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Two doctors start vaccinating patients at 7:30 am. Doctor A vaccinates 1 patient in 4 minutes, and Doctor B vaccinates 1 patient in 3 minutes. If they need to vaccinate 168 patients together, what tim

Mathematics

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Answer

12:18 pm\text{12:18 pm}

Step-by-step solution

5 steps

  1. Determine the individual vaccination rates of Doctor A and Doctor B.

    Doctor A vaccinates 1 patient in 4 minutes.
    Doctor B vaccinates 1 patient in 3 minutes.

  2. Calculate the combined vaccination rate of both doctors.

    Their combined rate is the sum of their individual rates:

    Combined Rate=14 patient/minute+13 patient/minute\text{Combined Rate} = \frac{1}{4} \text{ patient/minute} + \frac{1}{3} \text{ patient/minute} Combined Rate=312+412=712 patients/minute\text{Combined Rate} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12} \text{ patients/minute}

  3. Calculate the total time required to vaccinate 168 patients.

    Total Time=Total PatientsCombined Rate\text{Total Time} = \frac{\text{Total Patients}}{\text{Combined Rate}} Total Time=168 patients712 patients/minute\text{Total Time} = \frac{168 \text{ patients}}{\frac{7}{12} \text{ patients/minute}} Total Time=168×127 minutes\text{Total Time} = 168 \times \frac{12}{7} \text{ minutes} Total Time=24×12 minutes\text{Total Time} = 24 \times 12 \text{ minutes} Total Time=288 minutes\text{Total Time} = 288 \text{ minutes}

  4. Convert the total time from minutes to hours and minutes.

    There are 60 minutes in an hour.

    288 minutes=28860 hours=4 hours and 48 minutes288 \text{ minutes} = \frac{288}{60} \text{ hours} = 4 \text{ hours and } 48 \text{ minutes}

    (4×60=2404 \times 60 = 240, 288−240=48288 - 240 = 48)

  5. Calculate the finishing time by adding the total time to the starting time.

    Starting time: 7:30 am
    Add 4 hours: 7:30 am + 4 hours = 11:30 am
    Add 48 minutes: 11:30 am + 48 minutes = 12:18 pm

    The two doctors will finish vaccinating at 12:18 pm\text{12:18 pm}.

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