KCC's quiz AQQ274 about 2 sportsmen running and walking

As walking typically is slower than running, the sportsman who walks half of the way spends the longer time walking, than the one, who walks half the time. Therefore, the sportsman who walks half of the time only, reaches the target faster. 

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. 

Assuming running is faster than walking, B is faster. Because running half the time you cover more distance than running half the distance, which is what A is doing. Then the distance left for walking is shorter than half the distance. Thus quicker. B arrives first.

Assuming running is faster than walking, then sportsman B will be first.
Reasoning: 
 - The less time you are wasting with walking the quicker you are.
 - Sportsman B is spending half of his time walking while
 - Sportsman A is wasting more than half of his time reaching half distance, thus he will be later

Take: w the walking speed, r the running speed, d the total distance, t1 the time taken byt the first sportsman and t2, taken by the second sportsman.

First sportsman:  t1 = (d/2)/w  + (d/2)/r     so  t1=(d/2) ( 1/w +1/r)  = (d/2) (w+r)/(wr)
Second sportman:  w(t2/2) + r(t2/2) = d    so  t2 =                              2d / (w+r)

t1/t2  =  1/4  (w+r)^2  / wr.  If that value is 1.0, they both reach the end at the same time.  Less that 1, t1 is less than t2 so the first sportsman arrives first, and if greate than 1, the second sportsman arrives first.



Clearly, that depends on the ratio of w/r.  If w/r  =1,  the expression leads us to conclude that they both arrive at the same time.
If we assume running is faster than walking, the expression leads us to conclude that the first sportsman arrives last.

Have  a = r/w, t1/t2=1/4  (1+a)^2  / a.  When equals to 1, we have   4a = (1+a)^2, leading to a = 1, that is, if running equals walking, both arrive at the same time.
If we assume running is faster than walking, as with a=2 for example, the expression lead us  9/8, so to conclude that the first sportsman arrives last.  (its time, t1, is larger than t2) 

The answer is: sportsman-B. 

You can demonstrate it by algebraically calculating t_A and t_B or graphically on a time-space diagram. However, you can observe that sportsman-A will certainly spend more than half of the time by walking, while sportsman-B will spend just half of the time by walking. Basically, to cover the same distance, sportsman-B will spend more time by running, so he will arrive first.

Let R equal running speed and W equal walking speed.

Time for A to travel distance D:  D/2W+D/2R = (D/2)[1/W+1/R] = D(W+R)/2WR

Time for B to travel distance D:  2D/(W+R)

Let W=kR where 0< k <1:  Walking speed less than running speed

Ta/Tb = [(W+R)^2]/4WR = [(k+1)^2]/4k

If Ta/Tb < 1 then k^2+2k+1<4k or (k-1)^2<0 :FALSE

If Ta/Tb > 1 then k^2+2k+1>4k or (k-1)^2>0 :TRUE

Therefore, Tb is less then Ta and sportsman B finishes first.

Assuming Running speed > Walking speed

Sportsman B can start running earlier than A.

A has to spend more than half his time to get to half distance, then can start running, spending less than half time running to get to the endpoint.

B covers less distance in the first part of his trip, but can start running in the second half of the time, covering more distance in that second half, arriving before A

B finishes first.

A walks half distance and runs half distance

B walks half of the TIME and runs half of the TIME

We can assume that running is faster than walking

Total time for A is proportional to 0.5/W +0.5/R, where W and R are the speed of walking and running.

B walks half of the TIME and runs half of the TIME

Given the assumption that running is faster than walking, we can see that B will run for more than half the distance, since the running TIME is the same as the walking TIME

Since B is running for more than half the distance and running is faster than walking, B will finish the trip first, since A will run only half the distance and therefore for less time.

Sportsman B is faster. He runs for half of the time, while A runs for less than half of the time, because his walking and running distances are equal, therefore time cannot be equal and must be lower for running.