Today’s GMAT tip comes from Kaplan. In this article, Kaplan GMAT instructor Arthur Ahn offers advice on problems that call for the work formula:
On the GMAT, you will generally see at least one question that’s related to the work formula. This formula deals with questions such as “how long will it take x people to paint a room?”, or other questions concerning length of time to complete certain jobs or tasks.
First, let’s take a look at a straightforward example and see where the logic of the “work formula” that you may have memorized actually comes from:
Let’s say that there are two people tasked to paint a room. Let’s call one person A, and the other person B (you can see I’m really creative about picking names).
If person A takes 2 hours to paint a room by herself, and person B takes 4 hours to paint a room by himself, how long would it take for A and B to paint a room if they work together at their respective rates?
Notice we have two different times here: one for A, and one for B. The problem here is that these times can’t really be compared, especially for the question to be answered. “Two hours per room, and four hours per room, equals how many hours per room?” It feels a little strange. We need to look at these numbers in terms of rates.
To make these two numbers for A and B comparable, look first at how much of the work each person completes in one hour. For instance, if person A takes two hours to paint a room by herself, then in one hour she has completed half the room on her own. If person B takes four hours to paint a room by himself, then in one hour he has completed one-fourth of the room by himself. So as they’ve worked together, they’ve completed ¾ of the room in 1 hour. Remember that you’re looking for how much time it takes to complete the entire task: in this case, the entire task is 1 room, not just ¾. So you can set up the ratio as follows:
¾ / 1 = 1/ x, in which x represents the time it takes to complete 1 room.
(3/4)x = 1, and x is equal to 4/3.
Keep in mind the classic work formula, also leading to our answer of 4/3:
1/A + 1/B = 1/T.
(also commonly written as AB/(A+B) = T)
Notice that our reasoning matches the formula. We found the rate of work, found the sum of the two rates, and took the reciprocal of that to find our answer.
If you have a question with three or more people, just add on those people to the left side of the equation.
1/A + 1/B + 1/C + … = 1/T.
One of the common errors is to forget to find the reciprocal of the sum of those rates. 1/T represents how much of the work is completed in 1 hour, while T represents how much time it takes to complete the work. These are two very different things!
Best of luck!
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Read the full article: GMAT Tip: Working the Work Formula
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