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## GMAT Combined Work Problems

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Combined Work problems are among my favorites to teach: they look difficult, but most such problems on the GMAT are quite simple. Once you learn a couple of equations and remember a few traps to avoid, you'll have no problem acing these problems without spending too much time on them.

An example of a combined work problem is #140 in The Official Guide for GMAT Quantitative Review, on page 80:

Machines A and B always operate independently at their respective constant rates. When working alone, machine A can fill a production lot in 5 hours, and machine B can fill the same lot in x hours. When the two machines operate simultaneously to fill the production lot, it takes them 2 hours to complete the job. What is the value of x?

The variable isn't always in the same place--on easier problems, the variable will be the amount of time the two machines take when operating simultaneously--but the concept never changes. If all the numbers represent the same unit (in this case the number of hours it takes to do a certain amount of work), use the following equation:

1/A + 1/B = 1/T

where A is the amount of time it takes machine A (or person A, or whatever) to do the job, B is the amount of time for B, and T is the amount of time it takes working simultaneously.

In this problem, A = 5, B = x, and T = 2. Plugging in those values gives you:

1/5 + 1/x = ½

Manipulate the equation a bit, and you find that x = 3 1/3, choice (A).

Extending This Equation

Occasionally, especially on harder problems, the GMAT will ask you how much time it takes for three machines (or people, or whatever) to complete a job, working simultaneously. This is one of the benefits of remembering the basic equation: it's easy to extend it to include more variables. For three machines, it looks like this:

1/A + 1/B + 1/C = 1/T

For four or more machines, you can probably guess what the equation will be. However, I've never seen a GMAT question that asked you to go any farther than three inputs.

The Shortcut

If you tinker around with the original equation a bit and solve for T, you end up with this:

AB / (A+B) = T

The variables stand for the same things. In the case of our sample problem, where A = 5, B = x, and T = 2, it might be a little simpler:

5*x / (5 + x) = 2

Where this shortcut comes in particularly handy is on problems where you know the amount of time it takes for machine A and machine B, and you want to know the total time. If, for instance, you were told that machine A could do a job in 4 hours and machine B could do the same job in 5 hours, look how easy it is:

5*4 / (5 + 4) = 20/9

However, extending this shortcut is much more difficult. The three-variable version is

(ABC) / (AB + BC + AC) = T

It's not impossible to remember, but I vastly prefer the other three-variable approach.

Common Traps

Of course, if you can solve a problem with a simple two-variable equation, the GMAT will find ways to make that problem harder. Because combined work problems rarely contain much conceptual difficulty, the testmakers pull out their bag of tricks to keep you alert. For instance, the example question we looked at above could be reworded as follows:

Machines A and B always operate independently at their respective constant rates. When working alone, machine A can fill a production lot in 5 hours, and machine B can fill a separate production lot of the same size in x hours. When the two machines operate simultaneously to fill both production lots, it takes them 4 hours to complete the job. What is the value of x?

Once you figure out what they're talking about, this is the same question as the one above. Instead of telling you that one job takes 2 hours to complete when the machines are operating simultaneously, the question tells you it takes 4 hours to do two jobs. Same thing, but it's the kind of trap unwary GMAT students fall into all the time.

Just about every combined work trap is somehow related to that idea: instead of giving you the numbers that you can directly plug in to the equations above, they give you something you have to convert into those numbers. Some questions will tell you how long it takes machine A to do seven jobs, machine B to do three jobs, and then ask how long it would take them to do 12 jobs working simultaneously. There are extra steps in that problem, but if you're paying attention, the complexity doesn't increase.

The main thing to remember is that A, B, and T are times (not rates!), and they all must refer to the amount of time it takes for the same job. If you can apply that knowledge to every combined work problem and remember a couple of simple equations, you'll nail every such question on your GMAT.

About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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