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The question is about Circular races, combines with the idea of Number theory. CAT is known to test on multiple ideas in the same question, so you should be ready to face them. Time Speed and Distance is a favorite in CAT Exam, and appears more often than expected in the CAT Quantitative Aptitude section in the CAT Exam

Question 1: Two friends A and B simultaneously start running around a circular track . They run in the same direction. A travels at 6 m/s and B runs at b m/s. If they cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take?

  1. 3
  2. 4
  3. 7
  4. 5

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Explanatory Answer

Method of solving this CAT Question from Races: Understanding of relative speed would help.

Let track length be equal to T.
Time taken to meet for the first time = \\frac{T}{relative speed}\\) = \\frac{T}{6-b}\\) or \\frac{T}{b-6}\\)
Time taken for a lap for A = \\frac{T}{6}\\)
Time taken for a lap for B = \\frac{T}{b}\\)
So, time taken to meet for the first time at the starting point = LCM (\\frac{T}{6}\\),\\frac{T}{b}\\)) = \\frac{T}{HCF (6,b)}\\)
Number of meeting points on the track = Time taken to meet at starting point/Time taken for first meeting = Relative speed / HCF (6,b).
So, in essence we have to find values for b such that \\frac{6 - b}{HCF(6, b)}\\) = 2 or \\frac{b - 6}{HCF(6, b)}\\) = 2

The question is " If two people cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take?"
b = 2, 10, 18 satisfy this equation. So, there are three different values that b can take.

Hence, the answer is 3.

Choice A is the correct answer.


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