Circular Tracks Race

CAT 2020 Slot 2 · QA · Medium · Arithmetic

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Two circular tracks T1 and T2 of radii 100 m100\text{ m} and 20 m20\text{ m}, respectively touch at a point A. Starting from A at the same time, Ram and Rahim are walking on track T1 and track T2 at speeds 15 km/hr15\text{ km/hr} and 5 km/hr5\text{ km/hr} respectively. The number of full rounds that Ram will make before he meets Rahim again for the first time is

  1. A.

    5

  2. B.

    3

  3. C.

    4

  4. D.

    2

Answer

B

Explanation

The tracks touch at point A. They can only meet again at point A because the two circles share only one common point A. Circumference of T1 = 2π(100)=200π m2\pi(100) = 200\pi\text{ m}. Circumference of T2 = 2π(20)=40π m2\pi(20) = 40\pi\text{ m}. Time taken by Ram to complete 1 round of T1: tRam=200π15t_{\text{Ram}} = \frac{200\pi}{15} Time taken by Rahim to complete 1 round of T2: tRahim=40π5=8πt_{\text{Rahim}} = \frac{40\pi}{5} = 8\pi They will meet at A after LCM of these times. Ratio of times tRamtRahim=200π/158π=200120=53\frac{t_{\text{Ram}}}{t_{\text{Rahim}}} = \frac{200\pi / 15}{8\pi} = \frac{200}{120} = \frac{5}{3}. So 3×tRam=5×tRahim3 \times t_{\text{Ram}} = 5 \times t_{\text{Rahim}}. Therefore, Ram completes 3 full rounds when Rahim completes 5 full rounds.

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