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Math Help - Machines/ work rate

  1. #1
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    Machines/ work rate

    Six machines, working at the same constant rate together can complete a certain job in 12 days. How many additional machines, each working at the same constant rate are needed to complete the job in 8 days?

    What I did (and where am I wrong?):
    each machine does "2" of the job in 6 days (since each machine does 2 parts of the job). so rate of each machine is 2/6 which is 1/3.
    (1/3)*6 +(1/3) *x = 1/8.

    I understand I'm taking out the rate wrong...but what else should it be? Rate is always 1/ work done in a day.... so 12/6=2, even if i use rate of 1/2, the answer is still wrong. Im very confused.

    Thanks a lot!

    Answer is 3
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  2. #2
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    Re: Machines/ work rate

    Hello, usre123!

    Your reasoning is faulty . . .


    Six machines, working at the same constant rate together can complete a certain job in 12 days.
    How many additional machines, each working at the same constant rate, are needed to complete the job in 8 days?

    I baby-talked my way through the problem like this . . .

    \begin{array}{cccc}\text{6 machines} & \text{in 12 days} & \text{can do 1 whole job.} \\ \text{6 machines} & \text{in 1 day} & \text{ can do }\frac{1}{12}\text{ of the job.} \\ \\[-4mm] \text{1 machine} & \text{in 1 day} & \text{can do }\frac{1}{72}\text{ of the job.} \\ \\[-4mm] x\text{ machines} & \text{in 1 day} & \text{can do }\frac{x}{72}\text{ of the job.} \\ \\[-4mm] x\text{ machines} & \text{in 8 days} & \text{can do }\frac{8x}{72}\text{ of the job.} \end{array}

    That final fraction \frac{8x}{72} \,=\,\frac{x}{9} equals one whole job.
    . . Hence:- \frac{x}{9} \:=\:1 \quad\Rightarrow\quad x \,=\,9

    Nine machines are required.

    Therefore, 3 additional machines are needed.
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  3. #3
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    Re: Machines/ work rate

    if i was to do I would go like this, the method is called unitary method:
    If the whole work is to completed in 12 days we need number of machines = 6
    If the whole work is to completed in 1 day we need number of machines = 6 x 12 [ to do work in short time we need more machines ]
    If the whole work is to completed in 8 day we need number of machines = (6 x 12)/8 [ to do work in more time we need less machines ]
    = 9 machines
    Thus we need to have one more machine to complete the work in 8 days.

    ALTERNATELY
    6 machines complete the work in 12 days
    1 machine will complete the work in 12 x 6 = 72 days.
    Thus 1 machine in one day can finish 1/72 of work
    So i machine in 8 days would complete (1/72) x 8 = 1/9 work
    Hence 1 work will be completed by 8 machines in 9 days
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  4. #4
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    Re: Machines/ work rate

    If 6 machines do a certain job in 12 days then each machine does 1/6 of the job in 12 days and so each machine does (1/6)/12= 1/72 of the job in 1 day. If we have a total of X machines, working at that rate, they will do X/72 of the job each day. To finish the job in 8 days, we must have (X/72)(8)= X/9= 1 so X= 9. We need a total of 9 machines to do the job in 8 days. Since we already have 6 machines, we will need to add 9- 6= 3 new machines.
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