Welcome back to our series on rate problems. In the first article, we introduced the relationship between the variables of rate, work, and time – or speed, distance, and time. This article and the next will teach you how to approach combined work problems which involve two or three machines working together to complete a job.
Almost all of these problems ask for time: if one machine takes a hours to do the job alone and the other machine takes b hours, how many hours do they take to do the job when they work together? Or, less frequently, if one machine takes a hours to do the job alone and the two machines take c hours to do the job when they work together, how many hours does the other machine take to do the job alone?
GMAT Combined work Problems
These scenarios are all governed by the same formula. Using a, b, and c for the independent and combined times for the machines as in the example above:
(1 / c) = (1 / a) + (1 / b)
Recall from our first article that rate and time are inversely related. Since this is true, then when a is the amount of time to complete a given job, the reciprocal (1 / a) is the fraction of the job that is completed in a single unit of time. If a is expressed in hours, then (1 / a) is the fraction of the job completed by machine A in a single hour. The fractions (1 / a) and (1 / b) represent the rates for machines A and B in a unit we might call “jobs per hour.”
GMAT rates problems sometimes use another unit of time like days or seconds, but hours are common.
Since this is true, then the equation above simply says this: the combined rate of the machines, (1 / c), is equal to the sum of their independent rates, (1 / a) and (1 / b). Rates are “stackable” or addable! If in a single hour, you can paint ⅙ of a room and your friend can paint ⅛ of a room, then together you can ⅙ + ⅛ of the room in that hour. Simple enough, right? As you can see, these problems often involve finding a common denominator in order to add the fractions. But in the next article, we’ll learn a workaround to help you avoid that step.
Remember that the sum of your fractions is still in the rates unit of “jobs per hour.” In order to get back to the “hours per job” figure that most problems ask for, simply flip the fraction. In the example above, you and your friend paint at a combined rate of ⅙ + ⅛ = 4/24 + 3/24 = 7/24 “jobs per hour.” How many hours will it take you and your friend to paint the room? Flip the fraction to find 24/7, or 3 3/7 hours.
Official GMAT problems
Let’s try this out on an official GMAT problem:
Printing press X can print an edition of a newspaper in 12 hours, whereas press Y can print the same edition in 18 hours. What is the total number of hours that it will take the two presses, working together but independently of one another, to print the same edition?
(A) 15
(B) 7.4
(C) 7.2
(D) 7.0
(E) 6.8
Press X takes 12 hours to print the edition, meaning that it prints at a rate of 1/12 editions per hour. Press Y takes 18 hours to print the edition, meaning that it prints at a rate of 1/18 editions per hour.
(1 / c) = (1 / a) + (1 / b)
a = 12
b = 18
1 / c = 1/12 + 1/18 = 3/36 + 2/36 = 5/36 editions per hour
c = 36/5 hours per edition = 7.2 hours per edition
And the correct answer is C. All you have to do is plug in the values, add the fractions, and flip the sum.
Here’s another official problem:
Three printing presses, R, S, and T, working together at their respective constant rates, can do a certain printing job in 4 hours. S and T, working together at their respective constant rates, can do the same job in 5 hours. How many hours would it take R, working alone at its constant rate, to do the same job?
(A) 8
(B) 10
(C) 12
(D) 15
(E) 20
I guess the GMAT really likes printing presses. This time there are three of them, but don’t let that scare you! The way the problem is set up, presses S and T can be treated like a single printing press. We have the combined time (for all three presses) of 4 hours and the “press ST” time of 5 hours. We can plug these right into our “add the rates” equation to find the time for press ST.
(1 / c) = (1 / a) + (1 / b)
c = 4
a = 5
(1 / 4) = (1 / 5) + (1 / b)
(5/20) = (4/20) + (1/20)
b = 20
And the correct answer is E.
Data sufficiency problems
Let’s move into some data sufficiency problems. We won’t have to solve for anything, but the same formula will govern our thinking.
Machine R and machine S work at their respective constant rates. How much time does it take machine R, working alone, to complete a job?
