The quantitative section of the GMAT works by using simple mathematical relationships to create challenging problems. This makes rate problems – which depend on simple relationships that are surprisingly versatile in their presentation and arrangement – a favorite and frequent category for GMAT quant. While these problems may be a favorite for the writers of the test, they are rarely a favorite for test-takers! If you groan every time you encounter a problem that begins like this:
“Two machines, X and Y, working together at their respective constant rates…”
then this series is for you. With a solid grasp of the fundamentals, the right mathematical tools, and some practice, you’ll be able to solve these problems faster than you can say “respective constant rates.” Okay, I exaggerate. But only a little.
Success on GMAT rate problems begins with a solid grasp of the foundational rate, work, time relationship.
Rate = work / time
Time = work / rate
Work = rate * time
This is the same relationship and the same equation expressed in terms of each of the three variables. The first form, rate = work/time, is the most fundamental starting point because quite simply, work/time is what a rate is. There is no single unit for expressing a rate. Instead, we have to use both a work unit and a time unit, connected by a word laden with mathematical meaning: per.
As an exercise in units conversions, my high school physics teacher made us express the speed of light in furlongs per fortnight! A furlong is ⅛ of a mile, and a fortnight is two weeks or fourteen days. Of course, our starting point was the speed of light expressed in meters per second. But whether you use meters per second, furlongs per fortnight, or marathons per leap year, any expression of speed is “distance unit per time unit,” with the word “per” acting like a fraction bar.
This brings up an important point about rate problems: problems involving machines completing jobs or producing widgets are fundamentally the same as problems involving speed, distance, and time! If one type of problem makes more sense to you than the other, you can take advantage of this to help your comprehension of the other type of problem.
If you like machines, you can think of cars/trains/bikes/buses as machines that “produce” kilometers or miles, where the “work” done by a moving body is the distance it covers in a given time.
If you understand speed, distance, and time, you can remember that rate, work, and time are mathematically related in the exact same way. To say it another way, “rate” is the general term for something happening in a given amount of time, and “speed” is simply what we call a rate when the thing happening is motion (which needs distance units to be expressed).
Here are both forms of the formulas, in parallel:
Rate = Work / Time Speed = Distance / Time
Time = Work / Rate Time = Distance / Speed
Work = Rate * Time Distance = Speed * Time
It is not enough to know these simple formulas by rote: you need to understand the network of direct and inverse relationships between the variables. When a single variable is isolated, or solved for, or expressed in terms of the others (three ways of saying the same thing), it is directly related to the variables on the same side of any fraction bars and inversely related to the variables on the opposite side of any fraction bars.
Remember that if an isolated variable is not in a fraction at all, as in the left sides of the formulas above, it can be thought of as the numerator of a fraction with a denominator of 1. In other words, if it’s not a denominator, it’s a numerator.
Based on this system, the following direct and inverse relationships exist between the variables:
Rate and Time (or Speed and Time) are inversely related.
Work and Time (or Distance and Time) are directly related.
Work and Rate (or Distance and Speed) are directly related.
A direct relationship means that the values of variables increase together and decrease together. An inverse relationship means that the values of variables change in opposite directions: if the value of one variable increases, the value of the other decreases, and if the value of one variable decreases, the value of the other increases. Let’s apply this to the variables in the formula.
As rate increases, the amount of work done in a given time increases.
As rate increases, the amount of time to do a given amount of work decreases.
As time increases, the amount of work done at a given rate increases.
As time increases, the rate to complete a given amount of work decreases.
As work increases, the rate to complete the job in a given amount of time increases.
As work increases, the time to complete the job at a given rate increases.
Try “translating” these relationships into speed, distance, and time to help solidify the concepts.
Since the rate = work/time formula is “pure variables” and involves no exponents, coefficients, or constants, the change in any variable given the change in another is remarkably predictable and consistent. If the value of one variable changes by a given factor, the value of a directly-related variable changes by the same factor, and the value of an inversely-related variable is multiplied by the reciprocal of that factor.
For example, if a car’s average speed over a given distance increases from 60mph to 75mph (a multiplication of the value by a factor of 5/4), the time it takes to cover that distance becomes 4/5 (the reciprocal of 5/4) of what it was, because speed and time are inversely related.
If the time in which a machine needs to complete a job is halved, the rate at which the machine must work to finish in time doubles. Or if the time allowed to complete the job is multiplied by a factor of 3/2 (like an increase from 2 hours to 3 hours or from 2 days to 3 days), the machine may work at ⅔ of its prior rate. If the size of a job is multiplied by a factor of 5/3, a machine (or person) must work 5/3 as fast in order to finish in the same amount of time because work and rate are directly related. If the machine or person simply keeps working at the same rate, the job will take 5/3 as long as it would have taken, because work and time are also directly related.
Hopefully these relationships are starting to make sense. Let’s try them out on some official GMAT quant problems:
Carl averaged 2m miles per hour on a trip that took him h hours. If Ruth made the same trip in ⅔h hours, what was her average speed in miles per hour?
This one follows the form of the examples above. If Ruth completes the trip in ⅔ the time it took Carl, she must have been traveling 3/2 times as fast as Carl, on average, because time and rate (or speed) are inversely related. Carl’s speed was 2m, so Ruth’s speed was 3/2 * 2m = 3m. The correct answer is E. The inverse relationship of speed and time is all you needed for this problem.
Here’s another official problem:
If a certain machine produces bolts at a constant rate, how many seconds will it take the machine to produce 300 bolts?
- It takes the machine 56 seconds to produce 40 bolts.
- It takes the machine 1.4 seconds to produce 1 bolt.
(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.
The question asks “how many seconds,” so the variable we’re solving for is time to complete a given amount of work: 300 bolts. With the value for work filled in, the only remaining variables are time and rate. Therefore if a statement allows us to calculate a rate, time will be the only variable left, and that statement will be sufficient.
Each statement provides us with the amount of time it takes for the machine to complete a given amount of work: 56 seconds for 40 bolts or 1.4 seconds per bolt. Since rates don’t have their own units and can only be expressed as a given amount of work in a given amount of time, these statements are giving us the rate at which the machine produces bolts! We don’t have to calculate anything or actually find how many seconds this machine takes to product 300 bolts. Each statement provides the rate at which the machine will work to complete the 300-bolt job, so each statement is sufficient to determine the time in which the machine will complete this job. The correct answer is D.
Here’s one more problem for this introductory article:
Jonah drove the first half of a 100-mile trip in x hours and the second half in y hours. Which of the following is equal to Jonah’s average speed, in miles per hour, for the entire trip?
(A) 50 / (x + y)
(B) 100 / (x + y)
(C) 25/x + 25/y
(D) 50/x + 50/y
(E) 100/x + 100/y
How do you feel about problems with algebraic answers? Rate problems with algebraic answers will receive their own article in this series, but this problem is simple enough to be included here. I’ve seen test-takers get hung up on the detail about Jonah’s trip being split into two halves with times x and y. But this problem is only pretending to be complex.
A readiness to remember the speed = distance/time relationship makes this problem an easy one. Jonah’s average speed is nothing but the total distance he covered divided by the total time it took him to cover this distance. The distance is given as 100 miles. If Jonah drove the first half of this trip in x hours and the second half in y hours, then his total drive-time for the trip is x + y. Since average speed = total distance/total time, Jonah’s average speed was 100 / (x + y). The correct answer is B.
If these problems were easy for you, great! Stick with our series of articles on rate problems, and we’ll ratchet up the difficulty as we go. The next two articles will teach you how to handle problems where machines work together to complete a job.
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Contributor: Elijah Mize (Apex GMAT Instructor)