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distance vs time



James and Mike are running in a race.  They both walked and ran for part of the rate.  They each walked and ran at the same speed.  James ran for half the distance and walked for half the distance.  Mike ran for half of his time and walked for half of his time.  Who finished first?

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Mike. If Mike runs and walks for the same amount of time, he will have to run further than he walks. If they both start by running, James will stop running at the halfway (distance) point but Mike has to run past the halfway point to compensate for the slower walking speed.

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Mike finishes first

Both walk at rate w, and run at rate r.

Mike walks for half his time, runs for half the time

His distance is rt/2 + wt/2 = (r+w)t/2
And his total time was t

James is racing the same distance, half the distance is (r+w)t/4 , so his time running is distance / rate = (r+w)t/4r 

And his time walking is (r+w)t/4w

So his total time is (r+w)t/4r + (r+w)t/4w
= (4w+4R)(r+w)t/16rw 

James’ time is faster if t>(4w+4R)(r+w)t/16rw 
1 > (4w+4R)(r+w)/16rw 
1 > (4wr + 4w^2 + 4r^2 + 4wr) /16wr
1 > (1/2) + (w^2 + r^2)/4wr

2wr > w^2+ r^2

0 > w^2 -2wr + r^2 = (w-r)^2

But that’s not possible, unless w = r

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