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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]

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New post 15 May 2012, 13:47
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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
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Re: Two trains run in opposite directions on a circular track.  [#permalink]

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New post 15 May 2012, 23:49
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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. T  [#permalink]

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New post 15 May 2012, 16:50
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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. T  [#permalink]

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New post 15 May 2012, 14:27
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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.  [#permalink]

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New post 15 May 2012, 23:55
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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]

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New post 17 May 2012, 00:33
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.

:-D :-D
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Re: Two trains run in opposite directions on a circular track.  [#permalink]

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New post 17 May 2012, 02:20
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]

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New post 17 May 2012, 03:03
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]

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New post 16 Sep 2012, 22:23
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]

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New post 16 Sep 2012, 23:07
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]

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New post 17 Sep 2012, 02:45
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]

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New post 17 Sep 2012, 03:03
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]

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New post 17 Sep 2012, 03:48
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]

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New post 07 Sep 2014, 05:55
Bunuel,

Shouldn't we directly add the relative speeds of train A and B? Is circular approach different from linear one? Please explain
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Re: Two trains run in opposite directions on a circular track.  [#permalink]

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New post 17 Sep 2014, 22:30
I had the same doubt as above.
Shouldn't we add the relative speed?

But the first post mentions that the relative speed is subtracted ? :?
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Re: Two trains run in opposite directions on a circular track.  [#permalink]

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New post 17 Sep 2014, 23:16
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alphonsa wrote:
I had the same doubt as above.
Shouldn't we add the relative speed?

But the first post mentions that the relative speed is subtracted ? :?


Yes, when two objects run in opposite directions, their relative speed is given by adding the two speeds. But think what will \(12\pi/(4\pi + 6\pi)\) give us. It will give us the time taken by the two of them to complete one circle together. They will meet somewhere on the circle but not at the starting point in this much time.

For the two objects to meet again at the starting point, one object must complete one full circle more than the other object. In every hour, the train B - that runs at a speed of \(6\pi\) - covers \(2\pi\) extra miles compared with train A which runs at a speed of \(4\pi\). We want train B to complete one full circle more than train A. In how much time will train B cover \(12\pi\) (a full circle) more than train A? In \(12\pi/ 2\pi\) hrs = 6 hrs.

Or another way to think about it is this:

Time taken by train A to complete one full circle \(= 12\pi/4\pi = 3\) hrs
Time taken by train B to complete one full circle \(= 12\pi/6\pi = 2\) hrs

So every 3 hrs train A is at S and every 2 hrs train B is at S. When will they both be together at S?
train A at S -> 3 hrs, 6 hrs, 9 hrs
train B at S -> 2 hrs, 4 hrs, 6 hrs, 8 hrs

In 6 hrs - the first common time (the LCM of 3 and 2)

Answer (B)
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Re: Two trains run in opposite directions on a circular track.  [#permalink]

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New post 20 Sep 2014, 03:37
alphonsa wrote:
I had the same doubt as above.
Shouldn't we add the relative speed?

But the first post mentions that the relative speed is subtracted ? :?


Another thing - this method of dividing the distance by difference of the speed will not work in all cases.

Try the same question with speed of the trains as \(3\pi\) and \(5\pi\) and radius of the circle as 7.5 miles. The two methods give different answers - figure out why!
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Re: Two trains run in opposite directions on a circular track.  [#permalink]

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New post 20 Sep 2014, 07:23
Karishma,

The right method should be finding the LCM only for the trains have to meet at the same point. In the 3 Pi & 5 Pi problem, the trains will meet at the same point in 15 hrs.

Adding speeds can give the time at which they meet but not at the same meet e.g. in the 4 Pi & 6 Pi problem, the trains will first meet 1.2 hrs from the starting time i.e the 4 Pi train would have traveled 4.8Pi km & the 6Pi tarin would have traveled 7.2Pi kms [the total distance is 12Pi]


[Only if the trains travel in the same direction, the speeds should be subtracted...and again only the catch up time can be calculated from this. To get the time at which they will meet at the same point, LCM is the route]

??
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Re: Two trains run in opposite directions on a circular track.  [#permalink]

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New post 20 Sep 2014, 10:24
Ok...now what happens when the speed of the two trains are subtracted and the circumference distance is divided by that value.

Here what we are really finding out is the time taken for one of the trains to run one full circle more than the other train. i.e. when the speeds of trains are 4 Pi & 6 Pi...in 6 hrs [12 Pi / 2 Pi] one train travels 24 Pi miles and the other travels 36 Pi miles i.e. one train does 2 two rounds and the other train does 3 rounds.

Similarly when the speeds are 3 Pi & 5 Pi...in 7.5 hrs [15Pi/ 2 Pi] one train travels 22.5 miles and the other train travels 37.5 miles [15 miles more than the other] i.e. one train does 2.5 rounds and the other trains does 3.5 rounds.


It is only coincidental that this 6 hrs and the 6 hrs for the meeting time at the same point are equal....and hence LCM route is the right route.
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Re: Two trains run in opposite directions on a circular track.  [#permalink]

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New post 21 Sep 2014, 02:05
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Try the same question with speed of the trains as 3\pi and 5\pi and radius of the circle as 7.5 miles. The two methods give different answers - figure out why!

Method 1

Ratio of speed of Train A and Train B is 3:5.
Considering the circle as 8 units, the 1st meeting point is 3 units away from starting point.
For A & B to meet first time at starting point, we find lowest common multiple of 3 and 8 i.e. 24 units; thus, they must meet (24 units/3 units) 8 times.
Time for each meet = \((\frac{15}{8})hrs\)
Or time to meet 8th time=\((\frac{15}{8})*8=15 hrs.\)

Method 2

For the two objects to meet again at the starting point, one object must complete one full circle more than the other object .
However, in order to satisfy the meeting point to be same as starting point, it is necessary that the difference of distance covered is a exact multiple of full circular distance.
Relative speed per hr =\(2\pi\)
Total circular distance =\(15\pi\)
LCM of relative speed and circular distance =\(30\pi\)
Or, Time to meet at starting point = \(\frac{30\pi}{2\pi per hr} =15 hrs\)
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Re: Two trains run in opposite directions on a circular track.   [#permalink] 21 Sep 2014, 02:05

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