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Learn this trick to solve GMAT “Rate” problems faster

GMAT tip for Distance Rate GMAT problemsRemember those math word problems that gave you nightmares in high school algebra class? You know the ones — the problems where one train leaves Boston traveling west at a certain rate, and then another train leaves an hour later traveling on the same track at a faster rate, and then there’s some question you have to answer about distance traveled or time to impact or some other variation of distance/rate/time?

Well, I hate to break it to you, but you’re going to see those types of questions on the GMAT as well.

But I have good news for you. I’m about to make them a whole lot easier.

The Traditional Set-Up for GMAT Distance/Rate Problems

It’s useful to start by having a conceptual understanding of Distance/Rate problems on the GMAT from an algebraic perspective. As I’ve taught in other articles about GMAT Distance/Rate questions, you can use your understanding of the following formula to solve most of these question types algebraically:

Distance = Rate x Time

Distance/Rate problems on the GMAT generally come in one of three forms, and they’re effectively solved the same way. You can click on the following links to read a detailed explanation of each type:

  1. Motion in Opposite Directions (“Collision”)
  2. Motion in the Same Direction (“Catch-Up”)
  3. Round-Trip Questions

Try your hand at this sample “Catch-Up” question and then watch the video below for a detailed algebraic explanation of how to solve it.

Q: Sally left her home at 11:00AM, traveling along Route 1 at 30mph. At 1:00PM, her brother Tom left home and started after her on the same road going 45mph. At what time did Tom catch up to Sally?

(A) 2:45PM
(B) 3:00PM
(C) 3:30PM
(D) 4:15PM
(E) 5:00PM

The 700+ Shortcut

While it’s useful to know how to solve this sample GMAT question algebraically, this very common type of Distance/Rate problem can be done much more efficiently with logic than with algebra. In fact, on “catch up” problems, logic is always faster and easier than algebra.

The name of the game on the GMAT is learning to solve the same problems in less time after all, right?

Let’s take a look at how we’d do it.

Step 1: Determine how far behind the first person/train/car the second one (in this case, Tom) is before it starts. In this problem, Tom starts exactly two hours after his sister Sally. In two hours, the slower person/train/car (in this case, Sally) will have traveled 60 miles (2 hours * 30mph), so Tom is 60 miles behind Sally when he leaves home.

Step 2: Take the distance that the faster person/train/car is behind and divide it by the differential in rates between the two people/trains/cars. Remember, that’s how fast the slower vehicle will be closing the gap. In this problem, Tom is traveling at 45mph while his sister is only traveling at 30mph, so the difference in their rates is 15mph. In other words, Tom will “catch up” 15 miles every hour. Therefore, it will take exactly 4 hours (60 miles divided by 15mph) for Tom to cover the 60 miles he’s behind and catch up to Sally.

Thus, in the sample question above, the correct answer choice is E. If Tom leaves at 1:00PM and it takes him 4 hours to catch up to Sally, then he will do so at 5:00PM.

NOTE:¬†On harder variations of these types of problems, there may be wrinkles. Don’t panic. Use the same logic explained above and you’ll still be able to expertly get the more challenging questions correct in less time as well. For instance, what if the question above wanted to know at what time Tom would be exactly 30 miles ahead of Sally? Do the same calculations from Steps 1 and 2 and then realize it will take exactly two hours more for Tom to gain 30 extra miles on Sally (2 hours * 15mph).

Using Logic on GMAT “Collision” Problems

Logic is a useful compliment to algebra on “Collision” (opposite direction) Distance/Rate GMAT problems as well. It’s always easy to calculate the time to collision — i.e. the time each person/train/car travels* after they each start.

Step 1: Start by adding the rates of the two people/trains/cars to get the total speed with which they are covering the distance. If one train is traveling at 50mph and the other at 40mph, then together they are covering 90 miles every hour.

Step 2: Then divide the total distance between the two people/trains/cars by their combined speed. This will tell you how long (time) it will take for them to “collide.” For example if two trains are 450 miles apart when they start, and together they are covering 90 miles per hour as calculated in Step 1, then it will take them 5 hours (450 miles / 90mph) until they collide.

Step 3: Once you have the time to collision, go back to either person/train/car to see how far they traveled before “impact.” Simply use the Distance = Rate x Time formula.

Step 4: Finally, on harder problems, account for any distances traveled by the person/train/car that started first. If one train traveled 15 miles before they both started, for example, you would add that to the appropriate train’s distance as calculated in Step 3.

*If they start at the same time, their time traveled will necessarily be the same even if their distance traveled is different. If a more advanced problem delivers a wrinkle and tells you that the two vehicles are not starting at the same time, deal with it by accounting for the head start one vehicle has and then reset the equations and solve using the logical 4-step approach described above.

Application Example

Try answering the following sample GMAT “Collision” problem using logic and write your answer and thought process in the Comments area below:

Q: Two sixteen-wheeler transport trucks are 770 kilometers apart, sitting at two rest stops on opposite sides of the highway. Driver A begins heading down the highway driving at an average speed of 90 kilometers per hour. Exactly one hour later, Driver B starts down the highway toward Driver A, maintaining an average speed of 80 kilometers per hour. How many kilometers farther than Driver B will driver A have driven when they meet and pass each other on the highway?

(A) 90
(B) 130
(C) 150
(D) 320
(E) 450

Note: If you’d like to learn the algebraic approach to this problem as well, watch this video:

Leave your answer and comments/questions below: