A and B Collaborate to Complete a Work: A Case Study in Project Management

Project management involves coordinating various resources to achieve a common goal. This case study will demonstrate how two individuals with varying work efficiencies can collaborate effectively to complete a task that initially one could not do alone. Specifically, we will solve the problem where person A completes a work in 20 days, while person B does the same work in 25 days. Person A starts working alone, and after 2 days, person B joins him. We will determine how long it will take for both of them to complete the remaining work.

Understanding Work Rates and Efficiencies

To solve this problem, we first need to determine the work rates of A and B. The work rate is calculated by considering the rate at which each individual completes the work per day.

A's Work Rate:
A completes the work in 20 days, so the work rate of A is (frac{1}{20}) of the work per day.

B's Work Rate:
B completes the work in 25 days, so the work rate of B is (frac{1}{25}) of the work per day.

Work Done by A Alone

Person A started working alone, and after 2 days, B joined him. We first calculate the amount of work completed by A during these 2 days.

In 2 days, A will complete: [frac{1}{20} times 2 frac{2}{20} frac{1}{10}] of the work.

Remaining Work After 2 Days

The remaining work after A has worked for 2 days is:

[text{Remaining Work} 1 - frac{1}{10} frac{9}{10}] of the work.

Combined Work Rate of A and B:
When B joins A, their combined work rate is the sum of their individual work rates:

[frac{1}{20} frac{1}{25}]

To combine these fractions, we find a common denominator, which is 100:

[frac{1}{20} frac{5}{100} quad text{and} quad frac{1}{25} frac{4}{100}]

Therefore, their combined work rate is:

[frac{5}{100} frac{4}{100} frac{9}{100}] of the work per day.

Time to Complete the Remaining Work

To find out how many days it will take them to complete the remaining (frac{9}{10}) of the work, we divide the remaining work by their combined work rate:

[text{Time} frac{text{Remaining Work}}{text{Combined Work Rate}} frac{frac{9}{10}}{frac{9}{100}}]

This simplifies to:

[frac{9}{10} times frac{100}{9} 10 text{ days}]

Thus, A and B together will take 10 days to complete the remaining work.

Conclusion

This case study demonstrates the importance of collaboration in project management. By working together, A and B can achieve their goal more efficiently. Understanding work rates and combining individual efficiencies is crucial in estimating project timelines and resource allocation.