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Two workers, one who can complete a job in 12 hours and another who can complete the same job in 8 hours, work together. How long will it take them to complete the job?

Mathematics

ScanSolve AI Answer

Answer

4 hours 48 minutes\text{4 hours 48 minutes}

Step-by-step solution

3 steps
  1. Determine the individual work rates.

    The rate of work is the reciprocal of the time taken to complete the job.
    Worker 1's rate:

    R1=112 job/hourR_1 = \frac{1}{12} \text{ job/hour}

    Worker 2's rate:

    R2=18 job/hourR_2 = \frac{1}{8} \text{ job/hour}
  2. Calculate their combined work rate.

    When working together, their rates add up.

    Rcombined=R1+R2R_{\text{combined}} = R_1 + R_2 Rcombined=112+18R_{\text{combined}} = \frac{1}{12} + \frac{1}{8}

    To add these fractions, find a common denominator, which is 24.

    Rcombined=1×212×2+1×38×3R_{\text{combined}} = \frac{1 \times 2}{12 \times 2} + \frac{1 \times 3}{8 \times 3} Rcombined=224+324R_{\text{combined}} = \frac{2}{24} + \frac{3}{24} Rcombined=2+324R_{\text{combined}} = \frac{2+3}{24} Rcombined=524 job/hourR_{\text{combined}} = \frac{5}{24} \text{ job/hour}
  3. Calculate the total time taken to complete the job together.

    The total time is the reciprocal of the combined rate.

    Ttotal=1RcombinedT_{\text{total}} = \frac{1}{R_{\text{combined}}} Ttotal=1524T_{\text{total}} = \frac{1}{\frac{5}{24}} Ttotal=245 hoursT_{\text{total}} = \frac{24}{5} \text{ hours}

    To express this in hours and minutes:

    Ttotal=445 hoursT_{\text{total}} = 4 \frac{4}{5} \text{ hours}

    Convert the fractional part to minutes:

    45 hours×60 minutes/hour=4×12 minutes=48 minutes\frac{4}{5} \text{ hours} \times 60 \text{ minutes/hour} = 4 \times 12 \text{ minutes} = 48 \text{ minutes}

    So, the total time is 4 hours and 48 minutes.

    The total time it will take them to finish the job altogether is 4 hours 48 minutes\boxed{\text{4 hours 48 minutes}}.

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