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Welcome back to our series on GMAT quant rate problems. In the last article, we learned about problems involving fuel consumption rates. This article will address rates problems that incorporate geometry concepts. These are speed = distance/time problems where the distance is a measurement of a familiar shape – like a circle.

## Official GMAT Problem for Practice:

In a particular machine, there are 2 gears that interlock; one gear is larger in circumference than the other. The manufacturer of the gears guarantees that each gear will last for at least 6,000,000,000 revolutions. Assuming that there is no slippage between the 2 gears and that when one gear rotates the other gear also rotates, the larger gear is guaranteed to last how many days longer than the smaller gear?

1. The diameter of the larger gear is twice the diameter of the smaller gear.
2. The smaller gear revolves 600 times per minute.

(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.

Let’s think about the question being asked before we evaluate the statements. We need to know the difference in days of the lifespans of the gears, given the lifespan for each gear in revolutions (6 billion!). To answer this question, we will need to know the rate of each gear in revolutions per day.

Statement 1 provides the proportion of the gears’ diameters: a 2:1 ratio. To make use of this data, you need to know the basic fact that the circumference of a circle is directly related to its diameter (C = πD). If the gears’ diameters have a 2:1 ratio, then their circumferences do as well.

This circumference is the distance traveled by a tooth on the outside edge of the gear. Since the gears are interlocked, their teeth are “traveling” around their respective circumferences at the same speed. But since the teeth on the larger gear must travel twice as far to complete a revolution, each revolution of the larger gear takes twice as much time (again, distance and time are directly related).

So from statement 1, we know that the larger gear revolves once every time the smaller gear revolves twice. But we still don’t know the gears’ rates in revolutions per day. Statement 1 on its own is insufficient.

Statement 2 provides the rate of the smaller gear in revolutions per minute. For data sufficiency, this is just as good as revolutions per day, since we could multiply by the number of minutes in a day (1440) to convert this rate to revolutions per day. This is useful, but if we consider this data by itself, we don’t know anything about the relative rates of the larger and smaller gears. Statement 2 by itself is insufficient.

Combining the statements, we can find the rate of each gear in revolutions per day. Dividing the lifespan in revolutions of each gear (6 billion) by the difference between their “revolutions per day” rates would reveal the difference in days of the lifespans of the gears, answering the question asked in this problem. Statements 1 and 2 together are sufficient, and the correct answer is C.

## Official GMAT Problem for Practice Involving Rates and Circles:

A circular rim 28 inches in diameter rotates the same number of inches per second as a circular rim 35 inches in diameter. If the smaller rim makes x revolutions per second, how many revolutions per minute does the larger rim make in terms of x?

(A) 48π / x

(B) 75x

(C) 48x

(D) 24x

(E) x / 75

This problem, like the last, provides data about the diameters of the circles. A quick recognition of the ratio of these diameters – and therefore of the circumferences – helps solve the problem.

The smaller rim is 28 inches in diameter, and the larger rim is 35 inches in diameter. 28 = 4 * 7, and 35 = 5 * 7, so the circles have a diameter:diameter and circumference:circumference ratio of 4:5. We are told that these circular rims rotate at the same rate of inches per second. Since the larger rim has 5/4 as many inches to cover for each revolution, it takes 5/4 as much time to complete each revolution (because distance and time are directly related).

Or, in a given amount of time, the larger rim completes 4/5 as many revolutions as the smaller rim.  If, as we are told, the smaller rim completes x revolutions per second, the larger rim completes (4/5)x revolutions per second. This problem asked for the rate of the larger rim in revolutions per minute, so our final step is to multiply (4/5)x by 60. The result is 48x, and the correct answer is C.

If you come across GMAT quant problems that combine rates with geometry elements, lean on your fundamentals about the direct and inverse relationships between speed, distance, and time. The next article in the series will explore how rates function in data sufficiency inequalities problems.

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Contributor: Elijah Mize (Apex GMAT Instructor)