How to Solve Math Problems About Combined Work

You've probably seen a math question similar to this before: Jack can mow the lawn in 1 hour, and Meg can mow the lawn in 2.5 hours. How long will it take them to mow the lawn if they work together? Such problems involve a concept called "combined work," and this article will show you how to solve similar word problems. These tricks will be especially valuable if you are studying for the SAT, ACT, GRE, or GMAT.

The first step is to label all the time variables. For example, you can set J = the amount of time it takes Jack to mow the lawn, and M = the amount of time it takes for Meg to mow the lawn. In this particular math problem, we have J = 1 and M = 2.5.

Next, plug these numbers into the Combined Work Formula For Two People:
T = (JM)/(J+M)

In this example, we have (JM)/(J+M) = (1x2.5)/(1+2.5), which simplifies to 2.5/3.5 = .714. So Jack and Meg can mow the entire lawn together in .714 hours. (To figure out how many minutes that is, just multiply .714 times 60, which is about 43 minutes.)

The Combined Work Formula For Two People is easy to use for word problems involving two workers or two machines, but what about an equation for three people? As it turns out, the Combined Work Formula For Three People is quite similar to the previous formula. If you have 3 workers and their individual times are A, B, and C, their combined time is T = (ABC)/(AB+AC+BC) when they work together.

Let's try out this formula for three people named Ann, Betty, and Carol who can each weld a truck frame in 1 hour, 1.5 hours, and 2 hours respectively.

We first write A = 1, B = 1.5, and C = 2. And so ABC = (1)(1.5)(2) = 3, and AB+AC+BC = 1.5 + 2 + 3 = 6.5. And finally, 3/6.5 = .462. So it takes them only .462 hours to weld the whole truck frame when they work together. (This is about 28 minutes.)

There are similar combined work formulas for 4, 5, or more people:

T = (ABCD)/(ABC+ABD+ACD+BCD)
T = (ABCDE)/(ABCD+ABCE+ABDE+ACDE+BCDE)
...

© Had2Know 2010

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