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Two trains, X and Y, started simultaneously from opposite

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Two trains, X and Y, started simultaneously from opposite [#permalink]

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Two trains, X and Y, started simultaneously from opposite ends of a 100-mile route and traveled toward each other on parallel tracks. Train X, traveling at a constant rate, completed the 100-mile trip in 5 hours; Train Y, traveling at a constant rate, completed the 100-mile trip in 3 hours. How many miles had train X traveled when it met train Y?

(A) 37.5
(B) 40.0
(C) 60.0
(D) 62.5
(E) 77.5

OPEN DISCUSSION OF THIS QUESTION IS HERE: two-trains-x-and-y-started-simultaneously-from-opposite-en-168061.html
[Reveal] Spoiler: OA

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Last edited by Bunuel on 30 Apr 2014, 02:12, edited 2 times in total.
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Re: Converging train problem [#permalink]

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New post 30 Nov 2010, 06:40
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lets say they meet after X miles have been covered by T2..that means train 1 has travelled 100 -X miles in the same time

time taken by train 1 to cover 100 -X miles = (100 -X)/20
time taken by train 2 to cover X miles = X /(100/3) = 3X/100

(100 -X)/20 = 3X/100

which gives X = 62.5
which means train 1 had travelled 37.5
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Train A's Speed 100/5 == 20 miles/hr
B's Speed 100/3 = 33.3 miles/hr

Relative speed 53.3 miles/hr

To cover 100 miles with relative speed time taken 100/53.3 almost 135 % of an hour

Train A will cover 135% of 20 approx 37.6

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Re: Converging train problem [#permalink]

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Train A speed = t1 = 20 mph
Train B speed= t2 = 100/3 mph
Combined speed(travelling towards each other) = 160/3 mph

When trains meet each other, they've covered 100 miles together.

Thus time = 100/(160/3) = 30/16 = 15/8 hours

out of 100 miles, distance covered by Train A = 20mph * 15/8 hours = 37.5 miles

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Re: Converging train problem [#permalink]

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ajit257 wrote:
Two trains X and Y started simultaneously from opposite ends of a 100 mile route and travelled toward each other on parallel tracks. Train X travelling at a constant rate completed the 100 mile trip in 5 hours. Train Y travelling at constant rate completed the 100 mile trip in 3 hours. How many miles had train X travelled when it met train Y ?


Can someone please explain the concept behind this type of problem ? All help appreciated.


The concept used in these questions is Relative Speed.

If two people walk in opposite directions (either towards each other or away from each other), their speed relative to each other is the sum of their speeds. e.g. If you are walking away from me at a speed of 2 miles/hr and I am walking away from you at a speed of 1 mile/hr, together we are creating a distance of 3 miles in 1 hr between us so our relative speed is 2 + 1 = 3 miles/hr
On the other hand, when two people walk in the same direction, their relative speed is the difference between their speeds.
e.g. if you are walking away from me at 1 mile/hr and I am walking towards you at 2 miles/hr, my speed relative to you is 2-1 = 1 mile/hr.

Time taken to meet = Total distance traveled/Relative speed

Speed of train X = 100/5 = 20 miles/hr
Speed of train Y = 100/3 miles/hr
Relative Speed = 20 + 100/3 = 160/3 miles/hr
Distance between them = 100 miles
Time taken to meet = 100/(160/3) hr = 15/8 hrs

In this time, train X would have traveled 20 * (15/8) = 37.5 miles

Faster Alternate Approach using Ratios :

Time taken by train X : Time taken by train Y = 5:3
Then, Speed of train X:Speed of train Y = 3:5
Since they start simultaneously, they travel for same time. So the ratio of their distance covered should be same as ratio of their speeds.
Distance covered by train X : Distance covered by train Y = 3:5
3/8 *100 = 37.5 miles (Distance covered by train X)
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Two trains X and Y started simultaneously from opposite ends of a 100 [#permalink]

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Two trains X and Y started simultaneously from opposite ends of a 100 mile route and travelled toward each other on parallel tracks. Train X travelling at a constant rate completed the 100 mile trip in 5 hours. Train Y travelling at constant rate completed the 100 mile trip in 3 hours. How many miles had train X travelled when it met train Y ?

