Two trains run in opposite directions on a circular track.

Question

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

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

Bunuel · Accepted Answer

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.

gomennassai · Answer

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

RSG · Answer

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

ronr34 · Answer

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

saikarthikreddy · Answer

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 .

Carcass · Answer

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

sri30kanth · Answer

Bunuel,

Shouldn't we directly add the relative speeds of train A and B? Is circular approach different from linear one? Please explain

alphonsa · Answer

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 ?

KarishmaB · Answer

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)

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!

Vetrik · Answer

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

??

arunspanda · Answer

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

KarishmaB · Answer

The method of "dividing the distance by difference of the speed" is not necessarily wrong. In the  3\pi  and  5\pi  problem, the circumference of the circle is  15\pi . Train B gets ahead of train A by  2\pi  every hour. After 7.5 hours, it is  15\pi  ahead but it is not at S at that time because it reaches S in only integral hours. So train B needs to complete 2 full circles more than train A which it will do in 15 hours. In 15 hours, both trains will be at S.

kshitij89 · Answer

When two trains are running in opposite direction, the relative speeds are added . Am I correct here ?

generis · Answer

kshitij89  , yes, you are correct.

VeritasPrepKarishma

here:

https://gmatclub.com/forum/two-trains-run-in-opposite-directions-on-a-circular-track-132630.html#p1417394

See this post, too:

https://gmatclub.com/forum/two-trains-run-in-opposite-directions-on-a-circular-track-132630.html#p1418111

Maybe re-read the whole thread? Rates aren't the best method to solve this problem.

Hope that helps.

rishi02 · Answer

Sorry for dragging up an old post, but is this solution right? The objects are moving in opposite direction so relative speed should be sum of individual speeds right? Can relative speed be applied to solve such a problem?

EMPOWERgmatRichC   VeritasKarishma   ScottTargetTestPrep

KarishmaB · Answer

This solution is correct and you are right too.

Yes, since the trains are moving in opposite directions, their "relative speed" will be sum of their speeds. But the concept of relative speed does not help us here because the trains need to meet back at point S. We can find the the trains will cover 1 full round together in 12π/10π hrs but they will not meet at S in this case.

The best way is to figure at what time each train comes to S and the common point when they both will be at S.

What is done in this solution is this:

Assume the trains are moving in same direction. Then in 1 hr, they will create a gap of 2π miles between them. In 2 hrs, they will create a gap of 4π miles between them and so on till in 6 hrs, they create a gap of 12π miles between them. So they would create a gap of one full circle and would both be back at S. In this time, one train would have made 2 full rounds and the other would have made 3 full rounds. Now think - does it matter in which direction the trains were moving?

nayanq2001 · Answer

Hi! I religiously followed your videos and basis that, I did this, could you please help if my solution is right? Ratio of the speeds can be simplified to 3:4 i.e. they meet 7th time when they finally reach the starting point with 3 and 4 laps completed respectively, so the relative distance travelled is 7 *2 pi *r as the ratios are added to find out the number of times they have met in 1 full cycle. Relative distance/ Relative speed = 7* 2 pi*6/(6pi+8pi)= 6 hours.

Bixy34 · Answer

Bunuel   KarishmaB  Is it just a coincidence that subtracting their relative speeds instead of adding in this case is leading us to the correct answer or is there some logic behind it?

egmat · Answer

Hi Bixy34,

It's not a coincidence in the way you might fear, but it's also  not  the "opposite-direction rule" doing the work. What's really happening is that the numbers in this specific question let a  different  formula land on the right answer for the right reason. Let me separate the three things that are getting mixed up.

Three different events, three different formulas

For two objects on a circular track, there are actually  three  distinct questions you can ask:

-  When do they first meet each other, anywhere on the track? 
 - Opposite directions - use  sum  of speeds:  12π / (4π + 6π) = 1.2  hrs
 - Same direction - use  difference  of speeds:  12π / (6π − 4π) = 6  hrs (faster one laps the slower one)
-  When are they both back at the starting point S? 
 - Neither "sum" nor "difference" is the rule. You need each train to have completed a  whole number  of laps. Use  LCM  of individual return times:  LCM(3, 2) = 6  hrs.

So the  12π / (6π − 4π) = 6  you're seeing is the  same-direction catch-up  formula. It answers "when would the faster train lap the slower one if they moved together?" - a completely different physical event from what the question asks.

Why it still gives the right answer here

Because of the specific numbers. Watch the  6 -hour mark:

- Train A:  6 / 3 = 2  full laps - back at S
- Train B:  6 / 2 = 3  full laps - back at S

The catch-up time ( 6 )  happened  to equal the LCM of the return times ( 6 ). That's the coincidence - not that subtraction "worked," but that the two very different quantities collided at the same number.

The stress test

Change one number and the illusion breaks. Take speeds  3π  and  5π , radius  7.5  (circumference  15π ):

- Difference method:  15π / (5π − 3π) = 7.5  hrs
- But at  7.5  hrs, Train A has done  7.5 / 5 = 1.5  laps -  not  at S.
- True answer via LCM: return times are  5  and  3  hrs, so LCM =  15  hrs.

Same formula, wrong answer. That's the tell that subtraction was never the underlying principle.

The takeaway

For "meet again at S" on a circular track, always use the  LCM  of individual return times - it works in every case, regardless of direction. The sum/difference formulas answer  when they meet each other , not  when they're both home .

Answer: B

Two trains run in opposite directions on a circular track.

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Two trains run in opposite directions on a circular track.

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