Welcome back to our series on GMAT quant rate problems. The last two articles covered machines working together to complete a job. This article addresses problems that have two objects moving at given speeds instead of two machines working at given rates.

Recall from the first article in the series that speed/distance/time scenarios are mathematically identical to rate/work/time scenarios. When two objects are moving, the work done by them is the change in the distance between them. This leads to a somewhat counterintuitive fact: two objects traveling the same direction are technically working against each other, like one pump filling a pool and another pump emptying the pool at the same time. Any change in distance accomplished by one moving object will be partly nullified by the motion of the other object. While the getaway car works to increase the distance between itself and the police vehicles, the police vehicles work to decrease this distance.

When objects move in opposite directions – whether converging or diverging – they work together to change the distance between them – either to decrease it or to increase it.

In either scenario – travel in the same direction or travel in opposite directions – it is necessary to consider also the positions of the two objects. Imagine that two trains start out 100 miles apart, traveling towards each other, then meet and continue traveling opposite directions to end up 30 miles away from each other. The overall “work done” by the trains – the change in the distance between them – is 130 miles (100 + 30), not 70 miles (100 – 30). The trains did 100 miles of work to meet and then another 30 miles of work.

Imagine that Racecar A is trying to overtake Racecar B. If Racecar A starts out 50 meters behind Racecar B and ends up 20 meters ahead of Racecar B, the “work done” by Racecar A is 70 meters (50 + 20), not 30 meters (50 – 20).

Not every problem involving two moving objects will include this element of complexity. Often, they are simply about overtaking/catching up, not catching up, and then passing. In any case, what controls the problem is the sum or difference of the given speeds. When the two objects are traveling in opposite directions, we care about the sum of their speeds, or their combined speed. This is the rate of change of the distance between the objects. When the two objects are traveling in the same direction, we care about the difference between their speeds, or their net speed. Again, this is the rate of change of the distance between the objects.

Let’s try an official GMAT problem:

Maria left home ¼ hour after her husband and drove over the same route as he had in order to overtake him. From the time she left, how many hours did it take Maria to overtake her husband?

  1. Maria drove 60 miles before overtaking her husband.
  2. While overtaking her husband, Maria drove at an average rate of 60 miles per hour, which was 12 miles per hour faster than her husband’s average rate.

(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.

Statement 1 by itself can’t be sufficient. It provides the distance Maria traveled before catching up. But we don’t know how fast she was driving, so we can’t use time = distance/speed to answer how long it took Maria to catch up.

Statement 2 provides Maria’s speed and the difference between her speed and her husband’s. If she averaged 60 mph, and this was 12 mph faster than her husband’s average speed, then her husband averaged 48 mph.

Intuition might tell us (correctly) that the “catch-up rate” of 12 mph given by statement 2 is sufficient, but it could be tempting to select answer choice C, thinking that we also need the data from statement 1 about how far Maria traveled before overtaking her husband. In fact, we don’t need this distance, because we know that when Maria overtook her husband, each person had driven the same distance. Since distance = rate * time, the product rate * time is equal for Maria and her husband.

Maria’s distance = Husband’s distance

(Maria’s rate * Maria’s time) = (Husband’s rate * Husband’s time)

Statement 2 gives us the rates, and we know from the question stem that Maria’s husband’s time is ¼ hour greater than Maria’s time.

(Maria’s rate * Maria’s time) = (Husband’s rate * Husband’s time)

(60 * Maria’s time) = [(48) * (Maria’s time + ¼)]

Since Maria’s time in hours is the only variable in this equation, it is solvable and sufficient. We only used data from statement 2 and from the question stem, so the correct answer is B.

Here’s another official problem:

If Car X followed Car Y across a certain bridge that is ½ mile long, how many seconds did it take Car X to travel across the bridge?

  1. Car X drove onto the bridge exactly 3 seconds after Car Y drove onto the bridge and drove off the bridge exactly 2 seconds after Car Y drove off the bridge.
  2. Car Y traveled across the bridge at a constant speed of 30 miles per hour.

(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.

To restate the date from statement 1 in a more helpful way, Car X covered the ½ mile bridge 1 second faster than Car Y did.  Since we don’t know the speed or time for either car in absolute terms, this is insufficient by itself.

Statement 2 provides the speed at which Car Y covered the ½ bridge but tells us nothing about Car X. Either statement alone is insufficient, so let’s combine the data. The number of seconds it took Car Y to cross the bridge can be derived from time = distance / speed, since we have the distance of ½ mile and the speed of 30 mph. Once we know this number of seconds for Car Y, we simply add 1 second to obtain the time for Car X (using the data provided by statement 1). The correct answer is C: both statements together are sufficient.

Here’s a final “two moving objects” problem:

Pat’s watch gains an extra 10 seconds every 2 hours. Kim’s watch loses 5 seconds every 3 hours. If both watches are set to the correct time at 8 o’clock in the morning and run without interruption, after 72 hours, what will be the difference in time between Pat’s watch and Kim’s watch?

(A) 4 min

(B) 6 min

(C) 6 min 40 sec

(D) 7 min 30 sec

(E) 8 min

Okay, this isn’t technically a “two moving objects” problem, but it might as well be. What matters is that the two things (watches) are “moving” in opposite directions – one watch is gaining seconds while the other loses seconds. We find ourselves working with the bizarre time-time unit of “seconds per hour,” but why should we treat this any differently than miles per hour? Since the watches are running in opposite directions, we will find the combined rate via addition, which means we need a common denominator.

Pat’s watch → 10 seconds every 2 hours → 30 seconds every 6 hours

Kim’s watch -> 5 seconds every 3 hours -> 10 seconds every 6 hours

  40 seconds every 6 hours

For every 6 hours, the watches diverge by 40 seconds. 72 / 6 = 12, so there are 12 40-second “pieces of distance” between Pat’s watch and Kim’s watch at the end of the 72 hours. 12 * 40 = 480 seconds. 480 seconds / 60 = 8 minutes, and the correct answer is E.

This concludes our study of “two moving objects” problems. In the next article, we’ll look at GMAT quant problems that involve more than two or three machines and introduce the concept of “machine hours.”

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Contributor: Elijah Mize (Apex GMAT Instructor)