M03-09

Question

Two trains continuously travel between Washington D.C. and Baltimore which are 120 miles apart. The trains start simultaneously, with train A starting in Washington DC and train B starting in Baltimore, and travel at 30 and 90 mph respectively. If the station turnaround times are negligible, what is the distance between the point where the trains meet for the first time and the point where they meet for the second time?

A. 0
B. 30 miles
C. 60 miles
D. 90 miles
E. 120 miles

A. 0
B. 30 miles
C. 60 miles
D. 90 miles
E. 120 miles

Bunuel · Accepted Answer

Official Solution:

Two trains continuously travel between Washington D.C. and Baltimore which are 120 miles apart. The trains start simultaneously, with train A starting in Washington DC and train B starting in Baltimore, and travel at 30 and 90 mph respectively. If the station turnaround times are negligible, what is the distance between the point where the trains meet for the first time and the point where they meet for the second time?

A. 0
B. 30 miles
C. 60 miles
D. 90 miles
E. 120 miles

A. 0
B. 30 miles
C. 60 miles
D. 90 miles
E. 120 miles

Firstly, observe that the faster train from Baltimore can reach Washington D.C. and return to Baltimore more quickly (in 240/90 = 8/3 hours) than the slower train from Washington D.C. takes to get to Baltimore (120/30 = 4 hours). This means that the first encounter point occurs when they move toward each other, and the second encounter point happens when the faster train reaches Washington D.C., turns around, and catches up with the slower train on its journey to Baltimore.
 
The relative speed of the trains moving in opposite directions is 30 mph + 90 mph = 120 mph. Therefore, they first encounter each other in (time) = (distance)/(relative speed) = 120/120 = 1 hour. Hence, the first encounter point is 30 miles from Washington D.C.:
  
W - - - X - - - - - - - - - B
  
At this point, the catch-up distance between the trains is: 30 miles (from the encounter point to Washington D.C.) + 30 miles (from Washington D.C. to the encounter point) = 60 miles. Essentially, the distance between the trains at the first encounter point is simultaneously 0 miles and 60 miles.
 
For simplicity, we can consider the faster train to be 30 miles past Washington D.C.: 
  
(Faster Train) - - - W - - - (Slower Train at X) - - - - - - - - - B.
  
The time needed to compensate for this catch-up distance is (time) = (distance)/(relative speed) = 60/(90 - 30) = 1 hour (notice that in this case, the relative speed would be 90 - 30 = 60 miles per hour because the trains are moving in the same direction). In that hour, the slower train will cover 30 more miles from the first encounter point X. As a result, the second encounter point will be 30 miles away from the first one.

Answer: B

akhilbajaj · Answer

Untitled.png Distance between Washington (W) and Baltimore (B) = 120 Relative speed of trains A and B = 30+90=120 Therefore, the trains together cover a distance of 120 miles in 1 hr and thus meet every hour. At 1 hr, Train A travels 30 miles from W, Train B travels 90 miles from B. This is where the two trains meet (meeting pt 1). At 2 hr, Train A reaches 60 miles from W, Train B travels 30 miles to W, turns around in negligible time and reaches 60 miles from W. This is where the two trains meet (meeting pt 2). Distance between meeting pt 1 and pt2 = 60-30 = 30. Answer B.

Prices · Answer

To be pedantic, the distance between Washington, D.C. and Baltimore is actually only about 35 miles.

abani · Answer

Distance = 120 miles
A's speed : B's speed = 30:90 = 1:3
first meeting point -
A's distance = 1/4 x 120 = 30 
B's distance = 3/4 x 120 = 90
So first meeting point is at 30 miles from Washington

At each meeting point, distance by A: distance by B will always be multiple of 1:3
When A=40 B = 120 (not meeting)
When A=50 B = 150 = -30 (not meeting : '-' implies change in direction)
When A =60 B = 180 = -60 (meeting) 
So second meeting point is 60 miles from Washington.

Difference between meeting points is 60 -30 =30 miles

shanks2020 · Answer

How do we know they will be meeting every 1 hour?

AndrewN · Answer

Good question, shanks2020 . I have taken the liberty of highlighting the original text that states as much. If we picture each train on a parallel track, shuttling back and forth from city to city (although who would ever want to take the slow train?), we can model the position of each train per hour to test the statement. I am going to use color to make the graphical interpretation a bit easier on the eyes. Each highlighted portion will represent a meeting of the two trains. I will also use an arrow (--> or <--) to indicate the number of 30-mile movements of each train per hour, as well as the direction of travel. Train A (moves at 30 mph) 0---30--60---90---120 miles |----|----|----|----| Train B (moves at 90 mph) 120-90--60---30---0 miles |----|----|----|----| Again, the question states that the station turnaround times are negligible , so we can, for the sake of our simulation, assume constant motion. We must also ignore the rates at which each train would accelerate/decelerate, but with all of that said, let us take a look from the top: Hour 1: A: |---- | ----|----|----| --> B: |---- | ----|----|----| <-- <-- <-- Hour 2: A: |----| ---- | ----|----| --> B: |----|---- | ----|----| <-- --> --> Hour 3: A: |----|----| ---- | ----| --> B: |----|----| ---- | ----| --> --> <-- Hour 4: A: |----|----|----| ----| --> B: |----|----|---- |----| <-- <-- <-- As you can see, after 4 hours of travel, the two trains will be in opposite cities at the same time, and it would not be until hour 5 that they would cross paths again and repeat the mirror of the hour 1 position: Hour 5: A: |----|----|---- | ---- | <-- B: | ----|----|---- | ----| --> --> --> Thus, we can conclude that the trains will not meet every hour under the given conditions. Of course, this meta-analysis does not help at all with the problem at hand, but it does correct an assumption, the type that could get a test-taker into trouble on another (similar) question, and I hope it satisfies your curiosity. Good luck with your studies, and thank you for opening the door to my light-hearted response. (I miss physics schematics.) - Andrew

shanks2020 · Answer

Even we have to do a similar analysis till the faster turns around. 
Maybe Bunuel has a faster and direct way to tell something about the meeting points after the 1st meeting point. The least what i can think of is that we will have to find manually till the second meeting and then we can draw some pattern, based on the combined distance covered to meet once.

