Lesson: Shrinking and Expanding Gaps

Comment on Shrinking and Expanding Gaps

For sabi and gwyn problem, tried in formal approach.

In 4 hours sabi would travel 160 miles with 40 miles/hour speed, gwyn would travel 120 miles with 30 miles/hour.

distance gap between will be 40 miles.
greenlight-admin's picture


Hi Brent!
I am finding difficult in tackling distance rate time word problems, do you have any recommendation of other sources which will give more problems on same and some more theory.

Thank you.
greenlight-admin's picture

To find more distance/rate/time questions, you you can use the question-tagging tools at GRE Prep Club (http://gre.myprepclub.com/forum/viewforumtags.php?)

what if Sabi and Gwyn traveling in opposite direction? I mean away from each other. Do we add the expansion?
greenlight-admin's picture

That's correct.

So, with that example, the rate of expansion would be 70 miles per hour. That is, for each hour that passes, the distance between Sabi and Gwyn INCREASES by 70 miles.

Hi, for both 160(1) & 262 (14) ETS Question I'm having a hard time understanding.

-160(1) : I got the problem correct but when I read the explanation, I've noticed that my process was wrong.
Quant A: 30 x 3= 90
Quant B: 48 x 4= 192

But in the explanation book it stated that they compared with Quant A being 6x.
I don't know how they got that number. Confusing which method they used

-262(14) : I made a progress till rate difference of "x , x+8"
I dont know what to do after this method.

I will kindly wait for ur kind instructions
greenlight-admin's picture

Question link for page 106, question #1: https://gre.myprepclub.com/forum/machine-r-working-alone-at-a-constant-r...

The main problem with your solution is that you have not calculated each machine's RATE.
For example, we're told that machine R produces x units of a product in 30 minutes.
This means machine R produces x/30 units in ONE minutes
In other words, machine R's RATE is x/30 units PER MINUTE

The same goes for machine S's rate.

Here's my step-by-step solution: https://gre.myprepclub.com/forum/machine-r-working-alone-at-a-constant-r...


Page 262, question #14: Two cars started from the same point, and traveled on a straight course in opposite directions for exactly 2 hours, at which time they were 208 miles apart. If one car traveled, on average, 8 miles per hour faster than the other car, what was the average speed for each car for the 2-hour trip?

Let's start with a word equation.

How about (distance traveled in 2 hours by car #1) + (distance traveled in 2 hours by car #2) = 208 miles

At this point, we'll convert the word equation into an algebraic equation.
First, we need to know that: distance = (travel time)(travel speed)
We know that the travel time is 2 hours for each car, but we don't know their speeds.
So, let's assign some variables.

We're told that one car traveled 8 mph faster than the other car. So, let's say car #1 is the SLOWER car.

Let x = the speed (in miles per hour) of car #1
This means x + 8 = the speed (in miles per hour) of car #2

We're now ready to convert the word equation into an algebraic equation.

distance = (travel time)(travel speed)
So, we get: (2)(x) + (2)(x + 8) = 208
Expand: 2x + 2x + 16 = 208
Simplify: 4x + 16 = 208
So: 4x = 192
Solve: x = 48

This means car #1's speed = 48 mph
Since car #2's speed is 8 mph faster, car #2's speed = 56 mph

Answer: The speeds are 48 mph and 56 mph.


Understood perfectly. Thank you


Hi Brent,

For the URCH GRE Forum question (A & B travels in a circular track), can you please explain why you converted the Time from Hours to Minutes since all the variables were expressed in the same units? If one rate was expressed in hour and the other in minutes then it's a clear indication that the rates need to expressed uniformly, which is not the case here.

greenlight-admin's picture

Question link: http://www.urch.com/forums/gre-math/155431-time-distancea-problem.html

You're right; it wasn't 100% necessary to convert hours to minutes. We COULD have just that 1.5 elapse from 8:00am to 9:30am, and we COULD have noted that they cross paths every 0.2 hours.

