KCC's quiz AQQ274 about 2 sportsmen running and walking

Lets take the 2ndguy as basis. t is the time for the his first part. w=walking speed, r=running speed.
The distance would be t*w+t*r. Time needed 2*t. 

The 1st guy is travelling half distances for each action. 
First part  needs the time. (t*w+t*r) / ( 2*w ), second part needs the time (t*w+t*r) / (2*r)
Total time for 1st Guy then  t/2 + t*r/2*w + t*w/2 + t/2 = t + [ (t/2)* (r^2+w^2) / (r*w) ]

If r=w last portion of the equation would be 2. --> Then t+t/2 *2 = 2t. So first guy would also need 2*t as 2nd guy.

But normally r>w (running vs walking), and r>0 and w>0, so equation for (r^2+w^2) /(r*w)  will be always bigger than 2.  (1)
So 1st guy would need t+ t/2*(>2) more than 2*t he would need. 

Result: Second guy will be quicker. First walked and then run for same time.

(1) (r^2+w^2) /(r*w)  > 2 --> (r^2+w^2) > 2 (r*w)  --> (r-w)^2 > 0.... Correct. 

Edit: Readibility and addition of r>0, w>0. 

Lets take the 2ndguy as basis. t is the time for the his first part. w=walking speed, r=running speed.
The distance would be t*w+t*r. Time needed 2*t. 

The 1st guy is travelling half distances for each action. 
First part  needs the time. (t*w+t*r) / ( 2*w ), second part needs the time (t*w+t*r) / (2*r)
Total time for 1st Guy then  t/2 + t*r/2*w + t*w/2 + t/2 = t + [ (t/2)* (r^2+w^2) / (r*w) ]

If r=w last portion of the equation would be 2. --> Then t+t/2 *2 = 2t. So first guy would also need 2*t as 2nd guy.

But normally r>w (running vs walking), and r>0 and w>0, so equation for (r^2+w^2) /(r*w)  will be always bigger than 2.  (1)
So 1st guy would need t+ t/2*(>2) more than 2*t he would need. 

Result: Second guy will be quicker. First walked and then run for same time.

(1) (r^2+w^2) /(r*w)  > 2 --> (r^2+w^2) > 2 (r*w)  --> (r-w)^2 > 0.... Correct. 

Edit: Readibility and addition of r>0, w>0. 

Bravo hunl ! Correct answer and you use a solid development to get the result!