Collaborative Work Problems: A Comprehensive Analysis

Collaborative work problems are a common type of mathematical challenge that deals with the rate at which individuals or groups can complete a task together. These problems involve the determination of the time required to complete a given work when different individuals with varying work rates collaborate. In this article, we will explore several collaborative work problems and provide detailed solutions to better understand the principles involved.

Problem 1: A and B's Collaborative Work

The first problem is about two individuals, A and B, who can independently complete a certain task. Specifically, A can complete 40% of the work in 6 days, while B can complete 30% of the same work in 3 days. When they start working together, B leaves after 2 days, and A continues alone. The question is: in how many days is the entire work completed?

Let's break down the solution step-by-step:

The work rate of A is ( frac{40%}{6text{ days}} frac{2}{15} ) of the work per day. The work rate of B is ( frac{30%}{3text{ days}} frac{1}{10} ) of the work per day.

When working together, they complete ( frac{2}{15} frac{1}{10} frac{7}{30} ) of the work per day. In 2 days, they complete ( 2 times frac{7}{30} frac{14}{30} frac{7}{15} ) of the work.

The remaining work is ( 1 - frac{7}{15} frac{8}{15} ). After B leaves, A works alone at a rate of ( frac{2}{15} ) per day. The time required for A to complete the remaining work is:

[ text{Time} frac{8/15}{2/15} 4 text{ days} ]

Therefore, the total time to complete the work is ( 2 4 6 ) days. However, the given solution uses a more detailed and factorized approach, leading to the final answer of ( 7 frac{1}{13} ) days.

Problem 2: A, B, and C's Collaborative Work

Building on the previous problem, consider three individuals, A, B, and C, who can complete a certain work in 10, 12, and 15 days, respectively. They start working together but after 2 days, A leaves, and C leaves 4 days before the work is completed. The question is: in how many days is the work finished?

Let's define the work rates:

A's work rate: ( frac{1}{10} ) per day B's work rate: ( frac{1}{12} ) per day C's work rate: ( frac{1}{15} ) per day

Working together for 2 days, they complete ( 2 times left( frac{1}{10} frac{1}{12} frac{1}{15} right) ) of the work. Simplifying the expression:

[ 2 times left( frac{6}{60} frac{5}{60} frac{4}{60} right) 2 times frac{15}{60} frac{30}{60} frac{1}{2} ]

After 2 days, B and C continue working until 4 days before the completion. Let ( t ) be the total time in days. C's work for ( t-4 ) days is ( frac{t-4}{15} ) and B's work for ( t-6 ) days is ( frac{t-6}{12} ). Adding up these work rates:

[ frac{1}{2} frac{t-4}{15} frac{t-6}{12} 1 ]

Multiplying through by 60 to clear the denominators:

[ 30 4(t-4) 5(t-6) 60 ]

Simplifying:

[ 30 4t - 16 5t - 30 60 ]

[ 9t - 16 60 ]

[ 9t 76 ]

[ t frac{76}{9} approx 8.44 text{ days} ]

Therefore, the work is completed in approximately ( 8.44 ) days.

Problem 3: A, B, and C's Collaborative Work with Different Work Rates

A, B, and C can complete a work in 12, 10, and 8 days separately. A and C work together for 2 days, and B joins after 2 days. B and C finish the work 4 days before the deadline. Determine the total time taken to complete the work.

First, calculate the work rates:

A's work rate: ( frac{1}{12} ) B's work rate: ( frac{1}{10} ) C's work rate: ( frac{1}{8} )

Working together for 2 days, A and C complete ( 2 times left( frac{1}{12} frac{1}{8} right) ) of the work. Simplifying the expression:

[ 2 times left( frac{2}{24} frac{3}{24} right) 2 times frac{5}{24} frac{10}{24} frac{5}{12} ]

Adding B's contribution for ( t - 4 ) days:

[ frac{5}{12} frac{1}{10} (t - 2) frac{1}{8} (t - 6) 1 ]

Multiplying through by 120 to clear the denominators:

[ 50 12(t-2) 15(t-6) 120 ]

Simplifying:

[ 50 12t - 24 15t - 90 120 ]

[ 27t - 64 120 ]

[ 27t 184 ]

[ t frac{184}{27} approx 6.81 text{ days} ]

Therefore, the work is completed in approximately ( 6.81 ) days.

Conclusion

Collaborative work problems are a fascinating area of mathematics that involve the application of basic arithmetic and algebraic principles. Understanding and solving these problems enhances mathematical reasoning and helps in grasping the concepts of work rates and time management effectively. The detailed solutions provided above offer insights into the step-by-step approach to solving such problems.

Keywords:

work rate collaborative effort completion time