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Two trains run in opposite directions on a circular track.

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Two trains run in opposite directions on a circular track. [#permalink] New post 15 May 2012, 12:47
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Two trains run in opposite directions on a circular track. Train A travels at a rate of 4π miles per hour and Train B runs at a rate of 6π miles per hour. If the track has a radius of 6 miles and the trains both start from Point S at the same time, how long, in hours, after the trains depart will they again meet at Point S?

A. 3
B. 6
C. 9
D. 18
E. 22
[Reveal] Spoiler: OA
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Re: Two trains run in opposite directions on a circular track. T [#permalink] New post 15 May 2012, 13:27
The trains will travel the perimeter of the circle which is 12π miles. The relative speed of the trains is 2π miles per hour. Time taken to meet = 12π/2π = 6 hours
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Re: Two trains run in opposite directions on a circular track. T [#permalink] New post 15 May 2012, 14:32
RSG wrote:
The trains will travel the perimeter of the circle which is 12π miles. The relative speed of the trains is 2π miles per hour. Time taken to meet = 12π/2π = 6 hours



hello
please explain why are you looking for the relative speed in this case

thanks

best regards
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Re: Two trains run in opposite directions on a circular track. T [#permalink] New post 15 May 2012, 15:50
The question is not when the trains will meet but when will they meet at point s.

Heres how i think of it.

Train A 3 hours for a complete revolution (12pi/4pi)
Train B 2 hours for a complete revolution (12pi/6pi)

So every 6 hours the trains will meet at the starting point (LCM of 3,2)

So answer is b)6
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Re: Two trains run in opposite directions on a circular track. [#permalink] New post 15 May 2012, 22:49
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alexpavlos wrote:
Two trains run in opposite directions on a circular track. Train A travels at a rate of 4π miles per hour and Train B runs at a rate of 6π miles per hour. If the track has a radius of 6 miles and the trains both start from Point S at the same time, how long, in hours, after the trains depart will they again meet at Point S?

A. 3
B. 6
C. 9
D. 18
E. 22


The circumference of the track is 2\pi{r}=12\pi;

Train A will be at point S every \frac{12\pi}{4\pi}=3 hours;

Train B will be at point S every \frac{12\pi}{6\pi}=2 hours;

So, they will meet at point S for the first time in 6 hours (the least common multiple of 2 and 3).

Answer: B.
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Re: Two trains run in opposite directions on a circular track. [#permalink] New post 15 May 2012, 22:55
In cases like this I find it best to just calculate at what times
each train will reach point S and take the first common time.

a: 3 6 9
b: 2 4 6

so we can see that the first time they will meet back at point S is 6
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Re: Two trains run in opposite directions on a circular track. [#permalink] New post 16 May 2012, 23:33
When two objects travel in opposite direction, they are covering the given distance at a speed which is difference between their respective speeds.

Thus If train A travels at 6 km and train B travels at 4 km in opposite direction, every hour a gap of 2km is being bridged. Thus to cover a circumference of 12, they would need 6 hours i.e. 2 km every hour.

Point S is nothing but a point from where the trains start. Since trains are travelling in opposite directions, to reach to point S again, they will have to cover entire circumference. Therefore difference of relative speed is calculated.

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Re: Two trains run in opposite directions on a circular track. [#permalink] New post 17 May 2012, 01:20
Such Questions can be solved with the knowledge of LCM and in this case it is LCM of 3 and 2 which is the respective time taken by the trains.
Hence the ans is 6.
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Re: Two trains run in opposite directions on a circular track. [#permalink] New post 17 May 2012, 02:03
Lenght of the track = 2 * pi * r = 2* pi * 6 = 12pi

Time taken by first train to complete the track and to reach S = 12pi / 4pi = 3 hrs
Time taken by Second train to complete the track and to reach S = 12pi/ 6pi = 2 hrs


They will meet again at the LCM of time taken to complete one full circle i.e 6 hrs

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Re: Two trains run in opposite directions on a circular track. [#permalink] New post 16 Sep 2012, 21:23
My name is Brother Karamazov.
I don't agree with your solutions, first. And second the answer choices don't seem to contain the right answer.
My solution is as follows, and I ask anyone to correct me if I am wrong.

Solution 1
Let the distance covered by train A be X, thus that covered by the train B will be 12*3.14-X
dA = X dB = 12*3.14 - X
Times taken by A and B are
tA= X/4*3.14 , tB = (12*3.14 - X)/6*3.14 (ii)
Since they have been traveling for the same period of time, then
X/4*3.14 = (12*3.14 - X)/6*3.14
X/2 =(12*3.14 -X)/3
3X = 2(12*3.14 -X)
5X = 24*3.14
X = 24*3.14/5
Plugging that in either equation of (ii) yields t = 6/5

Solution 2

We add the speed of A and B: totalSpeed = 4*3.14 + 6*3.14 = 10*3.14
Total distance covered = 12*3.14
t ime = distance / speed = 12*3.14/10*3.14 = 6/5.

