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CAT 2021 Slot 2 — QA Question 9

TrainsEasy

Two trains A and B were moving in opposite directions, their speeds being in the ratio 5 : 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 

Answer & solution

  • A

    2 : 1

  • 3 : 2

  • C

    2 : 3

  • D

    5 : 3

Solution

Easy

The two "crossing" events both happen at the same relative speed a+ba+b. The first 4646 s covers train BB's length; the full 115115 s covers both lengths. Subtracting isolates train AA's length, and the relative speed cancels in the final ratio.

1

Front of A reaches rear of B (46 s). From when the front ends meet, the front of AA must travel past the whole of BB, i.e. length BB, at relative speed a+ba+b.

46=Ba+b⇒ B=46(a+b)…(1)\begin{aligned} &46=\tfrac{B}{a+b}\\ &\Rightarrow\ B=46(a+b) \quad\dots(1) \end{aligned}
2

Rear ends fully cross (46 + 69 = 115 s). From front ends meeting to rear ends clearing, the trains cover the combined length A+BA+B.

115=A+Ba+b⇒ A+B=115(a+b)…(2)\begin{aligned} &115=\tfrac{A+B}{a+b}\\ &\Rightarrow\ A+B=115(a+b) \quad\dots(2) \end{aligned}
3

Isolate AA. Subtract (1) from (2).

A=115(a+b)−46(a+b)=69(a+b)…(3)\begin{aligned} &A=115(a+b)-46(a+b)=69(a+b) \quad\dots(3) \end{aligned}
4

Take the ratio. Divide (3) by (1); (a+b)(a+b) cancels (the speed ratio 5:35:3 is not even needed).

AB=69(a+b)46(a+b)=6946=32\begin{aligned} &\frac{A}{B}=\frac{69(a+b)}{46(a+b)}=\frac{69}{46}=\frac{3}{2} \end{aligned}
A:B=3:2A:B=3:2
CAT 2021 Slot 2 QA Q9: Two trains A and B were moving in opposite directions, their speeds being in the ratio 5 : 3. The front end of — Solution | TheCATExam