Collaborative Work Efficiency: How Long Does It Take to Complete a Task When B and C Depart?

In the realm of work efficiency, understanding how different individuals can contribute to completing a task is crucial. This article discusses a common scenario where team members collaborate for a while and then leave, leading to questions about the total time required to complete a work project. Specifically, we will explore the scenario where B and C depart at different times during the task, and A, B, and C can individually complete the job in 10, 12, and 15 days respectively.

Problem and Solution

Let's consider the following scenario:

Welcome to the Problem Analysis and Calculation Formula and Steps Conclusion

Welcome to the Problem

Suppose A, B, and C start a project together but B and C leave the project 5 days and 2 days before its completion, respectively. A completes the project in 10 days, B in 12 days, and C in 15 days. We need to determine the total time required to complete the project under these conditions.

Analysis and Calculation

Step 1: Determine the Work Rates

The work rates of A, B, and C can be calculated as follows:

A's one day work: 1/10 B's one day work: 1/12 C's one day work: 1/15

Together, their one day work is:

1/10 1/12 1/15 6/60 5/60 4/60 15/60 1/4 of the total work per day.

Step 2: Work done by A, B, and C together

Let's assume the work is completed in T days. B leaves after 2 days and C leaves 5 days before the completion of the work. Therefore, B works for 2 days and C works for (T - 5) days.

The total work equation becomes:

2 × (1/4) (T - 2) × (1/10) (T - 7) × (1/15) 1

Formula and Steps

Let's solve the equation step-by-step.

Calculate the work done by A, B, and C together in 2 days:

2 × (1/4) 1/2 of the work

Calculate the work done by A and C in the remaining period:

(T - 7) × (1/15)

Calculate the work done by A alone in the period when both B and C left:

(T - 2) × (1/10)

Sum the work done to equal the total work (1):

1/2 (T - 2) × (1/10) (T - 7) × (1/15) 1

1/2 (T/10 - 1/5) (T/15 - 7/15) 1

1/2 T/10 - 4/10 - T/15 7/15 1

1/2 T/10 - 4/10 - T/15 14/30 1

3/6 3T/30 - 12/30 - 2T/30 14/30 1

3/6 T/30 1

T/30 1 - 3/6

T/30 1/2

T 15/2 7.5 days

Conclusion

The total time required to complete the project, under the given conditions, is 7.5 days. This calculation considers the individual work rates and the departure of B and C at their respective timings. By understanding such collaborative dynamics, managers can better plan and optimize the allocation of resources for project completion.

Final Conclusion

The project will be completed in 7.5 days, with A, B, and C contributing simultaneously for the initial period, followed by A continuing alone after B and C leave.

Keywords: Collaborative Work, Work Efficiency, Time Calculation, Work Completion