Cooperative Work Efficiency: A Study on Men, Women, and Children
This article explores the combined work rate of different individuals, specifically focusing on men, women, and children. By understanding their individual and collective work rates, we can determine how efficiently a group of 2 men, 4 women, and 10 children can complete a shared task. Let's delve into the mathematical calculations and practical implications of their work efficiency.
Introduction
The problem at hand involves determining how long it will take for a group of 2 men, 4 women, and 10 children to complete a specific task together. To solve this, we need to calculate each individual's work rate and then combine these rates to find the total work rate of the group.
Individual Work Rates
Men
According to the given information, a man can complete the work in 2 days.
Work rate of 1 man: (frac{1}{2}) work/day Work rate of 2 men: (2 times frac{1}{2} 1) work/dayWomen
4 women can complete the same work in 4 days.
Work rate of 4 women: (frac{1}{4}) work/day Work rate of 1 woman: (frac{1}{4} div 4 frac{1}{16}) work/day Work rate of 4 women: (4 times frac{1}{16} frac{1}{4}) work/dayChildren
5 children can complete the work in 4 days.
Work rate of 5 children: (frac{1}{4}) work/day Work rate of 1 child: (frac{1}{4} div 5 frac{1}{20}) work/day Work rate of 10 children: (10 times frac{1}{20} frac{1}{2}) work/dayCombined Work Rate
Now, we combine the work rates of 2 men, 4 women, and 10 children.
Total work rate: (1 frac{1}{4} frac{1}{2} frac{7}{4}) work/dayThe combined work rate of 2 men, 4 women, and 10 children is (frac{7}{4}) work/day.
Time to Complete Work
We know that the total work is considered as 1 unit (completing the entire job). Using the total work rate of (frac{7}{4}) work/day, we can calculate the time (t) required to complete 1 unit of work.
(t frac{1}{frac{7}{4}} frac{4}{7}) days
Therefore, if 2 men, 4 women, and 10 children work together, they will complete the work in (frac{4}{7}) days.
Conclusion
The work requires either 2 man-days or 16 woman-days or 20 child-days. When a man completes (frac{1}{2}) of the work in 1 day, a woman completes (frac{1}{16}) of the work in 1 day, and a child completes (frac{1}{20}) of the work in 1 day, collectively they can complete (frac{7}{4}) of the work in 1 day. This means they can complete (1 frac{3}{4} 1frac{3}{4}) times of the work in just 1 day.
Practical Implications
Understanding the work efficiency of different groups can help in task allocation and project management. By measuring the combined work rate, we can optimize resource utilization and ensure timely completion of tasks. This method can be applied in various scenarios, such as construction, manufacturing, and other collaborative projects.
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