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Crossing trains speed and length ratio

CAT 2021 Slot 2 · Quantitative Ability · Hard · Speed & Time

This is a hard Quantitative Ability question from the CAT 2021 Slot 2 paper. It tests Speed & Time. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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Two trains A and B were moving in opposite directions, their speeds being in the ratio 5:35 : 3. The front end of A crossed the rear end of B 46 seconds after the front ends of the trains had crossed each other. It took another 69 seconds for the rear ends of the trains to cross each other. The ratio of length of train A to that of train B is

  1. A.

    2 : 3

  2. B.

    2 : 1

  3. C.

    5 : 3

  4. D.

    3 : 2

Answer

D

Explanation

Let the speeds of trains A and B be 5s5s and 3s3s. Relative speed = 5s+3s=8s5s + 3s = 8s. Let LAL_A and LBL_B be the lengths of trains A and B respectively.

  1. Time from front ends crossing to front end of A crossing rear end of B: Distance covered in this time is LBL_B. LB=8s×46L_B = 8s \times 46

  2. Total time for the two trains to completely cross each other (rear ends crossing each other) is 46+69=11546 + 69 = 115 seconds. Total distance covered in complete crossing is LA+LBL_A + L_B. LA+LB=8s×115L_A + L_B = 8s \times 115

Thus: LA=(LA+LB)−LB=8s×(115−46)=8s×69L_A = (L_A + L_B) - L_B = 8s \times (115 - 46) = 8s \times 69

Ratio of lengths LA:LB=(8s×69):(8s×46)=69:46=3:2L_A : L_B = (8s \times 69) : (8s \times 46) = 69 : 46 = 3 : 2.

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