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Math Help - Work done problem

  1. #1
    xwy
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    Work done problem

    A carpenter worked alone on a job for two days and then hired a helper. The entire job was done in 5 5/7 days. Working alone, the carpenter could have done the job in 4 fewer days than the helper could have done the entire job alone. Find the time each would have needed to complete the job alone.

    Hello, I was wondering how do I differentiate the 2 days of work done by carpenter from the rest of the 3 5/7 days? I know that the carpenter could do 1/h+4 amount of work a day and the helper at the rate of 1/h. However, when I plucked it into the equation, which ultimately ends up as second degree equation, the answers are wrong.

    I would appreciate any help thanks!!
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  2. #2
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    Re: Work done problem

    Quote Originally Posted by xwy View Post
    A carpenter worked alone on a job for two days and then hired a helper. The entire job was done in 5 5/7 days.

    Working alone, the carpenter could have done the job in 4 fewer days than the helper could have done the entire job alone. Find the time each would have needed to complete the job alone.

    Hello, I was wondering how do I differentiate the 2 days of work done by carpenter from the rest of the 3 5/7 days? I know that the carpenter could do 1/h+4 amount of work a day and the helper at the rate of 1/h. However, when I plucked it into the equation, which ultimately ends up as second degree equation, the answers are wrong.

    I would appreciate any help thanks!!
    No. If the helper would take h days to do the job alone, his rate is 1/h "job per day". But the carpenter would take 4 less days to do it. The carpenter would take h- 4 days to do the job so his rate is 1/(h- 4) "job per day". (If you were to let h be the number of days the carpenter takes to do the job then the helper would take h+ 4 days and their rates would be 1/h and 1/(h+ 4). That may be where you got confused.)

    Working together, their rates add: working together they work at 1/h+ 1/(h- 4) "job per day".

    The carpenter works at rate 1/(h- 4) "job per day" so in the first two days he does work 2/(h- 4) of the job. Then then work together for an additional 3 and 5/7= 26/7 days, doing (26/7)(1/h+ 1/(h-4)) of the job. Since they completed the job, 2/(h-4)+ (26/7)(1/h+ 1/(h- 4))= 1. Solve that for h.
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  3. #3
    xwy
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    Re: Work done problem

    Hey would you mind explaining why is the equation =1?
    I'm confused because "work done" questions always follow this equation of (1/x)+(1/y)= 1/t? So when and how do we know which is =1 or 1/t?
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  4. #4
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    Re: Work done problem

    By thinking about what those mean. "job per hour" times "hours"= "job". If you work at 1/5 job per hour, in four hours you will complete 4(1/5)= 4/5 job.

    I said that the carpenter worked at a rate of 1/(h-4) "job per day" so that in the first 2 he completed 2/(h- 4) "job". The helper worked at a rate of 1/h job per day so that together they work at a rate of 1/(h-4)+ 1/h "job per day". In 3 and 5/7= 26/7 days working together, they complete (26/7)(1/(h-4)+ 1/h) "job". Including the two days the carpenter worked alone, they completed 2/(h-4)+ (26/7)(1/(h- 4)+ 1/h) "job".

    Now, the problem asked you to "find the time each would have needed to complete the job alone". We have already agreed that "h" is the number of hours it would take the helper to do the job and that h- 4 is the number of hours it would take the carpenter to do the job alone so we want to find "h". And "2/(h-4)+ (26/7)(1/(h-4)+ 1/h)" is there rates in "job per hour" times "hour". Since the question is about doing the job, that is equal to 1 "job".

    Your saying you had "1/x+ 1/y= 1/t" is meaning less if you don't say what "x", "y", and "t" mean. I suspect that x and y are the time it would take each of two people to do a specific job in, say, "hours per job" so that 1/x and 1/y have units "job per hour". And the problem asks "how long would it take the two people to do that job working together". If "t" is the time in hours to do the job working together (the answer to that question) then "job per hour" would be 1/t. But notice that, "how many hours working together?" is a different question than "how many hours would each take working alone?"

    Don't just memorize or copy formulas- think about what they mean.
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