CaptainLevi wrote:
P started running around a circular track of length 800 m, from the starting point A. When P reached the 600 m mark, Q started running around the track from A, in the same direction as P. When Q reached the 400 m mark, R started from A, in the same direction as Q. When R reached the 200 m mark, S started from A, in the same direction as R. If when P reached A for the first time, Q , R and S also reached A for the first time, what is the ratio of the speeds of P, Q, R and S?
(A) 3 : 12 : 24 : 32
(B) 1 : 2 : 3 : 4
(C) 1 : 4 : 8 : 16
(D) 1 : 4 : 16 : 64
(E) 1 : 4 : 18 : 39
First and foremost thing to observe that P is the slowest and S is the fastest among the four.
Let's divide the track in four parts and denote the points as A(0), B(200), (400), D(600) and E(800) where the numbers in bracket denote distance of that point from the start point A.
Now, When P reaches point D it needs to cover 200m to reach point A and Q starts from point A i.e. it needs to cover 800m to reach point A again. After Q starts, they take equal time to reach point A. Thus, \(\frac{200}{Sp} = \frac{800}{Sq}\) where Sp and Sq are the constant speeds of P and Q respectively.
\(\frac{Sp}{Sq} = \frac{1}{4}\) -
ISimilarly, When Q reaches point C it needs to cover 400m to reach point A and R starts from point A i.e. it needs to cover 800m to reach point A again. Thus, \(\frac{400}{Sq} = \frac{800}{Sr}\) where Sq and Sr are the constant speeds of Q and R respectively.
\(\frac{Sq}{Sr} = \frac{1}{2}\) -
IIWhen R reaches point B it needs to cover 600m to reach point A and S starts from point A i.e. it needs to cover 800m to reach point A again. Thus, \(\frac{600}{Sr} = \frac{800}{Ss}\) where Sr and Ss are the constant speeds of R and S respectively.
\(\frac{Sr}{Ss} = \frac{3}{4}\) -
IIIFrom I and II
\(\frac{Sp}{Sq} * \frac{Sq}{Sr} = \frac{1}{4} * \frac{1}{2}\)
\(\frac{Sp}{Sr} = \frac{1}{8}\) -
IVFrom III and IV
\(\frac{Sp}{Sr} * \frac{Sr}{Ss} = \frac{3}{4} * \frac{1}{8}\)
\(\frac{Sp}{Ss} = \frac{3}{32}\) -
VSince Sp is 3 in
V multiply fractions of I and IV by \(\frac{1}{3}\) to get respective ratio.
Hence from I, IV and V
\(Sp:Sq:Sr:Ss = 3:12:24:32\)
Answer A.
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