Optimizing Productivity: Calculating Work Completion Times

Productivity is a key factor in ensuring work is completed efficiently. This article dives into the mathematical methods that can be used to determine how long it will take a team of individuals to finish a project. We explore various scenarios and calculate the exact time required based on the individual productivity of each team member.

Scenario 1: Team Collaboration and Individual Contributions

Suppose A, B, and C can complete a work in 10, 8, and 12 days respectively. When they work together for a certain period, A and C leave. How much time does B require to finish the remaining work?

Let us assume the total work is represented as W.

First, calculate the one-day work done by each person: A’s one-day work 1/10 W B’s one-day work 1/8 W C’s one-day work 1/12 W

Together, A, B, and C complete 1/10 1/8 1/12 W in one day. Working together for 2 days, they complete 2(1/10 1/8 1/12) W.

The remaining work after 2 days is:

W - 2(1/10 1/8 1/12) W W - 2[12/120 15/120 10/120] W W - 2(37/120) W W - (37/60) W (23/60) W.

B alone needs to complete (23/60) W. As B’s one-day work is 1/8 W, the time required is:

(23/60) W / (1/8 W) (23/60) × 8 46/15 3 1/15 days.

Scenario 2: Team Collaboration and Completion Period

This time, let's assume A, B, and C can complete the same work in 10, 8, and 12 days respectively. They start working together but C leaves after 3 days, and B leaves 4 days before the work is completed.

The total work done in 3 days by A, B, and C is 3(1/10 1/8 1/12) W 3(6/60 7.5/60 5/60) W 3(18.5/60) W 55.5/60 W 111/120 W.

Let T be the total time in days required to complete the work. B leaves 4 days before the work is completed, so A and B work for (T - 4) days, and C works for 3 days. The work done by A, B, and C is:

T/10 (T - 4)/8 3/12 1.

Simplifying this equation:

T/10 (T/8 - 1/2) 1/4 1.

Multiplying through by 40 to clear the denominators:

4T 5T - 20 10 40.

9T - 10 40.

9T 50.

T 50/9 5 5/9 days.

Scenario 3: Team Collaboration and Completion Period (Cont.)

In this scenario, A, B, and C complete the work in 10, 12, and 15 days respectively. They start working together but C leaves after working for 3 days, and B leaves 4 days before the work is completed.

The total work done in 3 days by A, B, and C is 3(1/10 1/12 1/15) W. The least common multiple (LCM) of 10, 12, and 15 is 60.

A’s one-day work 6 units, B’s one-day work 5 units, and C’s one-day work 4 units.

In 3 days, they complete 3(6 5 4) W 3(15) W 45 W.

The remaining work is W - 45 W 15 W.

A and B work together for 3 days, so the work done is 3(11) W 33 W. Therefore, the remaining work is 15 W - 33 W 12 W.

The time required for A and B to complete 12 W is 12/11 days.

Since B leaves 4 days before the work is completed, the total time to complete the work is:

T 3 days 12/11 days 4 days 7 12/11 days 84/11 days.

Conclusion

By understanding and applying these mathematical methods, managers and teams can optimize their productivity and ensure work is completed efficiently and on time. Whether it is through collaboration of all team members or managing their individual contributions, these strategies provide significant insight into managing resources and time effectively.

Keywords: productivity, work completion, time management