Welcome back to our series on GMAT quant rate problems. In the last article, we learned about average speed and observed various ways that data sufficiency problems can try to “trick” test-takers with this concept. This article will contrast average speed against instantaneous speed. Whereas average speed is the quotient of the distance covered divided by the time it took to travel that distance, instantaneous speed is the speed a body is traveling at a given moment – or instant – of time.
GMAT quant rate problems: Contrast average speed against instantaneous speed
An average speed is made up of a series of instantaneous speeds which may vary widely. For example, an airplane traveling from gate 43 at LAX to gate 11 at JFK may have an average speed of about 500 mph for the whole trip from gate to gate, but the plane’s instantaneous speed as it rolls out of the gate at LAX or into the gate at JFK is very low.
Here’s a GMAT official data sufficiency problem that depends on knowledge about instantaneous speed
At what speed was a train traveling on a trip when it had completed half of the total distance of the trip?
- The trip was 460 miles long and took 4 hours to complete.
- The train traveled at an average rate of 115 miles per hour on the trip.
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.
Right away, you should recognize that statement 2 adds nothing to statement 1. We can already see from statement 1 that the train’s average speed for the trip was 115 miles per hour by taking 460/4 (since average speed = total distance/total time). Is any of this data sufficient to determine how fast the train was moving the instant it crossed the 230 mile mark? No! The average speed of 115 mph is made of an infinite number of instantaneous variables, none of which can be determined from the average speed alone. The correct answer is E.
Here’s a slightly more complex GMAT data sufficiency problem
Stations X and Y are connected by two separate, straight, parallel rail lines that are 250 miles long. Train P and Train Q simultaneously left Station X and Station Y, respectively, and each train traveled to the other’s point of departure. The two trains passed each other after traveling for 2 hours. When the two trains passed, which train was nearer to its destination?
- At the time when the two trains passed, train P had averaged a speed of 70 mph.
- Train Q averaged a speed of 55 miles per hour for the entire trip.
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.
After reading the question stem, you might be thinking something like, “Whichever train had the higher average speed up to the time when they passed each other was closer to its destination at that point.” This is a good thought, one informed by the fundamental fact that distance = speed * time, and it should help you recognize that statement 2 is useless.
Knowing Train Q’s overall average speed doesn’t help us. Since Train Q’s trip is a series of instantaneous speeds which may differ, the average speed for the entire trip is not necessarily equal to the average speed up to the point when the trains passed. Maybe Train Q averaged 90 mph until it passed Train P but then slowed down dramatically! Or maybe Train Q averaged only 30 mph until it passed Train P but then sped up dramatically! We simply can’t know.
Statement 1 is better than statement 2 because it provides an average speed up to the point of the trains passing. However, it only provides this data for one of the two trains. Is this sufficient? You could be forgiven for saying no, but you’d be wrong. How can data about only one of the two trains be sufficient? Well, since distance = speed * time, and since the question stem and statement 1 together provide the average speed and the travel time for train P up to the point of passing, we can calculate the distance Train P has traveled at this point. “Well and good,” you say, “but what about Train Q’s distance?”
This too is knowable, since the two trains together must travel 250 miles in order to meet. If Train P travels 140 miles to the meeting point (the actual distance from 2 hours at an average of 70 mph), then Train Q has traveled 250 – 140 = 110 miles. A subtle problem, and an instructive case study in the importance of considering every piece of data provided by a problem. Both the 250 mile distance and the travel time of 2 hours would be easy to overlook, but both are needed in order to recognize the sufficiency of statement 1. The correct answer is A.
This concludes our study of average speed and instantaneous speed on GMAT quant. In the next article, we’ll explore problems involving a change in the given speed of a vehicle.
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Contributor: Elijah Mize (Apex GMAT Instructor)