AnirudhaS · Answer

Hi  shanks2020 , I do not know what short-cut are you looking for.  Bunuel  has already given the most elegant solution. And  MentorTutoring  has explained how you visualise the solution.

There is a longer way to solve the problem, but you might take a few seconds more than 2 min to solve, hence not recommended. But this will help you understand the logic behind their solution.

So I am guessing you are okay with relative speeds so not going into that.
Step 1
Relative speed is 120mph and therefore they travel for 1 hour. Hence, the First meeting point is 30 miles away from Washington

Step 2
Now the faster train B has to reach the opposite station (Washington) at its respective speed (not relative speed here), as both trains have passed each other. B has 30 miles to traverse, so it takes =30/90 = 1/3 hour to do so. Now in this 1/3 hour the slower train A goes = 1/3 * 30 =  10 m  further. Therefore, now the train A is 40 m away from Washington.

Step 3
Now, relative speed is 90-30 = 60mph. To traverse 40m in this speed trains meet again in = 40/60 = 2/3 hour
In 2/3 hour, A goes = 2/3 * 30 =  20 m

In step 2, A went 10 miles, and in step 3, A went 20 miles (shown by bold text). Therefore total distance away from the first meeting point = 30m

Hope this is clear.

GKomoku · Answer

Prices  how?

AndrewN · Answer

I think  Prices  was talking about the real-life distance between the two cities, irrespective of the problem at hand. I used to live in Baltimore and traveled to D.C. from time to time. The driving distance via the Baltimore-Washington Parkway and MD-295 is under 40 miles, depending on where one lives in either place and where one is going. Regarding the GMAT™, I have seen some problems that use made-up names, perhaps to discourage any real-life knowledge from entering the picture (not that such knowledge would be useful), but other problems use actual locations, such as Chaco Canyon in the American southwest.

I am not sure any of this dialogue will lead to insights into the question at hand, but anyway, I guess my own life experience allowed me to understand exactly what  Prices  had meant.

- Andrew

hirakjd · Answer

Using combined speed we get the speed =90mph+30mph=120mph
Total time to complete 120 miles with combined speed is 1hr. Distance traveled by first train in 1 hr is 30 miles and the second train in 1hr is 90 miles.

So, they meet first time when first train traveled 30mils
Similarly in 2nd hour they meet again at second point as using combined speed they have traveled 120 miles. Distance traveled by first train in 2nd hour is 60 miles.

Difference in distance between meeting points = 60-30 =30 miles

Posted from my mobile device

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

KarishmaB · Answer

(30 mph) A -> ----------------120-----------------------<- B (90 mph) First Meeting: When they cover the 120 miles distance together. Time taken = 120/(30 + 90) = 1 hr (because their relative speed gets added). Hence 1st meeting is after 1 hr so in this time trains will be 30 miles from Washington . -------30----<-B,A->-------------90-------------- Now they continue and B covers the remaining 30 miles to Washington in 30/90 mins = 1/3 hr. In 1/3 hr, A reaches 30 * 1/3 = 10 miles further ahead B---------40-------A->----------80---------------- Now B turns around and B and A will travel in the same direction as A and will takeover A for its second meeting. How much time will B take to cover the 40 miles between B and A? Again use relative speed. Time taken = 40/(90 - 30) = 2/3 hrs So B takes 2/3 hrs to meet A in which time, B has covered a distance of 2/3 * 90 = 60 miles. Hence second meet is at 60 miles from Washington. Then distance between the two meetings is 30 miles. Answer (B)

jcrafer · Answer

I like the solution - it’s helpful. elegant solution!

mexsolares · Answer

I did not quite understand the solution. How do we know what relative speed is?

Bunuel · Answer

Relative speed depends on whether the trains are moving  toward  each other or  in the same direction .

When they move  toward each other , their speeds  add : 30 + 90 = 120 mph. 
 When they move  in the same direction  (after the faster one turns around), their speeds  subtract : 90 - 30 = 60 mph.

That’s why the first meeting uses 120 mph and the second uses 60 mph as the relative speed.

GMAT2025champ · Answer

Here is a lengthy but easier to understand method; surprised that this is a 600-level question, considering the time it takes to do.

Show Spoiler

Attachment:

GMAT-Club-Forum-rpa6tn4e.png [ 33.94 KiB | Viewed 1401 times ]

DarkAster · Answer

I did not quite understand the solution. Explanation for second meeting point is confusing.

Bunuel · Answer

1. Please go through the discussion above and check the posts that elaborate on the solution or offer entirely new approaches. 2. Check the alternative solutions in the other topic here: https://gmatclub.com/forum/two-trains-c ... 97862.html 3. If this still does not help, please explain what exactly was unclear in this solution so I can help more effectively.

M03-09

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