Four athletes P, Q, R, S are participating in a 800 m race. At the start P gives Q, R, S a lead of T seconds. P overtakes Q at the 160 m mark, R at the 200 m mark and S at the 400 m mark. P beats S by 20 sec.R completes the race in 3 minutes. By how many seconds does R beat Q?

Hi
Let velocity of P be Vp, Q be Vq, R be Vr and S be Vs and lead given by P to all of them be T seconds
Now R takes 180 seconds to complete the race
so his speed = 800/180 = 40/9 m/s
Now P beats R at 200 m mark
So we can say
200/Vp +T =200/Vr
Substituting Vr
we get
200/Vp +T =45 (1)
Now P overtakes S at 400 m mark
we get
400/Vp +T =400/Vs (2)
Now it is given that P beats S by 20 seconds
So as we know at 400m mark both P and S are together so in remaining 400 meters P covers those 400 meters 20 seconds faster than S
so we get 400/vp +20 =400/vs (3)
From (2) and (3)
we get T=20 seconds
Substituting in (1)
we get Vp= 8m/s
Now P beats Q at 160m mark
so we get
160/vp +T=160/Vq
we get 40=160/Vq
so Vq=4m/sec
So Time by Q to complete 800 m
800/4=200seconds
so R beat Q by 20 seconds