- The amount of time that it takes machine S, working alone, to complete the job is ¾ the amount of time that it takes machine R, working alone, to complete the job.
- Machine R and machine S, working together, take 12 minutes to complete the job.
(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 provides a factor of relationship between the times for machine S and machine R to complete the job. On its own, this isn’t sufficient. It is relative information comparing the two machines, but we were asked the absolute question of how long a single machine takes. There is nothing here to help “scale” our times. Maybe machine R takes 4 minutes, while machine S takes 3 minutes. Maybe machine R takes 4 days, while machine S takes 3 days. We don’t know.
Statement 2 provides the combined time for the machines working together. On its own, this is insufficient, because we don’t know how the machines relate to each other. Maybe S is much faster than R and basically doing all the work itself. Or maybe R is much faster than S. Or maybe they work at the same rate. The combined time of 12 minutes simply doesn’t tell us how the machines are splitting the work.
Together, the statements are sufficient. We can prove it with the formula. Let’s keep the variable c for combined time but use r and s instead of a and b.
(1 / c) = (1 / r) + (1 / s)
s = (¾ * r) (statement 1)
c = 12 (statement 2)
(1 / 12) = (1 / r) + [1 / (¾ * r)]
Since r is the only variable left in the equation, the two statements together are sufficient to solve for r, the time for machine R alone to complete the job.
Here’s a similar data sufficiency problem:
Working together, Rafael and Salvador can tabulate a certain set of data in 2 hours. In how many hours can Rafael tabulate the data working alone?
- Working alone, Rafael can tabulate the data in 3 hours less time than Salvador, working alone, can tabulate the data.
- Working alone, Rafael can tabulate the data in ½ the time that Salvador, working alone, can tabulate the data.
(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.
We can again use the variables r and s for the “working alone” times. But this problem, unlike the last, provides the combined time of 2 hours in the question stem. This means that any statement putting Rafael’s independent time r in terms of Salvador’s independent time s – or vice versa – is sufficient.
Statement 1 says that Rafael lakes 3 hours less than Salvador. r = s – 3, or s = r + 3. This is sufficient.
(1 / c) = [1 / (s – 3)] + (1 / s)
Statement 2 says that Rafael’s time r is ½ of Salvador’s time s. r = s/2, or s = 2r. Again, this is sufficient.
(1 / c) = (1 / r) + [1 / (2r)]
The correct answer is D: each statement on its own is sufficient.
Data sufficiency problem: Combined work
Here’s a final data sufficiency problem for this first combined work article:
If two copying machines work simultaneously at their respective constant rates, how many copies do they produce in 5 minutes?
- One of the machines produces copies at the constant rate of 250 copies per minute.
- One of the machines produces copies at twice the constant rate of the other machine.
(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.
We’ve moved on from printing presses but are still stuck in the early 2000s with copying machines. This question also differs from the previous ones in that it did not ask how much time it takes the machines to complete a given amount of work; it asked how much work the machines can complete in a given amount of time. Since work = rate * time, we are looking for statements that supply us with the rates at which the machines work.
Statement 1 is a good start, but it can’t be sufficient on its own because it only provides a rate for one of the two machines. Statement 2 can’t be sufficient on its own because it only tells us that one machine works twice as fast as the other – we know nothing about the speed of the machines in absolute terms.
The trap answer on this problem is C. Combining the statements, we have a lot of info. We know the rate of one of the machines, and we know that one machine is twice as fast as the other. But we don’t know whether the rate given of 250 copies per minute is for the fast machine or the slow machine! If it is for the fast machine, then the other machine produces 125 copies per minute. If it is for the slow machine, then the other machine produces 500 copies per minute. In the former case, the combined rate of the machines is 375 copies per minute. In the latter case, their combined rate is 750 copies per minute. Therefore the correct answer is E: both statements together are still insufficient.
Remember this distinction between the fast machine and the slow machine in a combined work problem.
It will be a major focus of the next article and a key to solving these problems very quickly.
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Contributor: Elijah Mize (Apex GMAT Instructor)