Welcome back to our series on GMAT quant rate problems. In the last article, we studied problems where two objects are moving, either in the same direction or in opposite directions. This article will return to machines and address problems that feature more than two or three machines.

## GMAT quant rates problems: Multiple machines and “Machine hours”

For this topic, the best way to begin is by viewing an official GMAT problem:

Working simultaneously and independently at an identical constant rate, four machines of a certain type can produce a total of x units of product P in 6 days. How many of these machines, working simultaneously and independently at this constant rate, can produce a total of 3x units of product P in 4 days?

(A) 24

(B) 18

(C) 16

(D) 12

(E) 8

The problem gives us some information about a group of four machines, and we can see from the answer choices that the group of machines we are asked about is even larger. These problems are different from the typical two-machine variety we addressed in articles 2 and 3. Problems like this one can be solved easily by thinking in “machine hours.”

Sometimes the term “man-hours” is used in the workplace. The term needs to be updated, but it is a useful concept. A “man hour” is the amount of work that can be done by one person in one hour. A 72 man-hour job may be completed in 1 hour if 72 people work simultaneously or in 6 hours if 12 people work simultaneously. The term refers to the total number of hours that must be worked to complete the job, regardless of how many people are sharing those hours.

We can apply this same concept to machines. In this problem, we are actually working in “machine days,” since the time unit in the scenario is days instead of hours. But machine days are just as good as machine hours. Let’s use what we’re given to calculate the number of machine hours required to produce x units of product P.

“. . . four machines of a certain type can produce a total of x units of product P in 6 days.”

Machine days = (# of machines) * (# of days of work)

Machine days = 4 * 6

Machine days = 24

Since it takes 24 machine days to produce x units of product P, it takes 3 * 24 = 72 machine days to produce 3x units of product P. The question asks how many machines it will take to produce these 3x units in only 4 days. We can simply apply the same formula:

Machine days = (# of machines) * (# of days of work)

72 = (# of machines) * 4

(# of machines) = 72 / 4 = 18

And the correct answer is B.

### Here’s another official “multiple machines” problem:

Five machines at a certain factory operate at the same constant rate. If four of these machines, operating simultaneously, take 30 hours to fill a certain production order, how many fewer hours does it take all five machines, operating simultaneously, to fill the same production order?

(A) 3

(B) 5

(C) 6

(D) 16

(E) 24

This problem can be solved with a machine hours approach or with the fundamentals we covered in article 1 of this series! Let’s solve it with machine hours first, and then we’ll check your memory of the fundamentals.

Machine hours = (# of machines) * (# of hours of work)

Machine hours = 4 * 30

Machine hours = 120

So how many hours does it take five machines to fill the order?

Machine hours = (# of machines) * (# of hours of work)

120 = 5 * (# of hours of work)

(# of hours of work) = 120 / 5 = 24

But E is not the correct answer! Remember that the problem asked how many fewer hours the five machines take. So our final step is to take 30 – 24 = 6. The correct answer is C.

That was pretty easy, but there’s an even easier way. Since, according to the problem, these machines operate at the same constant rate, the combined rate of the 5 machines is 5/4 the combined rate of the 4 machines. Now here come the fundamentals: since rate and time are inversely related, a 5/4 factor of change for rate means a ⅘ factor of change for the time to complete a given job (if this isn’t clear, review article 1 of this series). The 5 machines complete the job in ⅘ the time it takes 4 machines. Or, the 4-machine time is reduced by ⅕. ⅕ * 30 hours = 6 hours.

### Here’s a final “multiple machines” problem to wrap up the article:

Each machine at a toy factory assembles a certain kind of toy at a constant rate of one toy every 3 minutes. If 40 percent of the machines at the factory are to be replaced by new machines that assemble this kind of toy at a constant rate of one toy every 2 minutes, what will be the percent increase in the number of toys assembled in one hour by all the machines at the factory, working at their constant rates?

(A) 20%

(B) 25%

(C) 30%

(D) 40%

(E) 50%

This problem is different in that the number of machines is not specified. All of the values except for the production rates are relative (percentages). We are told that 40% of the machines are replaced, so there must be at least 5 machines in the factory. Given the usefulness of the direct and inverse relationships between rate, work, and time, let’s start by observing the factor of change in the problem. 40% of the machines will be replaced with machines that produce one toy in ⅔ as much time (2 minutes instead of 3 minutes).

We can say just as well that the production rate of these machines is multiplied by a factor of 3/2: the reciprocal of ⅔, since time and rate are inversely related. So in a given amount of time, these new machines do 3/2 as much work as the old machines – because work and rate are directly related. “3/2 as much work” means 1 ½ times as much work or 50% more work.

So, 40% of the machines increase their toy output by 50%. To find the overall percentage increase for all the machines in the factory, we can simply multiply these percentages together. But let’s use decimals for this, since percentages shouldn’t stand alone. 0.4 * 0.5 = 0.2, so the overall production increase for the machines in the factory is 20%. The correct answer is A.

We didn’t use machine hours for that last problem, but the concept is still useful for many problems featuring multiple machines. On these problems, either use machine hours, or apply your mastery of the fundamental inverse and direct relationships between rate, work, and time!

The next article in the series will focus on GMAT quant problems involving rates of fuel consumption.

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Contributor: Elijah Mize (Apex GMAT Instructor)