Introduction to Collaboration and Typing Efficiency

Efficiency and collaboration are key aspects of any successful project or task. This article takes a closer look at a specific case study where the typing speeds of individual workers and their collaborative effort are analyzed. Through a detailed examination of the given problem, we will derive the time B would take to write 42 pages on their own, drawing on the efficiency and productivity of both A and B working together.

Understanding the Problem

The problem at hand involves two typists, A and B, who have been observed under different work conditions. Initially, we are informed that A can write 75 pages in 25 hours. Subsequently, it is stated that A and B, working together, can write 135 pages in 27 hours. This data can be utilized to determine the individual and combined typing rates, which will help us find the answer to our question: 'In what time can B write 42 pages?'

Calculating the Typing Rate of A

We start by calculating the typing rate of A, which is given by:

A's rate dfrac{75, pages}{25, hours} 3, pages, per, hour

Determining the Combined Typing Rate

When A and B work together, their combined typing rate is calculated as:

Combined rate dfrac{135, pages}{27, hours} 5, pages, per, hour

Deriving B's Typing Rate

Let B's typing rate be denoted as (b) pages per hour. From the information given, we can set up the following equation:

A's rate B's rate Combined rate

3 b 5

Solving for (b), we get:

b 5 - 3 2, pages, per, hour

Calculating the Time for B to Write 42 Pages

Now that we know B's typing rate is 2 pages per hour, we can calculate the time it will take for B to write 42 pages:

Time dfrac{Pages}{Rate} dfrac{42, pages}{2, pages, per, hour} 21, hours

Hence, B can write 42 pages in 21 hours.

Example Calculations

Let's verify this with a few example calculations:

From the first solution, we see that if A can write 75 pages in 25 hours, A's rate is 3 pages per hour. For 27 hours, A would write 81 pages. B would thus write 135 - 81 54 pages in 27 hours, giving B's rate as 2 pages per hour. Therefore, 42 pages would take B 21 hours. Another method involves A typing at 75 pages in 25 hours, hence A's rate is 3 pages per hour. For 27 hours, A has written 81 pages. B’s contribution in this period is 135 - 81 54 pages, giving B’s rate as 2 pages per hour. Hence, B needs 21 hours to write 42 pages. The final solution also shows A’s efficiency at 75/25 3 pages per hour, and after 27 hours, A writes 81 pages. B then writes 135 - 81 54 pages, giving B's rate as 2 pages per hour. B needs 21 hours to write 42 pages.

Conclusion

This analysis confirms that B can write 42 pages in 21 hours with a typing rate of 2 pages per hour. This problem demonstrates how understanding individual and combined work rates can help in solving real-world efficiency and productivity problems. Such skills are invaluable in both professional and academic settings.

Further Explorations

For those interested in further explorations, consider applying similar methods to solve real-world problems involving work rates, time, and productivity. This approach can help optimize team performance and project management.