A, B, and C's Job Division and Completion: A Step-by-Step Guide

Given the scenario where A, B, and C start a job that they can complete in 2 days, with each individual having their own completion time, the challenge lies in determining how long it would take for A to complete the remaining work once B and C leave after 1 day. Let's break this down step-by-step.

Understanding the Work Rates

To solve this problem, we first need to determine the work rates of each individual. The work rate is the portion of the job that each person can complete in a single day.

Work Rates of A, B, and C

1. Combined Work Rate (for A, B, and C working together):

[ r_{A, B, C} frac{1}{2} text{jobs per day} ]

2. Individual Work Rate for B:

[ r_B frac{1}{5} text{jobs per day} ]

3. Individual Work Rate for C:

[ r_C frac{1}{4} text{jobs per day} ]

4. Work Rate of A:

[ r_A r_{A, B, C} - r_B - r_C ]

[ r_A frac{1}{2} - frac{1}{5} - frac{1}{4} ]

To subtract the fractions, we need a common denominator, which in this case is 20:

[ frac{1}{2} frac{10}{20}, frac{1}{5} frac{4}{20}, frac{1}{4} frac{5}{20} ]

[ r_A frac{10}{20} - frac{4}{20} - frac{5}{20} ]

[ r_A frac{10}{20} - frac{9}{20} ]

[ r_A frac{1}{20} text{jobs per day} ]

Calculating the Work Done and Remaining Work

Once the individual work rates are established, we can calculate the amount of work each person does in a day and any remaining work.

1. Total Work Done in 1 Day:

[ text{Work done in 1 day} frac{1}{2} text{jobs} ]

2. Remaining Work After 1 Day:

[ text{Remaining work} 1 - frac{1}{2} frac{1}{2} text{jobs} ]

Time for A to Complete the Remaining Work

Finally, we calculate how long it would take A to complete the remaining work based on their work rate.

1. Time Required for A:

[ t frac{text{Remaining work}}{r_A} ]

[ t frac{frac{1}{2}}{frac{1}{20}} ]

[ t frac{1}{2} times 20 10 text{days} ]

Therefore, it would take A 10 days to complete the remaining work.

Conclusion

Through this structured approach, we have successfully determined that A would take 10 days to complete the job after B and C have left. Understanding work rates and how they combine is crucial for solving such problems efficiently.