Circular Races: Concepts, Tricks, and Problems for CAT Exam: Circular races have always been an interesting and challenging subject for students preparing for the CAT exam. These problems typically involve finding the number of Distinct Meeting Points or the time the runners will meet at the starting point for the first time.

To solve circular races problems, there are three fundamental concepts that we mainly use: ratios, LCM, and relative speeds. I will discuss these concepts in detail in this article and provide tricks and explanations for solving circular race problems that will help to understand the Circular Races Concepts Tricks and Problems for CAT Exam

There are two categories of questions in circular races:

finding the time required to meet at the starting point and determining the number of distinct meeting points.

To find the time taken to meet at the starting point on a circular track, follow these steps:

1. Calculate the time taken by each runner to complete one round.
2. Find the LCM of these times, which will be the answer to the question.

To determine the number of distinct meeting points on a circular track, follow these steps:

1. Find the ratio of the runners’ speeds in its simplest form.
2. If the runners are moving in the same direction, find the absolute difference between the speeds.
3. This absolute difference will be the number of distinct meeting points on the circular track.

It is important to note that all the meeting points are evenly distributed on the track, and the starting point is always one of the meeting points and the last meeting point. However, if the difference between the speeds is odd, the runners will never meet at the diametrically opposite point.

I recently conducted an online session on YouTube where I discussed these concepts in detail. I recommend watching the video to grasp the concepts and tricks fully.

Now, let’s look at some solved problems to better understand the application of these concepts.

Problem 1: Rakesh and Brijesh start running on a circular track (length 420 meters) from the same point simultaneously with speeds of 3 m/s and 7 m/s. Answer the following questions:

a) After how much time from the start will they meet again at the starting point?
b) After how much time from the start will they meet for the first time anywhere on the track?
c) If they keep moving infinitely, at how many different points on the track will they meet?

Solution:

a) The time taken by Rakesh to complete one round = 420/3 = 140 seconds.
The time taken by Brijesh to complete one round = 420/7 = 60 seconds.
The LCM of 140 and 60 is 420 seconds. Therefore, they will meet at the starting point after 420 seconds from the start.

b) The ratio of Brijesh’s speed to Rakesh’s speed is 7:3.
The absolute difference between their speeds is 7 – 3 = 4 m/s.
They will meet for the first time when Brijesh has taken a lead of one round over Rakesh.
Since they meet at the starting point after 420 seconds, the time taken to meet for the first time is 420/4 = 105 seconds.

c) The number of distinct meeting points is equal to the absolute difference of their speeds, which is 4.
Therefore, they will meet at 4 different points on the track.

Problem 2: Rakesh and Brijesh start running on a circular track (length 420 meters) from the same point simultaneously in the opposite direction with 3 m/s and 7 m/s speeds. Answer the following questions:
a) After how much time from the start will they meet again at the starting point?
b) After how much time from the start will they meet for the first time anywhere on the track?
c) After how much time from the start will they meet for the sixth time anywhere on the track?
d) If they keep moving infinitely, at how many different points on the track will they meet?
e) Will they ever meet at the point diametrically opposite to the starting point?
f) When will they meet at a point 300 meters away from the starting point in the direction moved by Rakesh?

Solution:
a) The time taken to meet at the starting point remains the same, which is 420 seconds.
b) The number of distinct meeting points is equal to the sum of their speeds, which is 3 + 7 = 10.
The time taken to meet for the first time is 420/10 = 42 seconds.
c) The time taken to meet for the sixth time is 6 * 42 = 252 seconds.
The distance moved by Rakesh when they meet for the sixth time is 252 * 3 = 756 meters.
Subtracting one complete round (420 meters) from 756, we get 336 meters. Therefore, the sixth meeting happens at a distance of 336 meters from the starting point in the direction moved by Rakesh.
d) Since the number of distinct meeting points is even, they will meet at the point diametrically opposite to the starting point.
e) According to the rule, they meet for the first time at a distance of 126 meters in the direction of Rakesh. As the distance between consecutive meeting points is 126 meters, they will never meet at a point 300 meters away from the starting point.

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