A Is 50% More Efficient Than B A Work Efficiency Problem

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This article delves into a problem concerning work efficiency and time management, a common topic in mathematics and project management. We'll explore a scenario where two individuals, A and B, work together and independently on a task, with varying efficiencies and time allocations. The core challenge lies in deciphering the individual work rates and subsequently calculating the time taken to complete the job under changing circumstances. Let's break down the problem step-by-step, ensuring clarity and a comprehensive understanding of the solution.

Problem Statement

Person A is stated to be 50% more efficient than person B. This implies that A completes work at a faster rate than B. Initially, A and B collaborate on a job for 5 days. Following this, A departs, and B continues the work alone for a certain duration. Eventually, A returns to substitute B and concludes the remaining 10% of the job. The entire job spans a total of 7 days. The objective is to determine the time duration for which B worked alone.

Defining Efficiency and Work Rate

To solve this, we must first understand the concept of efficiency in the context of work. Efficiency is directly related to the rate at which work is completed. If A is 50% more efficient than B, it means A's work rate is 1.5 times B's work rate. We can represent this mathematically to simplify the calculations. Let's denote B's work rate as b (the fraction of the job B completes in one day) and A's work rate as a. Therefore, a = 1.5b. This relationship is crucial in formulating equations that will help us solve the problem. By defining the variables and establishing the relationship between their efficiencies, we create a solid foundation for the subsequent steps in the problem-solving process. This methodical approach is essential in tackling complex work-rate problems.

Setting Up the Equations

Now that we understand the relationship between A and B's work rates, let's translate the problem's information into mathematical equations. This involves expressing the work done in different phases of the project in terms of a, b, and the time spent. For the first 5 days, A and B worked together. Their combined work rate is a + b. So, the fraction of work completed in these 5 days is 5(a + b). After A left, B worked alone for a certain number of days, which we'll denote as x. The work done by B during this period is x * b*. Finally, A returned and completed the remaining 10% of the job. Since we've already accounted for the work done by A in the first 5 days, this remaining work was done by A after B stopped working alone. The fact that the remaining work is 10% is a crucial piece of information. It provides us with a fixed value to incorporate into our equations. The sum of the work done in all phases must equal 1 (representing the completion of the entire job). This gives us the primary equation: 5(a + b) + x * b* + (work done by A to complete 10%) = 1. The strategic formation of this equation is a pivotal step toward solving the problem. This allows us to mathematically represent the relationships described in the problem statement.

Accounting for the Remaining Work

The problem states that A finishes the remaining 10% of the job. To calculate the time A took to complete this, we need to know A's work rate. We already established that A's work rate is 1.5b. Let's denote the time A worked alone as t. The work done by A in this time is t * a* which is t * 1.5b. According to the problem, this work represents 10% or 0.1 of the total job. Therefore, we have the equation t * 1.5b = 0.1. We now have a separate equation that helps us isolate t in terms of b. This is a crucial step in simplifying the primary equation, as it allows us to express all variables in terms of b. The isolation of variables is a common strategy in solving multi-variable equations. By isolating t and expressing it in terms of b, we reduce the complexity of the original equation.

Total Time Constraint

The problem also provides the constraint that the total time taken to finish the job is 7 days. This piece of information is vital because it links the time A and B worked together, the time B worked alone, and the time A worked alone. We know A and B worked together for 5 days, B worked alone for x days, and A worked alone for t days. Therefore, the equation representing the total time is 5 + x + t = 7. This equation simplifies to x + t = 2. Now, we have a simple linear equation relating x and t. This equation, combined with the previous equations we've established, forms a system of equations that we can solve to find the value of x, which represents the time B worked alone. The incorporation of the total time constraint is a key step in solving the problem, as it provides the necessary link between the various phases of the job.

Solving the System of Equations

Now we have a system of equations:

  1. a = 1.5b
  2. 5(a + b) + x * b* + t * a* = 1
  3. t * 1.5b = 0.1
  4. x + t = 2

Substituting a = 1.5b in equation 2, we get:

5(1.5b + b) + x * b* + t * 1.5b = 1

  1. 5b + x * b* + 1.5t * b* = 1

From equation 3, we can express t as t = 0.1 / (1.5b) = 1 / (15b). Substituting this in equation 4, we get x = 2 - t = 2 - 1 / (15b). Now substitute t in equation 5:

  1. 5b + x * b* + 1.5 * (1 / (15b)) * b* = 1

  2. 5b + x * b* + 0.1 = 1

Substitute x = 2 - 1 / (15b) in the above equation:

  1. 5b + (2 - 1 / (15b)) * b* + 0.1 = 1

  2. 5b + 2b - 1/15 + 0.1 = 1

  3. 5b + 2b = 1 - 0.1 + 1/15

  4. 5b = 0.9 + 1/15

  5. 5b = 9/10 + 1/15

  6. 5b = (27 + 2) / 30

  7. 5b = 29/30

b = 29 / (30 * 7.5) = 29/225

Now we can find t:

t = 1 / (15 * (29/225)) = 1 / (29/15) = 15/29

Finally, we can find x:

x = 2 - t = 2 - 15/29 = (58 - 15) / 29 = 43/29

Thus, x, the time B worked alone, is 43/29 days, which is approximately 1.48 days.

Final Answer

Therefore, B worked alone for approximately 1.48 days. This comprehensive solution demonstrates how to break down a complex work-rate problem into manageable steps, using mathematical equations and logical reasoning. Each step, from defining variables to solving the system of equations, plays a crucial role in arriving at the final answer.

Conclusion

This problem illustrates the importance of understanding the relationship between work rate, time, and efficiency. By carefully translating the problem statement into mathematical equations and systematically solving them, we can determine the time each person spent working on the job. This approach is applicable to various scenarios involving work distribution and time management. The key is to break down the problem into smaller, manageable parts and use the given information to establish relationships between the variables.

Work Efficiency Problem Solving A is 50% More Efficient Than B