(1.5)/(0.2) = 7.5, so A and B cross paths 7 times.

I must have felt it would be easier to think in terms of minutes.



I approached this question in a different way. I listed the time with respect to their speed. For example A does 2 rounds per hour to therefore would be completing the rounds at 8:30 9:00 & 9:30. And B would be completing one round at 8:20 8:40 9:00 & 9:20. As all the timings are different that means that they would see each other at different times. Does this strategy work or I got lucky?
greenlight-admin's picture

Question link: http://www.urch.com/forums/gre-math/155431-time-distancea-problem.html

I'm having a hard time determining why that approach would work. What role does the "different times" play in your approach?

Ask, because 9:00 appears in both sets of times.
Also, if they BOTH able to complete 2 rounds per hour, both of their times would be the same, BUT they would still meet each other several times.


Hi, could you explain how the strategy would change in cases where one person starts a few hours later,with an example may be
greenlight-admin's picture

That kind of question is covered in this lesson: https://www.greenlighttestprep.com/module/gre-word-problems/video/916

If one person (say Person A) traveled for 1 hour before Person B started traveling, our word equation could look like this:
(Person A's travel time) = (Person B's travel time) + 1 hour

Here's a question to practice with: https://gre.myprepclub.com/forum/trains-a-and-b-are-190-miles-apart-1300...


Hi Brent,

Great explanations! I wanted to know what happens if two people leave from one place (let's call it A) at the same time, in 2 opposite directions.
<---- A ---->

According to the above theory, their distance will expand and their expansion rate will be the sum of the rates of the two travelers right? I do not know if such a question will pop up at the GRE but I thought it is better to ask. Thanks
greenlight-admin's picture

You're 100% correct!
It's conceivable that such a question could arise on the GRE. So, it's good to know that principle.


Hi Brent,

I am working on this question: https://gre.myprepclub.com/forum/rachel-and-rob-live-190-miles-apart-they-both-drive-in-a-st-16735.html

I am looking at your solution, and I get that we definitely need to convert 1 hour to 45 minutes. However, I am confused about why all we need to do is multiply 120*0.75. Wouldn't that tell us the distance apart they are 45 minutes *after they started traveling*, not necessarily the same as 45 minutes *from reaching their destination* unless the trip takes them exactly an hour and a half? I know I'm probably not thinking about this right, can you explain this?
greenlight-admin's picture

Question link: https://gre.myprepclub.com/forum/rachel-and-rob-live-190-miles-apart-the...
We know that the gap between Rachel and Bob decreases at a rate of 120 miles per hour.

The question asked us to determine "how many miles apart are they exactly 45 minutes BEFORE they meet"
When they meet, they are 0 miles apart.
So, 1 hour BEFORE they meet, they must be 120 miles apart (since the gap decreases by 120 miles every hour)
So, 30 minutes BEFORE they meet, they must be 60 miles apart (since the gap decreases by 120 miles every hour)
Likewise, 45 minutes BEFORE they meet, they must be 90 miles apart

Does that help?


I think I get it now, thanks!

At exactly 12 noon Train A is 500 miles due East of Hotdogsville travelling toward the city at a constant speed of 70 mph. Also at 12 noon train B leaves Hotdogsville and heads due east at 30 mph. At what time will the trains past each other?

For this question I set the times equal to each other since they both left at 12 noon. and said (d+500)/70 = d/30 solve for d and then we can get t through d = r*t. Is this the right equation? I solved this the relative rate way and got a different answer which was correct, I am trying to see if i can use word equations to solve it as well.Thanks!
greenlight-admin's picture

In your solution, you let d = the distance train B travels until they meet.
Since the two trains start 500 miles apart and are travelling towards each other, the distance train A travels until they meet = 500-x
This way, the total distance traveled by both trains = d + (500-d) = 500

In this problem https://gre.myprepclub.com/forum/a-gang-of-criminals-hijacked-a-train-heading-due-south-at-e-9706.html I realized this was a shrinking gap rate problem, however I was under the impression that since the distance is shrinking we would add the rates. However in the problem in the video between yolanda and Bob it is also shrinking but we are adding the rates this time, while in the problem in the link I posted (gang and police) it is a shrinking rate problem but we are subtracting the rates. How does one figure out when to add or subtract the rates?
greenlight-admin's picture

Question link: https://gre.myprepclub.com/forum/a-gang-of-criminals-hijacked-a-train-he...

