Calculating Work Efficiency: A and B Together and Alone
In many real-world scenarios, understanding the work rate and efficiency of individuals working together or separately is crucial. This article will explore the relationship between two workers, A and B, as they undertake a piece of work. We'll delve into the calculation of their individual and combined work rates, and how these rates determine the total time required to complete a task.
Introduction to Work Rates
Let's denote the total work as W. A and B together can finish the work in 30 days. If we denote A's work rate as a (work per day) and B's work rate as b (work per day), then their combined work rate is:
W 30 work per dayThis combined work rate tells us that together, A and B can finish W units of work in 30 days.
Combining Work for 20 Days
For the first 20 days, A and B work together. During this period, they complete:
20 30 WAfter 20 days, B leaves, and A continues working alone for 15 more days to finish the remaining work. We denote A's work rate as a.
Remaining Work Calculation
The remaining work after 20 days is:
W - 20 30 W W 3A completes this remaining W/3 units of work in 15 days. Using the work rate formula, we can express this as:
15 a W 3 #8748; a 1 45This means A's work rate is 1/45 work per day. Therefore, A alone can finish the entire work in:
Time W 1 45 45 daysHence, A alone can finish the job in 45 days.
Alternative Calculation for B
Using a different approach, let's see how B can complete the work alone.
1 a 1 b 1 30From the equation above, we can solve for B's work rate:
1 b 1 30 - 1 aAfter 20 days, the remaining work is:
W - 20 30 W W 3Solving for B, we find:
b 1528 7 #8748; 60 daysHence, B alone can finish the work in 60 days.
Conclusion
In conclusion, the calculations demonstrate how the combined and individual work rates can be used to determine the time required for a task to be completed by two workers. Whether A or B, both can complete the work alone, with A taking 45 days and B taking 60 days.