*Monday, August 5th, 2019*

**1. In a circular race of 1200m, A and B start from the same point and at the same time with speeds of 27km/hr and 45km/hr respectively. Find when will they meet again for the first time on the track when they are running**

i) In the same direction

ii) In the opposite directions.

**Sol:** Length of the track, L =1200m

Speed of A= 27 x 5/18 = 7.5m/s

Speed of B= 45 x 5/18 = 12.5m/s

i) Same direction:

Time = L/(Relative Speed)=1200/((12.5-7.5)) = 240 seconds

ii) Opposite direction:

Time= L/(Relative Speed)= 1200/((12.5+7.5))= 60 seconds

It is a simple formula based question, in which we have used the concept of relative speed. In same direction case, we take difference, and in opposite directions, we took the sum of the speeds.

Formulae used are: L/(Relative Speed) ; Relative Speed= S1+/- S2 (+ for Opposite direction and â€“ for same direction)

**2. In a circular race of 1200m length, A and B start with speeds of 18km/hr and 27km/hr respectively at the same time from the same point. When will they meet for the first time at the starting point when running,**

i) In the same direction

ii) In the opposite direction

**Sol:** L=1200m

Speed of A= 18 x 5/18 = 5m/s

Speed of B= 27 x 5/18 = 7.5m/s

Time taken by A to complete one round = 1200/5 =240 seconds

Time taken by B to complete one round = 1200/7.5 = 160 seconds

i) Same direction:

They will meet at the starting point at a time which is the LCM of the timings taken by each of them to complete one full round, i.e., the LCM of 160 and 240 seconds which is 480 seconds.

ii) Opposite direction:

The time taken by them to meet for the first time at the starting point doesnâ€™t change whether they are running in the same direction or in the opposite direction, hence the answer is 480 seconds.

The only difference between the 1st question and this question is that here we have to find the time when they will meet at starting point. For this we find the individual time of each and then took the LCM.

Formula used: LCM of L/a and L/b

**3. A, B, and C run simultaneously, starting from a point, around a circular track of length 1200m, with respective speeds of 2m/s, 4m/s, and 6m/s. A and B run in the same direction while C runs in the opposite direction to the other two. After how much time will they meet for the first time? (in sec)**

**Sol:** Given Aâ€™s speed is 2m/s.

Bâ€™s speed is 4m/s

Câ€™s speed is 6m/s

Time after which A and B meet for the first time = 1200/(4-2) = 600 seconds

Time after which A and C meet for the first time = 1200 /(2+6) (since they run in the opposite direction)

=150 seconds

So time after which all three meet for the first time = LCM(600,150) = 600 seconds

This question is also similar to the 1st question, however in this, there are 3 runners, among which 1 runner is running in the opposite direction. So we have to find the time of meeting of the same direction runners and time of meeting of any two opposite direction runners, and then we took the LCM like in the above question.

**Note:** If instead of A and C, we would have taken B & C, the answer wouldn’t change.

Formula used: LCM of L/a & L/b and then LCM of L/(a or b) and L/c

**4. Two men, B and C run around a circular track of length 500m in opposite directions with initial speeds of 4m/s and 1m/s respectively starting from the same point simultaneously. Whenever they meet, Benâ€™s speed halves and Carlâ€™s speed doubles, after how much time will they meet for the 3rd time?**

**Sol:** Given, Bâ€™s initial speed is 4m/sec

Câ€™s initial speed is 1m/sec

Time take for them to meet for the 1st time =500/(4+1) = 100 seconds

Time taken, after the 1st meet for them to meet for 2nd time = 500/((4/2+1*2)) =500/4 = 125 seconds

Time taken, after the 2nd meet for them to meet for 3rd time = 500/((4/4+1*4)) =500/5 =100 seconds

So total time taken= 100+125+100=325 seconds

In this question, we have to find time for each case. The speed is changing every time they meet, hence we have use relative speed formula with keeping in mind that speed is changing.

Formula used: B & C meeting first time anywhere on track (opposite direction): L/(a+b)

**5. Ram and Shyam run a 10km race on a circular track of length 1000m. They complete one round in 200 seconds and 400 seconds respectively. After how much time from start will the faster person meet the slower person for the last time?**

**Sol:** Let us say, Ram meets Shyam after â€˜tâ€™ seconds he starts from the starting point. Speeds of Ram and Shyam are 5m/s and 2.5m/s respectively.

t=1000/(5-2.5)

=400 seconds

Distance travelled by Ram when meets Shyam for the first time =2000m i.e., after 2 complete revolutions. Ram can make 10 complete revolutions. As he meets Shyam after every two revolutions he would meet Shyam after the 10th revolution for the last time i.e., after 2000seconds.

Formula Used: L/(a-b)

**6. Having started from the same pint and at the same time, two runners A and B are running around a circular track of length 300m in opposite directions with speeds of 5m/s and 8m/s respectively. If they exchange their speeds after meeting for the first time, who will reach the starting point first?**

**Sol:Â **

Let S be the starting point and let them meet after K units of time. Let M be the first meeting point.

Arc SPM: Arc SQM =5:8 as speeds are 5 and 8 metres/sec.

At M, A and B interchange their speeds.

As there is no change in the direction, A covers MQS at speed of 8m/s, and hence take K units of time to reach S.

B continues in the direction of MPS, at a speed of 5m/s, Hence, B takes K units of time to reach S.

Therefore, A and B both will take same amount of time to reach S. Hence, no one will reach first.

**7. Two athletes P and Q are running around a circular track of length 1200m at speeds of 6m/s and 3m/s respectively. Both of them start simultaneously from the same point in the same direction but P reverses his direction every time he completes one round. After how much time from the start will they meet for the first time?**

**Sol:** Time taken by P to reach the starting point for the first time =1200/6= 200 seconds.

Distance between P and Q when P reaches the starting point =1200-600 = 600m.

At this point P reverses his direction, i..e., he runs towards Q. So time taken for P and Q to meet =200+ 600/(3+6)

= 200 + 600/9

=200 + 66.66 = 266.66 seconds.

In this question, firstly we find the time taken by P to complete a round as he is running fast. Within this time Q has covered a certain distance as per his speed. So at the moment when P reverses his direction, we find the difference between P and Q.

Then it becomes a concept of relative speed question in which distance and speeds are given and we have to find out the time.

Formula used: 1st: Time= L/a and L/(a+b) as P and Q are in opposite directions

**8. A and B are running on a circular track at a rate of 44 m/min and 22 m/min respectively. They start on the same point and run in opposite directions. The diameter of the track is 140 m. When they both meet for the 12th Time, what is the distance that A would have covered over B?**

**Sol:** This problem can be solved by noticing that the speeds are in a ratio of 2:1.

The distances covered will, therefore, be in a ratio of 2:1.

When they meet for the 8th Time, they would have covered the entire distance of the track 8 times = 8 (2Ï€r).

A would have covered 8 Times the distance

B would have covered 4 Times the Distance

(Total= 12, ratio 2:1)

Difference = 4(2Ï€r) = 4x 2 x (22/7)x(140/2)= 1760.

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