Points to Remember:
- This is a rate-based problem involving work done by individuals and their combined work rate.
- We need to find the combined work rate and then use it to calculate the time taken when working together.
- The key is to express the work rate of each individual as a fraction of the work completed per day.
Introduction:
This question is a classic example of a work-rate problem. It involves determining the combined efficiency of individuals working together to complete a task. The fundamental concept is that the rate of work is inversely proportional to the time taken. If A can complete a work in 4 days, their work rate is 1/4 of the work per day. Similarly, B’s work rate is 1/5 per day, and C’s is 1/10 per day. We will use these individual rates to calculate their combined rate and subsequently the time taken to complete the work together.
Body:
1. Individual Work Rates:
- A’s work rate: 1/4 of the work per day
- B’s work rate: 1/5 of the work per day
- C’s work rate: 1/10 of the work per day
2. Combined Work Rate:
To find the combined work rate, we simply add the individual work rates:
Combined work rate = A’s work rate + B’s work rate + C’s work rate
= 1/4 + 1/5 + 1/10
= (5 + 4 + 2) / 20
= 11/20 of the work per day
3. Time Taken Working Together:
If they complete 11/20 of the work in one day, the time taken to complete the entire work (which is represented by 1) is:
Time taken = Total work / Combined work rate
= 1 / (11/20)
= 20/11 days
Therefore, it will take A, B, and C 20/11 days, or approximately 1.82 days, to complete the work together.
Conclusion:
In summary, by calculating the individual work rates of A, B, and C and then summing them to find their combined work rate, we determined that it would take them 20/11 days to complete the work together. This problem highlights the concept of combined efficiency and how individual contributions can be aggregated to achieve a common goal more quickly. This simple model can be applied to various real-world scenarios, such as project management, where understanding individual and team productivity is crucial for efficient task completion. Further analysis could incorporate factors like varying work efficiency over time or potential disruptions to the workflow, but this basic calculation provides a valuable starting point for understanding collaborative work dynamics. The result emphasizes the synergistic effect of teamwork, leading to faster completion of tasks compared to individual efforts.
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