Circular Races

Circular Races : Circular races are on circular tracks where one can meet other person more than once. 

When two persons A & B starts from same point at same time on a circular track then we can find

I. after how much time they meet for first time :- they meet for first time when one covers one more lap than other person. Relative distance would be length of track & using relative speed, time taken can be found. 

II. After how much time they will meet for first time at starting point : this can be find out by taking LCM of time taken  by individual to cover one lap.

Q1. Two person X & Y start from the same point and move along a circular track of 60 m. Speed of X is 5 m/s & speed of Y is 7 m/s. After how much time will they meet for the first time ?

(1) 30 s (2) 15 s (3) 12 s (4) 16 s (5) Can not be determined

Solution:-

Since we don't know the directions of X & Y we can not determined answer. It is possible that they are running in same direction or they might be running in opposite direction. Hence answer option 5.

Q2. Two friends Raj & Rahul start a race on circular track of 240 m from same point in same direction at same time. Speed of the Raj is 20 m/s & that of Rahul is 25 m/s. After how much time will they meet for first time ?

(1) 10 s (2) 12 s (3) 24 s (4) 36 s (5) 48 s

Solution:-

Since same direction is same their related speed is : 25 - 20 = 5 m/s

Related distance to meet for first time is one lap of track = 240 m

Time taken : 240 / 5 = 48 seconds.

Q3. In above question, what would be time taken if they are running in opposite direction ?

(1) 3.33 s (2) 5.33 s (3) 8.33 s (4) 10 s (5) 12 s

Solution:-

Since opposite direction is same their related speed is : 25 + 20 = 45 m/s

Related distance to meet for first time is one lap of track = 240 m

Time taken : 240 / 45 = 5.33 seconds.

Q4. Two friends Raj & Rahul start a race on circular track of 500 m from same point in same direction at same time. Speed of the Raj is 20 m/s & that of Rahul is 25 m/s. After how much time will they meet for first time at starting point?

(1) 20 s (2) 25 s (3) 100 s (4) 200 s (5) 500 s

Solution:- 

Time taken to meet at starting point would be when both complete laps at same time. That is LCM of their time taken to complete track. 

Time taken by Raj to complete track = 500/20 = 25 s

Time taken by Rahul to complete track = 500/25 = 20 s

Time taken to meet for first time at starting point = LCM(20,25) = 100 s.

Q5. In above question, what would be time taken to meet for first time if they are moving along circular track in opposite direction ?

(1) 20 s (2) 25 s (3) 100 s (4) 200 s (5) 500 s

Solution:- 

Time taken to meet at starting point would be when both complete laps at same time. That is LCM of their time taken to complete track. 

Time taken by Raj to complete track = 500/20 = 25 s

Time taken by Rahul to complete track = 500/25 = 20 s

Time taken to meet for first time at starting point = LCM(20,25) = 100 s.

Meeting at starting point in circular races is independent of direction.

when more than 2 people are running in circular track. For e.g. 3 persons X, Y & Z.

I. after how much time they meet for first time :- It can be found by determining the time taken between two people & then between three.

II. After how much time they will meet for first time at starting point : LCM of time taken by X, Y & Z.

Q6. If X, Y & Z are starts their race by moving along a circular track of length 120 m from same point at same time in same direction. Find the time taken for them to meet for first time if speed of X is 2m/s, Y is 3m/s & that of Z is 5 m/s.
(1) 40s (2) 60s (3) 100 s (4) 120s (5) 240s

Solution:- 
All three will meet only when X meets Y.
relative distance = 120 m
relative speed of X & Y = 3 - 2 = 1 m/s
time taken for them to meet for first time = 120 s


All three will meet only when X meets Z.
relative distance = 120 m
relative speed of X & Z = 5 - 2 = 3 m/s
time taken for them to meet for first time = 120 /3 =  40 s

All three will meet for first time when X meets Y & Z together for first time : LCM (120,40)= 120 s


Q7. in above question, what time they will meet for first time at starting point ?
(1) 40s (2) 60s (3) 100 s (4) 120s (5) 240s

Solution:- 

Time taken to meet at starting point would be when all complete laps at same time. That is LCM of their time taken to complete track. 

Time taken by X to complete track = 120/2 = 60 s

Time taken by Y to complete track = 120/3 = 40 s
Time taken by Z to complete track = 120/5 = 24 s

Time taken to meet for first time at starting point = LCM(60,40,24) = 120 s.
It is independent of directions. Let them run in any direction. You don't worry whenever we are finding their first meet at starting point.


Q8. If X, Y & Z are starts their race by moving along a circular track of length 120 m from same point at same time. Y & Z are running in same direction while X is running in opposite direction. Find the time taken for them to meet for first time if speed of X is 2m/s, Y is 3m/s & that of Z is 5 m/s.
(1) 40s (2) 60s (3) 100 s (4) 120s (5) 240s

Solution:-

All three will meet only when X meets Y.
relative distance = 120 m
relative speed of X & Y = 3 + 2 = 5 m/s
time taken for them to meet for first time = 120/5 = 24 s


All three will meet only when Y meets Z.
relative distance = 120 m
relative speed of X & Z = 5 - 3 = 2 m/s
time taken for them to meet for first time = 120 /2 =  60 s

All three will meet for first time when Y meets X & Z together for first time : LCM (24,60)= 120 s.

Q9. If A overtakes B for the first time in the middle of 6th lap. Find ratio of speed of A to B. We know they started their race from same point at same time.
(1) 6:5 (2) 11:9 (3) 5:6 (4) 9:11 (5) Can not be determined.

Solution:-
Overtakes means same direction.
A overtakes B for first time when he covers 5.5 laps. Same time A would cover 4.5 laps. 
Ratio of speeds = ratio of distance covered = 5.5 : 4.5 = 55:45 = 11:9
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