile track is also impressive.

But there is no need to put in specific numbers, which might restrict its application- how do you know that different numbers wouldn't give different results? Although trying specific numbers to get a "feel" for the problem, and then do it more generally is very good strategy.

Two runners are running on a track of length d. X runs the whole track with an average speed of s.

"speed= distance/time" s= d/t. Then st= d and t= d/s.

Y runs at a speed of s/2 for half the track and then runs at a speed of 2s for the rest of the way.

Y runs distance d/2 at speed s/2 and so takes time t= (d/2)/(s/2)= (d/2)(2/s)= d/s- the same time X took to run the entire distance.

Who will win and where is one when the other wins?

Well, that's pretty obvious isn't it? In the time, d/s, when X is just finishing the race, Y has just finished half the race.

 

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Thanks!