Introduction

This article explores the intriguing topic of collaborative work through the lens of a mathematical problem involving individuals A, B, and C working together to complete a task. We will tackle the problem step-by-step, exploring how their individual work rates and varying work durations impact the overall completion time. This analysis not only satisfies the mathematical curiosity but also provides insights into real-world collaborative efforts.

Understanding Work Rates

For clarity, let us first define the work rates of A, B, and C:

A can complete the task in 15 days, hence A's work rate is (frac{1}{15}) of the work per day. B can complete the task in 30 days, hence B's work rate is (frac{1}{30}) of the work per day. C can complete the task in 40 days, hence C's work rate is (frac{1}{40}) of the work per day.

Combined Work Rate

When A, B, and C work together, their combined work rate is the sum of their individual work rates:

(text{Combined rate} frac{1}{15} frac{1}{30} frac{1}{40})

To add these fractions, we find a common denominator, which is 120:

(frac{1}{15} frac{8}{120}), (frac{1}{30} frac{4}{120}), (frac{1}{40} frac{3}{120})

Thus, the combined work rate is:

(text{Combined rate} frac{8 4 3}{120} frac{15}{120} frac{1}{8})

This means that together, A, B, and C can complete (frac{1}{8}) of the work per day.

Critical Moments in the Work Process

The problem specifies that A leaves 2 days before the completion, while B leaves 4 days before. This means:

A worked for (x - 2) days, where (x) is the total number of days the work was completed in. B worked for (x - 4) days. C worked for all (x) days.

To find (x), we set up the equation based on the total work done:

(left(x - 2right) cdot frac{1}{15} left(x - 4right) cdot frac{1}{30} x cdot frac{1}{40} 1)

Multiplying through by 120 to eliminate fractions:

(120 left(left(x - 2right) cdot frac{1}{15} left(x - 4right) cdot frac{1}{30} x cdot frac{1}{40}right) 120)

This simplifies to:

(8x - 16 4x - 16 3x 120)

Distributing and combining like terms:

(15x - 32 120)

Solving for (x):

(15x 152)

(x frac{152}{15} approx 10.13) days

Since the number of days must be a whole number, we round to 10 days and confirm the solution to be 10 days.

Conclusion

The problem demonstrates how to calculate the total time needed to complete a task by individuals with varying work rates, working together but at different stages of the project. This study not only helps in understanding the principles of collaborative work but also provides practical insights into project management and team dynamics.