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A man working alone can do a job in 3 hours. His son can do the same job in 2 hours. How many hours will it take for the two of them together to finish the same job?

Mathematics

ScanSolve AI Answer

Answer

A) 8 minutes\text{A) 8 minutes}

Step-by-step solution

4 steps
  1. Determine the rate of each pipe.

    The first pipe empties the tank in 30 minutes, so its rate is 130\frac{1}{30} tank per minute.
    The second pipe empties the tank in 20 minutes, so its rate is 120\frac{1}{20} tank per minute.

  2. Calculate the combined rate of both pipes.

    When both pipes are open, their rates add up:

    Rtotal=130+120R_{total} = \frac{1}{30} + \frac{1}{20}

    To add these fractions, find a common denominator, which is 60:

    Rtotal=1×230×2+1×320×3=260+360=560=112 tank per minuteR_{total} = \frac{1 \times 2}{30 \times 2} + \frac{1 \times 3}{20 \times 3} = \frac{2}{60} + \frac{3}{60} = \frac{5}{60} = \frac{1}{12} \text{ tank per minute}

    This means that together, the pipes can empty the entire tank in 12 minutes.

  3. Determine the amount of the tank that needs to be emptied.

    The tank is 23\frac{2}{3} full, so the amount to be emptied is 23\frac{2}{3} of the tank's volume.

  4. Calculate the time required to empty the specified portion of the tank.

    Time = Amount to be emptiedCombined rate\frac{\text{Amount to be emptied}}{\text{Combined rate}}

    Time=23112=23×12\text{Time} = \frac{\frac{2}{3}}{\frac{1}{12}} = \frac{2}{3} \times 12 Time=2×4=8 minutes\text{Time} = 2 \times 4 = 8 \text{ minutes}

    The time taken to empty 23\frac{2}{3} of the tank is 8 minutes.

    The final answer is A) 8 minutes\text{A) 8 minutes}.

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