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555-605 (Medium)|   Distance and Speed Problems|               
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Two trains, X and Y, started simultaneously from opposite ends of a 10 13 Mar 2018, 05:15
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Bunuel
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
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Greetings everyone :)
Can anyone share some useful formulas like relative speed etc ? :)

many thanks! :-)
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Two trains, X and Y, started simultaneously from opposite ends of a 10 Updated on: 08 Jun 2025, 13:56
Originally posted by KarishmaB on 13 Mar 2018, 11:39.
Last edited by Bunuel on 08 Jun 2025, 13:56, edited 2 times in total.
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dave13
Bunuel
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

Greetings everyone :)
Can anyone share some useful formulas like relative speed etc ? :)

many thanks! :-)
Show more

Check out these two videos:
https://youtu.be/wrYxeZ2WsEM
https://youtu.be/EDsHa_CULmg
They discuss relative speed in detail.
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Two trains, X and Y, started simultaneously from opposite ends of a 10 14 Apr 2019, 19:44
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Step 1/3: Observe the outcome after 1 unit of time
After 1 hour, train X have traveled for 20 miles and train Y have traveled for 33.3 miles.

Step 2/3: Determine the shrink / expansion rate
The shrink rate is 20+33.3 = 53.3
It means that the distance between train X and train Y will shrink 53.3 miles, every 1 hour travel.

Step 3/3: Apply the shrink rate to the question
The question: How many miles had train X traveled when it met train Y?
If the shrink rate is 53.3 miles per hour, the total distance of 100 miles shall be shrunk in a little less than 2 hours. Train X is traveling at 20 miles per hour. A little less than 2 hours means a little less than 40 miles.
Only options A is possible.
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Two trains, X and Y, started simultaneously from opposite ends of a 10 01 Oct 2019, 18:31
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relative speed of both
100/5+100/3 ; 800/15
so time taken ; 100/800 * 15 ; 15/8 hrs
so for train A distance ; 20*15/8 ; 75/2 ; 37.5 miles
IMO A



Bunuel
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
Show more
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Two trains, X and Y, started simultaneously from opposite ends of a 10 09 Dec 2019, 13:04
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Bunuel
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
Show more


Time taken to meet = Total distance travelled/Relative speed

Speed of train X = 100/5 = 20 miles/hr
Speed of train Y = 100/3 = 33.3 miles/hr
Relative Speed = 20 + 33= 53 miles/hr
Distance between them = 100 miles
Time taken to meet = 100/(53) hr = 1.88 HR

bACK TO TARGET QUESTION'
How many miles had Train X traveled when it met Train Y?

dISTANCE TRAVELLED BY TRAIN X= 20 TIME =1.88
ANS= 20*1.88 =37.7 ANS A

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.
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Two trains, X and Y, started simultaneously from opposite ends of a 10 18 Aug 2020, 11:12
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I used estimation to narrow down the answer.
Speed of X = 100/5 = 20
Speed of Y= 100/3 = 33.33
Relative Speed = 53.33

Relative time taken = 100/53.33 or we can estimate 100/50= 2 hours (approx.) (Actually, it will be less than 2 hours)

In 2 hours X will cover = 20*2=40 miles(approx). Since the time it will take will be slightly less than 2, the distance covered will be slightly less than 40 miles which ends up pointing towards one answer only - A.
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Two trains, X and Y, started simultaneously from opposite ends of a 10 24 Aug 2021, 17:53
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Two trains, X and Y, started simultaneously from opposite ends of a 10 24 Aug 2021, 18:13
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both train X and train Y leave at the same instant towards each other with a 100 mile Gap Distance between them.

When X and Y meet, the TIME each traveled will be the SAME.

when the Time Traveled is Constant: Speed is Directly Proportional to Distance Traveled

this means that the RATIO of the Speeds traveled at will be Equivalent to the RATIO of the Distance each travels when they meet each other.


