Two trains continuously travel between Washington D.C. and Baltimore

Question

Two trains continuously travel between Washington D.C. and Baltimore, which are 120 miles apart. The trains start simultaneously, with Train A departing from Washington D.C. and Train B departing from Baltimore. They travel at constant speeds of 30 miles per hour and 90 miles per hour, respectively. Assuming that the turnaround times at the stations are negligible, what is the distance between the points where the trains encounter each other for the first and second times?

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

time	A	B	tot miles
1h	30 miles	90 miles	120
2h	60 miles	120 miles	240

first meeting at t=1
second meeting at t=2
60-30 = 30 miles

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

M03-09

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 departing from Washington D.C. and Train B departing from Baltimore. They travel at constant speeds of 30 miles per hour and 90 miles per hour, respectively. Assuming that the turnaround times at the stations are negligible, what is the distance between the points where the trains encounter each other for the first and second times?

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

time	A	B	tot miles
1h	30 miles	90 miles	120
2h	60 miles	120 miles	240

first meeting at t=1
second meeting at t=2
60-30 = 30 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

Lampard42 · Answer

First meet will be  30  miles (120/(90+30)=1 hr (Same direction so speed will be added); B will reach by 120/90*60=80 min by which time A will have traveled (30*80)/60 = 40 Miles; So B has to catch 40 miles @ 60 mph (90-30 as they travel in opposite directions now) (40*60)/60 = 40 min. So they will meet at (90*40)/60 = 60 Miles which is 30 miles away from first point

Archit3110 · Answer

relative speed ; 90+30 ; 120
distance= 120 miles so they meet after 1 hr ; viz 120/120 ; 1 hr

in 1 hr A will over = 30 miles 
time for B to cover 30 miles ; 90*60/90 =1 hrs ; 
train B shall reach Washington by 80 mins by which A would have covered ; 30*80/60 = 40 miles
now train B has to cover 40 miles ; the relative speed now is in same direction ; 90-30 ; 60 mph
so B will meet A at 40*60/60 = 40 mins 
so distance covered by B in 40 mins ; 90*40/60 = 60 miles
distance gap b/w two points ; 60-30 ; 30 miles
IMO B

nick1816 · Answer

Time taken by both trains to meet for the first time=  \frac{120}{30+90} =1hr
Time taken by both trains to meet for the second time=  \frac{240}{30+90} =2hrs

Distance of train A from Washington D.C, when both trains meet for the first time and the point= 30*1=30
Distance of train A from Washington D.C, when both trains meet for the second time and the point= 30*2=60

Distance between the point where the trains meet for the first time and the point where they meet for the second time=60-30=30 miles

ScottTargetTestPrep · Answer

The point where the two trains meet for the  first time  is the point when they together travel the distance between Washington DC and Baltimore  once . So let t be the time when they together travel the distance between Washington DC and Baltimore once and we can create the equation:

30t + 90t = 120

120t = 120

t = 1

Therefore, the meeting point for the first time is 30 miles from Washington DC (or 90 miles from Baltimore).

Similarly, the point where the two trains meet for the  second time  is the point when they together travel the distance between Washington DC and Baltimore  twice . So let m be the time when they together travel the distance between Washington DC and Baltimore twice and we can create the equation:

30t + 90t = 240

120t = 240

t = 2

Therefore, the meeting point for the second time is 60 miles from Washington DC (or 60 miles from Baltimore).

Therefore, the distance between the two meeting points is 60 - 30 = 30 (or 90 - 60 = 30) miles.

Answer: B

GianKR · Answer

As the trains are traveling towards each other,  they would require 1 hour to meet for the first time : t= Gap distance/relative speed= 120/(30+90)= 1 hour.

At 1 hour time, train A would have traveled a distance of 30 miles: D= Rate (train A) x 1 hour= 30 x 1= 30 miles  (First meeting point is 30 miles away from D.C.) .

In order to meet twice,  the two trains will have to cover together a distance of 120 miles twice= 240 miles, which  they will do in 2 hours : t=240miles/120mph= 2 hours.

In two hours time, train A would have traveled a distance of 60 miles: D= 30mph x 2hours= 60 miles  (Second meeting point is 60 miles away from D.C.).

Therefore, the distance between the points where the trains encounter each other 1st and 2nd time is: 60-30= 30 miles.

Correct answer is B

demawtf · Answer

Two trains continuously travel between Washington D.C. and Baltimore, which are 120 miles apart. The trains start simultaneously, with Train A departing from Washington D.C. and Train B departing from Baltimore. They travel at constant speeds of 30 miles per hour and 90 miles per hour, respectively. Assuming that the turnaround times at the stations are negligible, what is the distance between the points where the trains encounter each other for the first and second times?

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

time	A	B	tot miles
1h	30 miles	90 miles	120
2h	60 miles	120 miles	240

first meeting at t=1
second meeting at t=2
60-30 = 30 miles

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

time   
   A   
   B   
   tot miles   
 
   1h  
  30 miles  
  90 miles   
  120  
 
   2h  
  60 miles  
  120 miles   
  240

first meeting at t=1 
second meeting at t=2
60-30 = 30 miles

inciduntveniam · Answer

The first meeting point was 30 miles from DC and and if in the next 2 hours they travel 240 miles then A's distance travelled is 2*30= 60 miles but isn't the difference from the first meeting point to baltimore 120- 30 = 90 miles so in 2 hours A wouldn't even have reached baltimore because he travelled only 60 miles?

egmat · Answer

Hi inciduntveniam,

Your arithmetic is actually perfect. In  2  hours Train A really does travel only  30 × 2 = 60 miles , and yes, that's short of the  90 miles  from the first meeting point to Baltimore. So your numbers are right - the issue is what you expect the  second meeting  to look like.

The hidden assumption to drop:  you're picturing the second meeting as A having to push all the way to Baltimore. It doesn't.  Train A never reaches Baltimore by the second meeting  - and it doesn't need to.

Here's what actually happens. Train B is the fast one ( 90 mph ):

- By  t = 4/3 hr , B has covered its full  120 miles  and arrives at Washington D.C.
- B turns around there and races back toward Baltimore.
- Meanwhile A is crawling forward, only  40 miles  from D.C. at that moment.
- B then  catches up to A from behind  and overtakes it.

That catch-up  is  the second meeting. It's not a head-on meeting like the first one - it's the faster train lapping back and reaching the slower train.

Check the positions at t = 2 hr (measuring from D.C.):

- Train A:  60 miles  from D.C. ( 30 × 2 ).
- Train B: traveled  180 miles  total ( 90 × 2 ) - that's  120 miles  to D.C., then  60 miles  back - so it's also at  60 miles  from D.C.

Same spot,  60 miles  from D.C. ✓

That's why the "combined  240 miles " shortcut works:  together  they've covered the route twice, but A only owes  60  of those miles and B owes the other  180 . The distance between the two meeting points is  60 − 30 = 30 miles  - answer  B .

Answer: B

Two trains continuously travel between Washington D.C. and Baltimore

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Two trains continuously travel between Washington D.C. and Baltimore

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