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The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

Problem Solving
Question: 119
Category: Algebra Applied problems
Page: 77
Difficulty: 600


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SOLUTION

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.
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Re: Two trains, X and Y, started simultaneously from opposite en [#permalink]

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Train X's Average speed = 100/5 = 20 mph
Train Y's Average speed = 100/3 mph
Relative speed when trains travel toward each other = 20 + 100/3 = 160/3 mph.

Note that the 2 trains together would have traveled 100 miles at their meeting junction. Also, the time traveled for both trains would be the same at the meeting point.

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

Of the 100 miles, Distance covered by Train A = 20mph * 15/8 hours = 37.5 miles.

Ans is (A).

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

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Answer = (A) 37.5

Explaination:

Let the point where they meet is x kms from Point X

So it is (100-x) from point Y

Ratio of distance = Ratio of time taken

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

Solving the above equation, we get x = 37.5
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SOLUTION

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.
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I figured out the time they met 't' and then calculated the distance.
D(X) + D(Y) = total Distance = 100
100t/5 + 100t/3 = 100 ; 300t + 500t = 1500 ; t = 15/8
Substitute t in any of the two distance equations. Lets say X:
D = st = 100/5*15/8 = 75/2 = 37.5

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As per the OG explanations:

To solve this problem, use the formula distance = rate x time and its two equivalent forms rate = distance and time = distance. Train X time rate traveled 100 miles in 5 hours so ts rate was 100/5 = 20 miles per hour. Train Y traveled 100 miles in 3 hours so its rate was 1003 miles per hour. If t represents the number of hours the trains took to meet, then when the trains met, Train X had traveled a distance of 20t miles and Train Y had traveled a distance of 100/3 t miles.

How can "t" represent the time for both the trains? Because they have different rates, doesn't that mean each will take different time to meet?

Please explain.
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Re: Two trains, X and Y, started simultaneously from opposite en [#permalink]

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New post 30 May 2015, 14:55
Hi Pretz,

In this question, the variable "T" represents "amount of time that each train was moving" - since both trains started moving SIMULTANEOUSLY, the "T" can be used in both calculations ("T" does NOT represent the "time of day"). Since the trains are moving at DIFFERENT RATES, the DISTANCE that each train will travel will be different.

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

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New post 31 May 2015, 09:21
Bunuel wrote:
SOLUTION

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.


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Pretz wrote:
As per the OG explanations:

To solve this problem, use the formula distance = rate x time and its two equivalent forms rate = distance and time = distance. Train X time rate traveled 100 miles in 5 hours so ts rate was 100/5 = 20 miles per hour. Train Y traveled 100 miles in 3 hours so its rate was 1003 miles per hour. If t represents the number of hours the trains took to meet, then when the trains met, Train X had traveled a distance of 20t miles and Train Y had traveled a distance of 100/3 t miles.

How can "t" represent the time for both the trains? Because they have different rates, doesn't that mean each will take different time to meet?

Please explain.



Think of it this way:

You are at your home and your friend is at his home. You both decide to meet. You leave your respective homes at exactly 12:00 and then travel toward each other's homes at your own speeds. You meet i.e. reach the same point, at say, 12:20. Have you traveled for the same amount of time? Sure. You both have traveled for exactly 20 mins. You traveled at your own speeds: say you are very fast and your friend is very slow. So how does this impact the entire equation? You would have covered much more distance than your friend in the same 20 mins. So higher speed will lead to more distance covered but the time for which the two of you would have traveled would be the same. Similarly, since the trains start at the same time, when they meet, the time elapsed would be the same. They will cover different distances due to their different speeds.
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Re: Two trains, X and Y, started simultaneously from opposite en [#permalink]

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New post 17 Jun 2015, 10:21
Bunuel wrote:
SOLUTION

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.



Can you please explain why ratio X to Y is 3 to 5. I understand the coverage is 3/8 vs 5/8 but how can I detect in the beginning that the ratio of the rates is the opposite of the hours they need ?