Please tell me what I am doing wrong

thanks
tA =( X/4*3.14) =(24*3.14/5)/4*3.14 = 6/5.
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Re: Two trains run in opposite directions on a circular track. [#permalink] New post 16 Sep 2012, 22:07
Ousmane wrote:
My name is Brother Karamazov.
I don't agree with your solutions, first. And second the answer choices don't seem to contain the right answer.
My solution is as follows, and I ask anyone to correct me if I am wrong.

Solution 1
Let the distance covered by train A be X, thus that covered by the train B will be 12*3.14-X
dA = X dB = 12*3.14 - X
Times taken by A and B are
tA= X/4*3.14 , tB = (12*3.14 - X)/6*3.14 (ii)
Since they have been traveling for the same period of time, then
X/4*3.14 = (12*3.14 - X)/6*3.14
X/2 =(12*3.14 -X)/3
3X = 2(12*3.14 -X)
5X = 24*3.14
X = 24*3.14/5
Plugging that in either equation of (ii) yields t = 6/5

Solution 2

We add the speed of A and B: totalSpeed = 4*3.14 + 6*3.14 = 10*3.14
Total distance covered = 12*3.14
t ime = distance / speed = 12*3.14/10*3.14 = 6/5.

Please tell me what I am doing wrong

thanks
tA =( X/4*3.14) =(24*3.14/5)/4*3.14 = 6/5.


You calculated the time it takes them to meet somewhere on the circumference for the first time and not at point S again.
First train travels 4\pi miles in an hour, the other train travels 6\pi miles in an hour. The total distance covered by them in 6/5 hours is (24/5 + 36/5)\pi=60/5\pi=12\pi miles, which is exactly the length of one circumference.

The question was when do they meet again at point S?
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Re: Two trains run in opposite directions on a circular track. [#permalink] New post 17 Sep 2012, 01:45
Hi,
what you are doing wrong is this :

In both the solutions you have missed out on the point that question says "how long, in hours, after the trains depart will they again meet at Point S?"

So the answers that you are arriving at is the time when both trains meet each other.

Vips



EvaJager wrote:
Ousmane wrote:
My name is Brother Karamazov.
I don't agree with your solutions, first. And second the answer choices don't seem to contain the right answer.
My solution is as follows, and I ask anyone to correct me if I am wrong.

Solution 1
Let the distance covered by train A be X, thus that covered by the train B will be 12*3.14-X
dA = X dB = 12*3.14 - X
Times taken by A and B are
tA= X/4*3.14 , tB = (12*3.14 - X)/6*3.14 (ii)
Since they have been traveling for the same period of time, then
X/4*3.14 = (12*3.14 - X)/6*3.14
X/2 =(12*3.14 -X)/3
3X = 2(12*3.14 -X)
5X = 24*3.14
X = 24*3.14/5
Plugging that in either equation of (ii) yields t = 6/5

Solution 2

We add the speed of A and B: totalSpeed = 4*3.14 + 6*3.14 = 10*3.14
Total distance covered = 12*3.14
t ime = distance / speed = 12*3.14/10*3.14 = 6/5.

Please tell me what I am doing wrong

thanks
tA =( X/4*3.14) =(24*3.14/5)/4*3.14 = 6/5.


You calculated the time it takes them to meet somewhere on the circumference for the first time and not at point S again.
First train travels 4\pi miles in an hour, the other train travels 6\pi miles in an hour. The total distance covered by them in 6/5 hours is (24/5 + 36/5)\pi=60/5\pi=12\pi miles, which is exactly the length of one circumference.

The question was when do they meet again at point S?

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Re: Two trains run in opposite directions on a circular track. [#permalink] New post 17 Sep 2012, 02:03
Just wanted to clarify .I think some of the pals are confused about the relative speed concepts.

When bodies move in the same direction ,there relative speeds must be subtracted.
When bodies move in the opposite direction their relative speeds must be added.

Some of them have applied it wrongly .It can lead to errors in the exam .
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Re: Two trains run in opposite directions on a circular track. [#permalink] New post 17 Sep 2012, 02:48
Expert's post
Sorry but move in the same direction is not the same to say: move toward each other and eventually crash ?? and in this case the relative speed is not the sum of the respective rates ??

here we do not have the situation move toward but meet at some point after rouded a circle.

Please some expert can clarify this situation of relative speed toward and relative speed in the problem at end ???

Thanks
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Re: Two trains run in opposite directions on a circular track. [#permalink] New post 07 Dec 2013, 23:42
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Re: Two trains run in opposite directions on a circular track.   [#permalink] 07 Dec 2013, 23:42
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