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Math Help - A basic arithmatic problem

  1. #1
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    A basic arithmatic problem

    Can someone explain why do we need to determine the amount of work done in 1 hour (as indicated by the underlined part). Please explain the logic behind taking the ratio of the number of hours. Also, does "1" in the numerators show a job or "1 hour".

    CB
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  2. #2
    Senior Member tukeywilliams's Avatar
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    We want to find the combined rate. So in the example, if machines X and Y work together we know it will take less than 4 hours intuitively. The question is how long? So after 4 hours machine X completes 100% of the task. After 5 hours, machine Y completes 100% of the task. So to compare apples to apples, how much will machines X and Y complete in 1 hour? It is  \frac{1}{4} and  \frac{1}{5} of the task respectively. To find how much of the task they complete together in 1 hour, we add the two and get  \frac{9}{20} of the task completed. But this is only the amount they have finished in one hour. Then we set up a ratio and proportion to find how long it will take both of them to complete 100% of the task. You don't have to use 1 hour. We could have done the following:

    How much will machines X and Y complete in 2 hours? It is  \frac{1}{2} and  \frac{2}{5} respectively. Then together in 2 hours they can complete  \frac{9}{10} of the task. Then we set up a ratio and proportion to find out how long it would take both of them to complete 100% of the task. The key idea is that we want to compare apples to apples (i.e. keep times constant to calculate rates).
    Last edited by tukeywilliams; July 30th 2007 at 11:26 PM.
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  3. #3
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    Hello, CB!

    I use a different approach to "Work" problems.
    . . See if you like it . . .


    A can do a job in 4 hours.
    B can do the same job in 5 hours.
    How long will it take them to complete job if they work together?
    A can do the job in 4 hours.
    . . In one hour, he can do \frac{1}{4} of the job.
    . . In x hours, he can do \frac{x}{4} of the job.

    B can do the job in 5 hours.
    . . In one hour, he can do \frac{1}{5} of the job.
    . . In x hours, he can do \frac{x}{5} of the job.

    In x hours, working together, they can do . \frac{x}{4} + \frac{x}{5} .of the job.


    But in x hours, we expect them to compete the job (one job).

    . . There is our equation! . . . . . \frac{x}{4} + \frac{x}{5}\;=\;1


    Multiply by 20: . 5x + 4x \:=\:20\quad\Rightarrow\quad 9x \:=\:20\quad\Rightarrow\quad x \,=\,\frac{20}{9}

    . . Therefore, working together, it will take them 2\frac{2}{9} hours.

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