Speed X : Speed Y = (100/5) : (100/3)

..................................... *15............*15

Speed X : Speed Y = 300 : 500 = 3x : 5x

Distance X : Distance Y = 3x : 5x

when X and Y Meet: X will have covered (3x) / (8x) = (3/8) of the entire Gap Distance Between them

(3/8) * 100 miles = 37.5 miles

answer (A)





Bunuel
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
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Two trains, X and Y, started simultaneously from opposite ends of a 10 31 Aug 2021, 02:38
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Given
• 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, 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.
To find
• The number of miles Train X had travelled when it met Train Y

Approach and Working out
Suppose the trains meet after T hours. So, both trains will have travelled for T hours till they meet.
• Train X:
    o Speed = 100/5 = 20 mph.
    o Time = T
    o So, distance travelled by train X in T hours = 20T miles
• Train Y:
    o Speed = (100/3) mph.
    o Time = T
    o So, distance travelled by train Y in T hours = 100T/3 miles.
• Now, distance travelled by train X + distance travelled by train Y = initial distance between them = 100 miles.
    o So, 20T + (100T/3) = 100
    o T = 15/8 hours
• Finally, distance travelled by Train X in 15/8 hours = 20 * (15/8) = 37.5 miles

Correct Answer: Option A
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Two trains, X and Y, started simultaneously from opposite ends of a 10 08 Nov 2021, 11:44
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Video solution from Quant Reasoning starts at 0:26
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1
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Two trains, X and Y, started simultaneously from opposite ends of a 10 09 Nov 2022, 12:27
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BrentGMATPrepNow
Bunuel
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

Train X completed the 100-mile trip in 5 hours
Speed = distance/time = 100/5 = 20 mph

Train Y completed the 100-mile trip in 3 hours
Speed = distance/time = 100/3 ≈ 33 mph [This approximation is close enough. You'll see why shortly]

How many miles had Train X traveled when it met Train Y?
Let's start with a word equation.
When the two trains meet, each train will have been traveling for the same amount of time
So, we can write: Train X's travel time = Train Y's travel time

Let d = the distance train X travels
So, 100-d = the distance train Y travels (since their COMBINED travel distance must add to 100 miles)
time = distance/speed
Plug our values into the word equation to get: d/20 = (100 - d)/33
Cross multiply to get: (33)(d) = (20)(100 - d)
Expand: 33d = 2000 - 20d
Add 20d to both sides: 53d = 2000
So, d = 2000/53

IMPORTANT: Before you start performing any long division, first notice that 2000/50 = 40
Since the denominator (53) is greater than 50, we can conclude that 2000/53 is LESS THAN 40
Since only one answer choice is less than 40, the correct answer must be A

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Hi Brent BrentGMATPrepNow, if using the shrinking concept, once we found out the rate and add the two rates together (20 + 33 = 55 mph). So time = 100/55 = 2 something. So the answer is not right. Not sure what did I miss here? Thanks Brent
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Two trains, X and Y, started simultaneously from opposite ends of a 10 09 Nov 2022, 15:11
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Kimberly77


Hi Brent BrentGMATPrepNow, if using the shrinking concept, once we found out the rate and add the two rates together (20 + 33 = 55 mph). So time = 100/55 = 2 something. So the answer is not right. Not sure what did I miss here? Thanks Brent
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First off, 100/55 is less than 2, which means it can't equal 2.something
In other words 100/55 = 1.something

So the distance train X travels = (speed)(time) = (20 mph)(1.something) = some number less than 40
Answer: A
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Two trains, X and Y, started simultaneously from opposite ends of a 10 09 Nov 2022, 15:35
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BrentGMATPrepNow
Kimberly77


Hi Brent BrentGMATPrepNow, if using the shrinking concept, once we found out the rate and add the two rates together (20 + 33 = 55 mph). So time = 100/55 = 2 something. So the answer is not right. Not sure what did I miss here? Thanks Brent

First off, 100/55 is less than 2, which means it can't equal 2.something
In other words 100/55 = 1.something

So the distance train X travels = (speed)(time) = (20 mph)(1.something) = some number less than 40
Answer: A
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Thanks BrentGMATPrepNow for the clarificatin. So the concept is right but my silly calculation here then :cry: . Noted.
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Two trains, X and Y, started simultaneously from opposite ends of a 10 02 Mar 2026, 09:59
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Bunuel
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
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