Efficiency and Collaboration: Determining Time to Complete a Job

Introduction:

Understanding the efficiency levels of individuals and determining how their combined efforts can impact the completion of a job is a crucial aspect of project management. This article explores a scenario where individual and combined work rates are considered to find out how much time A and B will take to complete a job, given that A is 30% more efficient than B.

Determining Work Rates

Let's begin by determining the individual work rates of A and B.

Work Rate of A

It is given that A can complete the job in 15 days. Therefore, the work rate of A is:

Work rate of A 1 job/15 days 1/15 jobs per day

Work Rate of B

Since A is 30% more efficient than B, we can set the work rate of B as x jobs per day. Consequently, the work rate of A can be expressed as 1.3 x.

Work rate of A 1.3 x

We know that the work rate of A is 1/15, so we can set up the equation:

1.3 x 1/15

Solving for x gives:

x 1/15 x 1.3 1/19.5 jobs per day

Combined Work Rate of A and B

The combined work rate of A and B can be calculated by adding their individual work rates. The least common multiple (LCM) of 15 and 19.5 is 195.

[text{Combined work rate of A and B} frac{1}{15} frac{1}{19.5}]

Converting to a common denominator (195):

[frac{1}{15} frac{13}{195} quad text{and} quad frac{1}{19.5} frac{10}{195}]

[text{Combined work rate} frac{13}{195} frac{10}{195} frac{23}{195} text{ jobs per day}]

Calculating Time Taken

Now, to find the time taken by A and B together to complete the job, we take the reciprocal of the combined work rate:

[text{Time} frac{1 text{ job}}{frac{23}{195} text{ jobs per day}} frac{195}{23} text{ days} approx 8.52 text{ days}]

Therefore, it will take approximately 8.52 days for A and B to complete the job together.

Conclusion:

This analysis demonstrates how the efficiency of individuals and their combined efforts can be quantified to determine the time required to complete a job. Understanding these principles is essential for effective project management and resource allocation.