# GMAT Tip: Understanding Rate Problems

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This tip on improving your GMAT score was provided by Brian Galvin at Veritas Prep.

Most GMAT test-takers report seeing at least one work/rate problem on their exam. Those who scored a little lower than they’d have liked usually report seeing one work/rate problem that completely threw them for a loop, and, yes, slowed their rate-per-question down enough to give them pacing problems.  So it’s important to be ready for challenging rate problems on the GMAT.  And you’re in luck—there are three major themes that will help you navigate most any rate problem you’ll see:

Rate = Output/Time
There are plenty of ways that people choose to remember the “rate formula,” but if you really think about it you don’t need to memorize it…you just know it.  Most of you will drive to the test center, and when you do you’ll see a few dozen speed limit signs along the way.  Speeds are rates, and how do they measure speeds?  Miles per hour—or “distance divided by time.”  Rates equal the output (a distance, a quantity of work, etc.) divided by the time it takes to produce that output.  Rate = Output/Time.

If two (or more) entities are working together to accomplish the same output, you can add their rates together.  For example, if I can paint a house in 8 hours and you can paint a house in 4 hours, then my rate is ⅛ and yours is ¼.  We can add those together to see that combined we work at a rate of 3 houses / 8 hours.

Most work/rate problems involve combined rates, and even some distance/rate problems involve them.  But there are exceptions. If you and I are running against each other in a race, we’re not working together—I have to run my 100 meters and you have to run yours.  We only work together on a distance problem if by my covering part of the distance you don’t have to.  So in the famous “train problem” (a train leaves New York and another leaves Los Angeles, when will they meet…) you can combine their rates, since every mile that the NY-bound train travels to the rendezvous point is a mile that the LA-bound train doesn’t have to.

When conditions change, your rates must too
This is where these problems tend to get tricky.  If a train leaves New York and another train leaves Los Angeles at the same time, and the NY-bound train travels 60 miles per hour and the LA-bound train travels 40 miles per hour, they’ll approach each other at 100 miles per hour.  But when the test wants to make things difficult, it can change the situation ever so slightly:

Train A leaves New York at 9 a.m. Eastern Time on Monday, headed for Los Angeles at a constant rate of 40 miles per hour.  On the same 3,000-mile stretch of track, Train B leaves Los Angeles at noon Eastern Time on Monday, traveling to New York at its constant rate of 60 miles per hour, with one exception: Train B is delayed for exactly 2 hours in Las Vegas (approximately 200 miles from Los Angeles) due to track maintenance.  Assuming the time it takes for Train B to decelerate and re-accelerate when stopping and resuming in Las Vegas is negligible, at what time (Eastern Time) will the two trains meet?

(A) 3:00 p.m. on Tuesday

(B) 4:30 p.m. on Tuesday

(C) 5:00 p.m. on Tuesday

(D) 6:00 p.m. on Tuesday

(E) 7:30 p.m. on Tuesday

On this question, you cannot simply use the combined rate of 100 miles per hour, as Train B doesn’t travel for a full 5 hours of the time that Train A is making its way toward Los Angeles.  So you need to use two different rates: 40 miles per hour for the five hours that Train A travels alone, and the combined rate of 100 miles per hour for the remainder of the time that both trains are traveling toward each other.  So your calculation will look like:

Solve for Distance = 200 miles

This means that of the 3,000 miles total, 2,800 of them will take place with both trains traveling together.  That combined rate is:

Solve for the number of hours here and you’ll find that it takes 28 hours.  That means that the trains traveled 28 hours together and 5 with Train A alone, so the time that it takes from the moment—9 a.m. on Monday—that Train A departed is 33 hours.  Twenty-four of those hours get you from Monday to Tuesday, leaving nine additional hours.  That means that the trains meet at 6pm on Tuesday.

The important takeaway here is that you recognize that, when conditions change, you need to account for different rates.  And those conditions can change in two ways. Two or more entities can work together (or alone) for a certain amount of time, or they can work together (or alone) for a certain amount of output (for example, John and Steve will work together to paint a fence, but John will stop when the project is half completed).  Your job on these challenging problems is to determine when/how those conditions change.

Be prepared for tricky rate problems on test day, and remember that the trickiest of all problems usually involve a combination of combined rates…but only for a limited time.

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