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Welcome back to our series on GMAT quant rate problems. The last article covered rate problems with geometry elements. In this article, we’ll see how rates are used in data sufficiency inequalities problems. Inequalities are a common form of data sufficiency problems, and rates add an extra layer of complexity to these problems. Let’s learn how to solve these problems confidently using rate fundamentals.

## Official GMAT Problem for Practice:

A conveyor belt moves bottles at a constant speed of 120 centimeters per second. If the conveyor belt moves a bottle from a loading dock to an unloading dock, is the distance that the conveyor belt moves the bottle less than 90 meters? (1 meter = 100 centimeters)

(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.

Problems like this one should be handled like any other data sufficiency inequalities problem: by finding the “cutoff point” for the answer to the question that was asked. This problem asks whether the distance the conveyor belt moves the bottle is less than 90 meters. Since the speed is constant and since distance = speed * time, the answer to this question depends on the time it takes for the bottle to cover the span between the docks. The two statements, of course, provide data about this time in the form of inequalities.

Like many rate problems, this one involves some unit conversions. We are given the speed of the bottles on the conveyor belt in centimeters per second. But the question asks about the distance in meters (not centimeters), and the statements provide times in minutes (not seconds).

We should start by converting the given speed of 120 centimeters per second to meters per minute. 120 centimeters is 1.2 meters, so the bottle moves at a speed of 1.2 meters per second. Multiplying 1.2 by 60, we recognize a speed of 72 meters per minute. Using this speed, we can find the “cutoff point” time for the distance of 90 meters.

Time = distance/speed

Time = 90 meters / 72 meters/minute

Time = (90 / 72) minutes

Time = 5/4 minutes = 1.25 minutes

1.25 minutes is our “cutoff point” for this question. Since distance and time are directly related, then the distance is greater than 90 meters whenever the time is greater than 1.25 minutes and less than 90 meters whenever the time is less than 1.25 minutes. If the time is equal to or greater than 1.25 minutes, the answer to the question that was asked is, “No, the distance is not less than 90 meters.” If the time is less than 1.25 minutes, the answer to the question is, “Yes, the distance is less than 90 meters.”

Here’s how a “cutoff point” functions on a data sufficiency inequalities problem: if it is outside the range defined by the statement, that statement is sufficient. All of the possible values will lead to the same answer (yes or no) to the question.  If the cutoff point is inside the range defined by the statement, that statement is insufficient. Some allowed values will lead to a “yes” answer to the question, and some allowed values will lead to a “no” answer.

Let’s apply this principle to the problem, one statement at a time.

The cutoff point of 1.25 minutes is outside the range defined by this statement. All times less than 1.2 minutes will lead to a dock-to-dock distance of less than 90 meters and an answer of, “Yes, the distance is less than 90 meters.” Statement 1 is sufficient.

The cutoff point of 1.25 minutes is inside the range defined by this statement. Some allowed values will lead to an answer of, “Yes, the distance is less than 90 meters,” while other allowed values will lead to an answer of, “No, the distance is not less than 90 meters.” If the time is 1.15 minutes, the distance will be less than 90 meters and the answer will be, “Yes.” But if the time is 1.35 minutes, the distance will be greater than 90 meters and the answer will be, “No.” So statement 2 is insufficient. The correct answer is A.

## Official GMAT Problem for Practice:

If Paula drove the distance from her home to her college at an average speed that was greater than 70 kilometers per hour, did it take her less than 3 hours to drive this distance?

1. The distance that Paula drove from her home to her college was greater than 200 kilometers.
2. The distance that Paula drove from her home to her college was less than 205 kilometers.

(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.

In the last problem, the question stem provided a speed and then asked about distance, with the statements providing data about time. In this problem, the question stem again provides a speed, but it asks about a time, with the statements providing data about distance. Given the relationships between speed, distance, and time, this isn’t a significant difference between the problems. But there is a significant difference: the last problem defined a fixed speed, while this one uses an inequality to define the speed as greater than 70 kilometers per hour.

We can still apply the “cutoff point” approach, using the asked time of 3 hours and a speed of 70 kilometers per hour.

Distance = speed * time

Distance = 70 km/h * 3 hours

Distance = 210 km

If Paula averages exactly 70 km/h, then 210 kilometers is the cutoff point for the asked-about time of 3 hours. But her speed is not necessarily 70 km/h – it could be anything faster than this. This means that if the distance is greater than 210 kilometers, she may or may not complete the trip in less than 3 hours. But if the distance is less than 210 kilometers, she will certainly complete the trip in less than 3 hours, because she will certainly drive at least the minimum speed to achieve that time. Therefore, if a statement establishes the distance as less than 210 kilometers, it is sufficient. And any statement that doesn’t do this is insufficient.

As before, let’s take the statements one at a time:

1. The distance that Paula drove from her home to her college was greater than 200 kilometers.

This allows some distances less than 210 kilometers but also any distance greater than 210 kilometers, meaning Paula may or may not complete her trip in less than 3 hours. Insufficient.

2. The distance that Paula drove from her home to her college was less than 205 kilometers.

Every distance allowed by this statement is less than the cutoff point of 210 kilometers, which means that Paula completes her trip in less than 3 hours in every possible scenario. Sufficient. The correct answer is B.

Imagine that one of the statements had said, “The distance that Paula drove from her home to her college was greater than 215 kilometers.” If Paula’s speed had been defined as exactly 70 kilometers per hour, this statement would be sufficient. The cutoff point of 210 kilometers would be outside the range defined by the statement. But since Paula’s speed was only defined as greater than 70 km/h, distances greater than 210 are all insufficient. This kind of the cutoff point – one defined by its own inequality – only works on one side.

The next article in the series will explore the concept of average speed in GMAT quant rate problems.

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Contributor: Elijah Mize (Apex GMAT Instructor)