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"Work" Word Problems (page 1 of 4)


"Work" problems involve situations such as two people working together to paint a house. You are usually told how long each person takes to paint a similarly-sized house, and you are asked how long it will take the two of them to paint the house when they work together. Many of these problems are not terribly realistic (since when do two laser printers work together on printing one report?), but it's the technique that they want you to learn, not the applicability to "real life".

The method of solution for work problems is not obvious, so don't feel bad if you're totally lost at the moment. There is a "trick" to doing work problems: you have to think of the problem in terms of how much each person / machine / whatever does in a given unit of time. For instance:

  • Suppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours. How long would it take the two painters together to paint the house?

    If the first painter can do the entire job in twelve hours and the second painter can do it in eight hours, then (this here is the trick!) the first guy can do 1/12 of the job per hour, and the second guy can do 1/8 per hour. How much then can they do per hour if they work together?

    To find out how much they can do together per hour, I add together what they can do individually per hour: 1/12 + 1/8 = 5/24. They can do 5/24 of the job per hour. Now I'll let "t" stand for how long they take to do the job together. Then they can do 1/t per hour, so 5/24 = 1/t. Flip the equation, and you get that t = 24/5 = 4.8 hours. That is:

      hours to complete job:
        first painter:
      12
        second painter:
      8
        together:
      t

      completed per hour:
        first painter:
      1/12
        second painter:
      1/8
        together:
      1/t

      adding their labor: Copyright Elizabeth Stapel 1999-2011 All Rights Reserved

        1/12 + 1/8 = 1/t

        5/24 = 1/t

        24/5 = t

    They can complete the job together in just under five hours.

As you can see in the above example, "work" problems commonly create rational equations. But the equations themselves are usually pretty simple.

  • One pipe can fill a pool 1.25 times faster than a second pipe. When both pipes are opened, they fill the pool in five hours. How long would it take to fill the pool if only the slower pipe is used?

    Convert to rates:

 

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      hours to complete job:
        fast pipe:
      f
        slow pipe:
      1.25f
        together:
      5

      completed per hour:
        fast pipe:
      1/f
        slow pipe:
      1/1.25f
        together:
      1/5

      adding their labor:

        1/f + 1/1.25f = 1/5

      multiplying through by 5f:

        5 + 5/1.25 = f
        5 + 4 = f = 9

    Then 1.25f = 11.25, so the slower pipe takes 11.25 hours.

If you're not sure how I derived the rate for the slow pipe, think about it this way: If someone goes twice as fast as you, then you take twice as long as he does; if he goes three times as fast, then you take three times as long. In this case, if he goes 1.25 times as fast, then you take 1.25 times as long.

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Cite this article as:

Stapel, Elizabeth. "'Work' Word Problems." Purplemath. Available from
    http://www.purplemath.com/modules/workprob.htm. Accessed
 

 



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