There are two kinds of shrinking gap questions:
1) The two travelers are moving towards each other.
In this case, the shrink rate = the SUM of the two speeds

2) The two travelers are moving in the same direction, and the traveler who started behind the other traveler is traveling faster.
In this case, the shrink rate = the positive DIFFERENCE of the two speeds

If we're not sure how to determine the shrink rate, we can always see what happens after 1 hour elapses.

In the Bob and Yolanda question (in the video above), the two travelers are 42 miles apart at noon.
One hour later, Yolanda has traveled 4 miles toward Bob, and Bob has traveled 4 miles toward Yolanda.
So, after 1 hour, the two travelers are now 35 miles apart.
In other words, the gap has shrunk by 7 miles in 1 hour.
This tells us that the shrink rate is 7 miles per hour.

In the linked question (above), the train and the police are both traveling in the same direction (south).
At the beginning (say 2 pm), the train is 50 miles ahead of the police.
After 1 hour, the train has traveled 50 miles, and the police have traveled 80 miles.
Since the police have traveled 30 miles further than the train (in one hour), we can see that the gap has shrunk by 30 miles.
In other words, the shrink rate is 30 miles per hour.

I hope that helps


For these two problems I am a bit confused on when it becomes useful to approach rate problems using relative rate or when to use word equations. I noticed in your solutions that for the Rachel and rob problem you used relative rate (not word equations) and for the trains X and train Y you used word equations and not relative rate. May you please clarify when to use relative rate vs. word equations or if these problems can be solved in either or form?
greenlight-admin's picture

Question #1: https://gre.myprepclub.com/forum/rachel-and-rob-live-190-miles-apart-the...
Question #2: https://gre.myprepclub.com/forum/two-trains-x-and-y-started-simultaneous...

Both questions can be solved in at least two different ways.

I went back and solved Question #2 using the same (relative rate) approach I used for Question #1.
Here it is: https://gre.myprepclub.com/forum/two-trains-x-and-y-started-simultaneous...

In general, we can use the relative rate approach to solve most shrinking/expanding gap questions.
I chose to use a word equation for Question #2 because I realized that, if I let d = the distance Train X travels, then solving the corresponding equation for d would immediately answer the question.

for this question I have a doubt on the way the question is posed. I was able to use word equation and solve this out with the equations, however I got hung up on the phrase Train A leaves 1 hour before train A. Wouldnt this mean that if Train B leaves at 10, then Train A leaves at 9 so Train A time = Train B time -1 ? Not +1
greenlight-admin's picture

Question link: https://gre.myprepclub.com/forum/trains-a-and-b-are-190-miles-apart-1300...

Be careful; in distance-rate-time questions, "time" refers to TRAVEL TIME (not departure or arrival times)
Let's say, for example, Train A leaves at 9 am, Train B leaves at 10 am, and the trains meet at noon.
This means Train A's TRAVEL TIME is 3 hours and Train B's TRAVEL TIME is 2 hours.
To make the travel times equal, we must add 1 hour to Train B's travel time.

In other word: (Train A's travel time) = (Train B's travel time) + 1 hour

Hey Brent are there any relative speed questions?
greenlight-admin's picture

That partially depend on how you define "relative speed questions."
I believe you'll find several in the linked questions above as well as here: https://www.greenlighttestprep.com/module/gre-word-problems/video/916

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