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

Problem Solving
Question: 119
Category: Algebra Applied problems
Page: 77
Difficulty: 600


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Is this approach recommendable?:

speed x = 100/5 = 20mph
speed y = 100/3 = 33,3mph

after 1h: X = 20miles, Y = 33,3 miles
after 2h: X= 40 miles, Y = 66,6 miles --> together 106,6 miles ... total dist = 100 miles..they have met in this time frame!
as 106,6 miles, X traveled not 40 but slightly lower --> 37,5 --> Answer A

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LaxAvenger wrote:
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

Problem Solving
Question: 119
Category: Algebra Applied problems
Page: 77
Difficulty: 600


GMAT Club is introducing a new project: The Official Guide For GMAT® Quantitative Review, 2ND Edition - Quantitative Questions Project

Each week we'll be posting several questions from The Official Guide For GMAT® Quantitative Review, 2ND Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

We'll be glad if you participate in development of this project:
1. Please provide your solutions to the questions;
2. Please vote for the best solutions by pressing Kudos button;
3. Please vote for the questions themselves by pressing Kudos button;
4. Please share your views on difficulty level of the questions, so that we have most precise evaluation.

Thank you!



Is this approach recommendable?:

speed x = 100/5 = 20mph
speed y = 100/3 = 33,3mph

after 1h: X = 20miles, Y = 33,3 miles
after 2h: X= 40 miles, Y = 66,6 miles --> together 106,6 miles ... total dist = 100 miles..they have met in this time frame!
as 106,6 miles, X traveled not 40 but slightly lower --> 37,5 --> Answer A


Yes, It's absolutely correct approximation with the given options.

Instead you could have done another thing here

Distance of X/Distance of Y = speed of X/speed of Y

a/(100-a) = 20/33.3

a/(100-a) = 6/10

10a = 600-6a

16a = 600

a = 37.5
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No calculations approach.
By the end of hr1 A would’ve traveled 20 miles and B 33 1/3 miles. So they have not met yet
By the end of hr2 A would’ve traveled 40 miles and B 66 2/3 miles. So they have already crossed. Thus A traveled less than 40 miles when it first met B. The only choice less than 40 miles is A
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100/(20+33 1/3)=15/8 hours until trains meet
train x's distance to meeting=(20 mph)(15/8 hours)=37.5 miles

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

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New post 07 Feb 2017, 04:26
Bunuel isn't the ratio of X to Y suppose to be 5 : 3 and not 3 to 5?

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iliavko wrote:
Bunuel isn't the ratio of X to Y suppose to be 5 : 3 and not 3 to 5?


Train X, travelling at a constant rate, completed the 100-mile trip in 5 hours --> rate of X = (distance)/(time) = 100/5 =20 miles per hour;
Train Y, traveling at a constant rate, completed the 100-mile trip in 3 hours --> rate of Y = (distance)/(time) = 100/3 miles per hour;

(rate of X)/(rate of Y) = 20/(100/3) = 20*3/100 = 3/5.
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New post 07 Feb 2017, 17:28
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

100 miles/(20+33 1/3) mph=1.875 hours
1.875 hours*20 mph=37.5 miles X had traveled when it met Y
A

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

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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


We are given that train X completed the the 100-mile trip in 5 hours, and that train Y completed the 100-mile trip in 3 hours.

Since rate = distance/time, the rate of train X is 100/5 = 20 mph and the rate of train Y is 100/3 mph.

Since the trains left at the same time, we can let the time of each train = t.

We need to determine the distance traveled by train X when it met train Y. Since the two trains are “converging” we can use the formula:

distance of train X + distance of train Y = total distance

20t + (100/3)t = 100

Multiplying the entire equation by 3, we have:

60t + 100t = 300

160t = 300

t = 300/160 = 30/16 = 15/8.

Thus, train X and Y met each other after 15/8 hours.

Since distance = rate x time, the distance traveled by train X when it met train Y was:

15/8 x 20 = 300/8 = 75/2 = 37.5 miles.

Answer: A
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Two trains, X and Y, started simultaneously from opposite en [#permalink]

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New post 25 May 2017, 18:16
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

Problem Solving
Question: 119
Category: Algebra Applied problems
Page: 77
Difficulty: 600


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There are two ways to solve this problem. In either case, find the rates of each train: Rate of train x = 20 and rate of train y = 100/3.

You can now realize that the ratio of x:y is 20/100/3 or 60/100 = 6/10 = 3/5.

In the first step, if train x is moving at 3/5 the total rate, then its coverage of distance will be 3/8* 100 or 37.5

The other method is to find t then solve for d in terms of train x.

100/3*t + 20t = 100 (because the two trains are collectively traveling 100 miles)

t = 15/8

20*15/8 = 37.5.

Last edited by mbaapp1234 on 02 Jun 2017, 08:09, edited 1 time in total.

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Two trains, X and Y, started simultaneously from opposite en   [#permalink] 25 May 2017, 18:16

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