(A) 37.5
(B) 40.0
(C) 60.0
(D) 62.5
(E) 77.5

I solved as under
Time taken to meet = Total distance traveled/Relative speed

Speed of train X = 100/5 = 20 miles/hr
Speed of train Y = 100/3 miles/hr
Relative Speed = 20 + 100/3 = 160/3 miles/hr
Distance between them = 100 miles
Time taken to meet = 100/(160/3) hr = 15/8 hrs

In this time, train X would have traveled 20 * (15/8) = 37.5 miles
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Re: Two trains X and Y started simultaneously from opposite ends of a 100 [#permalink]

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New post 17 Feb 2011, 18:42
Alternate Approach:

Given:

Speed of X = 20 mph
Speed of Y = 33.33 mph

Let the two trains meet at time 't'

Distance travelled by X in time 't' be Dx
Distance travelled by y in time 't' be Dy

Since the time is equal, we can equate it:

Dx/20 = Dy/33.33

or Dx*100/3 = Dy * 20
or 100Dx = 60Dy

Dx+Dy=100
Multiply above equation with 60

60Dx + 60Dy = 6000
60Dx + 100Dx = 6000 [Since, 60Dy = 100Dx]

160Dx=6000
or Dx=6000/160 = 600/16 = (25*6)/4 = (25*3)/2 = 75/2 = 37.5
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Re: Two trains X and Y started simultaneously from opposite ends of a 100 [#permalink]

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New post 17 Feb 2011, 18:44
Your approach is faster than the one I suggested :)

But I just wanted to highlight the concept of equating time.
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Re: Two trains X and Y started simultaneously from opposite ends of a 100 [#permalink]

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Baten80 wrote:
Two trains X and Y started simultaneously from opposite ends of a 100 mile route and travelled toward each other on parallel tracks. Train X travelling at a constant rate completed the 100 mile trip in 5 hours. Train Y travelling at constant rate completed the 100 mile trip in 3 hours. How many miles had train X travelled when it met train Y ?
(A) 37.5
(B) 40.0
(C) 60.0
(D) 62.5
(E) 77.5

I solved as under
Time taken to meet = Total distance traveled/Relative speed

Speed of train X = 100/5 = 20 miles/hr
Speed of train Y = 100/3 miles/hr
Relative Speed = 20 + 100/3 = 160/3 miles/hr
Distance between them = 100 miles
Time taken to meet = 100/(160/3) hr = 15/8 hrs

In this time, train X would have traveled 20 * (15/8) = 37.5 miles


Another approach:
Time taken by train X to cover 100 miles : Time taken by train Y to cover 100 miles = 5:3
Therefore, Speed of X: Speed of Y = 3:5
So, when they meet, X would have covered (3/8)th of the total distance of 100 miles.
Distance covered by X = (3/8)*100 = 37.5 miles
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Re: Two trains X and Y started simultaneously from opposite ends of a 100 [#permalink]

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New post 17 Feb 2011, 19:47
Good one Karishma :)

Didn't think of applying proportionality :D
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Re: Two trains X and Y started simultaneously from opposite ends of a 100 [#permalink]

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New post 18 Feb 2011, 02:01
Let X traveled x miles at speed=(100/5) miles/h when they both met.
Y would have traveled (100-x) miles at speed=(100/3) miles/h

time spent by X = time spent by Y

time = Distance/speed

\(\frac{x}{\frac{100}{5}} = \frac{100-x}{\frac{100}{3}}\)

\(\frac{5x}{100} = \frac{3(100-x)}{100}\)

\(5x=300-3x\)

\(8x=300\)

\(x=37.5 miles\)

Ans: "A"
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Re: Converging train problem [#permalink]

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Faster Alternate Approach using Ratios :

Time taken by train X : Time taken by train Y = 5:3
Then, Speed of train X:Speed of train Y = 3:5
Since they start simultaneously, they travel for same time. So the ratio of their distance covered should be same as ratio of their speeds.
Distance covered by train X : Distance covered by train Y = 3:5
3/8 *100 = 37.5 miles (Distance covered by train X)


This is mind blowing :)
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Re: Two trains, X and Y, started simultaneously from opposite [#permalink]

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As I can see, I notice different approachs that are good but not so efficient

\(relative rates : 5 + 3 = 8\)

\(RT= D\)----> \(\frac{100}{8}\) \(= 12.5\)

Now this is a problem where the 2 times are dissimilar, is like a weigthed average in some how: what is the train that have more weigth in this scenario: the train with the 3 hours of trip.

So : \(12.5 * 3 = 37.5\)

A is the answer. This is the fastest approach you can do with these tricky problems. that's it
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Two trains, X and Y, started simultaneously from opposite ends of a 100-mile route and traveled toward each other on parallel tracks. Train X, traveling at a constant rate, completed the 100-mile trip in 5 hours; Train Y, traveling at a constant rate, completed the 100-mile trip in 3 hours. How many miles had train X traveled when it met train Y?

(A) 37.5
(B) 40.0
(C) 60.0
(D) 62.5
(E) 77.5

As the ratio of the rates of X and Y is 3 to 5 then the distance covered at the time of the meeting (so after traveling the same time interval) would also be in that ratio, which means that X would cover 3/(3+5)=3/8 of 100 miles: 100*3/8=37.5 miles.

Answer: A.

OPEN DISCUSSION OF THIS QUESTION IS HERE: two-trains-x-and-y-started-simultaneously-from-opposite-en-168061.html
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Re: Two trains X and Y started simultaneously from opposite ends of a 100   [#permalink] 25 Feb 2017, 12:16
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