Collaborative Work Rates: Calculating Joint Effort for Project Completion

When two individuals work together on a project, their combined work rate can significantly reduce the time required to complete the task. Understanding work rates and how to calculate them is crucial in project management and team collaboration.

Introduction to Work Rates

Work rates are the measure of how quickly an individual or a team can complete a task. These rates are typically expressed in terms of the fraction of the work completed per unit of time. For example, if a person can complete a job in 24 days, their work rate is 1/24 of the job per day. Similarly, if another person can complete the job in 30 days, their work rate is 1/30 of the job per day.

Calculating Combined Work Rates

To find the combined work rate of two individuals working together, we first need to calculate their individual work rates and then add them.

Step-by-Step Calculation

Calculate individual work rates: A's work rate: 1/24 of the work per day B's work rate: 1/30 of the work per day Combine their work rates:

The combined work rate is derived by adding the individual work rates: [text{Combined work rate} frac{1}{24} frac{1}{30}]

Find a common denominator:

The least common multiple (LCM) of 24 and 30 is 120. We convert the fractions to have the same denominator:

[frac{1}{24} frac{5}{120}; quad frac{1}{30} frac{4}{120}] Add the fractions: [text{Combined work rate} frac{5}{120} frac{4}{120} frac{9}{120} frac{3}{40}] Calculate the time taken:

Since their combined work rate is 3/40 of the work per day, the time required to complete the entire work is: [T frac{1}{frac{3}{40}} frac{40}{3} approx 13.33 text{ days}]

This can be converted to a mixed number: 13 frac{1}{3} days or 13.33 days.

Additional Examples

Let's explore additional examples to solidify the understanding:

Example 1

A alone can do (frac{1}{15}) of the work in one day. B alone can do (frac{1}{12}) of the work in one day. Combined, they can do (frac{1}{15} frac{1}{12} frac{4}{60} frac{5}{60} frac{9}{60} frac{3}{20}) of the work in one day. Therefore, they can complete the work in approximately (6.67) days.

Example 2

Let's consider a different scenario where the work rate calculation is altered:

Let the total work be 1. Per day, A can complete (frac{1}{24}) of the work, and B can complete (frac{1}{30}) of the work.

Assume they work together for (x) days:

[1 frac{x}{24} frac{x}{30}]

Find the LCM of 24 and 30, which is 120:

[frac{x}{24} frac{5x}{120}; quad frac{x}{30} frac{4x}{120}]

Adding these:

[1 frac{5x 4x}{120} frac{9x}{120} frac{3x}{40}]

Solving for (x):

[frac{3x}{40} 1 implies x frac{40}{3} approx 13.33 text{ days}]

Complex Scenario Considerations

In some situations, the plan changes, and a process takes a detour. For example, if initially, A and B were to work together for 24 days, but then the plan changed to 24 days of working together plus 8 days of A working alone, we can calculate the time taken as follows:

With 24 days of working together, 4/5 of the work is completed, and the remaining 1/5 of the work is done by A alone in 8 days. If A were to complete the work alone from the start, it would take 5 times longer than 8 days, i.e., 40 days.

Conclusion

Understanding and applying work rates is essential for efficient project management and team collaboration. By calculating the combined work rate, managers can accurately estimate the time required to complete projects and make informed decisions.