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Two doctors, A and B, can vaccinate patients. Doctor A vaccinates 1 patient in 4 minutes, and Doctor B vaccinates 1 patient in 3 minutes. If they work together, how long will it take them to vaccinate

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}

    To add these fractions, find a common denominator, which is 12.

    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}}

    Substitute the values:

    Total Time=168 patients712 patients/minute\text{Total Time} = \frac{168 \text{ patients}}{\frac{7}{12} \text{ patients/minute}}

    To divide by a fraction, multiply by its reciprocal:

    Total Time=168×127 minutes\text{Total Time} = 168 \times \frac{12}{7} \text{ minutes}

    Simplify the multiplication:

    Total Time=(168÷7)×12 minutes\text{Total Time} = (168 \div 7) \times 12 \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 hours288 \text{ minutes} = \frac{288}{60} \text{ hours} 288÷60=4 with a remainder of 48288 \div 60 = 4 \text{ with a remainder of } 48

    So, 288 minutes=4 hours and 48 minutes288 \text{ minutes} = 4 \text{ hours and } 48 \text{ minutes}.

  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 am7:30 \text{ am} + 4 \text{ hours} = 11:30 \text{ am}
    Add 48 minutes: 11:30 am+48 minutes=12:18 pm11:30 \text{ am} + 48 \text{ minutes} = 12:18 \text{ pm}

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

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