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CAT 2020 Slot 2 — QA Question 10

Relative SpeedEasy

A and B are two points on a straight line. Ram runs from A to B while Rahim runs from B to A. After crossing each other, Ram and Rahim reach their destinations in one minute and four minutes, respectively. If they start at the same time, then the ratio of Ram's speed to Rahim's speed is

Answer & solution

  • A

    √2

  • B

    2√2

  • C

    1/2

  • 2

Solution

Easy

Use the meeting-point relation: after meeting, each runner covers the other runner's pre-meeting distance. The standard result is that the time to meet equals the geometric mean of the two post-meeting times, which makes the speed ratio t2/t1\sqrt{t_2/t_1}.

Let them meet at point MM after time tt. After meeting, Ram covers BMBM in 11 min and Rahim covers AMAM in 44 min. Speeds: Ram =a=a, Rahim =b=b.

1

Set up the two distances. AMAM is covered by Ram in time tt and by Rahim in 44 min; BMBM by Rahim in tt and by Ram in 11 min.

AM=a⋅t=b⋅4(1)BM=b⋅t=a⋅1(2)\begin{aligned} &AM = a\cdot t = b\cdot 4 \quad (1)\\ &BM = b\cdot t = a\cdot 1 \quad (2) \end{aligned}
2

Find tt. Multiply (1) and (2) to cancel aa and bb.

(at)(bt)=(4b)(a)[(1)×(2)]⇒ ab t2=4ab⇒ t2=4⇒t=2 min\begin{aligned} &(a t)(b t) = (4b)(a) \quad\text{[(1)×(2)]}\\ &\Rightarrow\ ab\,t^2 = 4ab\\ &\Rightarrow\ t^2 = 4 \Rightarrow t = 2\ \text{min} \end{aligned}
3

Speed ratio. Substitute t=2t=2 into (1).

a⋅2=b⋅4⇒ ab=42=2\begin{aligned} &a\cdot 2 = b\cdot 4\\ &\Rightarrow\ \frac{a}{b} = \frac{4}{2} = 2 \end{aligned}
ab=2\frac{a}{b} = 2

Ratio of speeds =tRahimtRam=41=2=\sqrt{\dfrac{t_{\text{Rahim}}}{t_{\text{Ram}}}}=\sqrt{\dfrac{4}{1}}=2.

CAT 2020 Slot 2 QA Q10: A and B are two points on a straight line. Ram runs from A to B while Rahim runs from B to A. After crossing e — Solution | TheCATExam