Time And Work Problem Solving A And B's Completion Time Explained

In the realm of mathematics, particularly in problems concerning time and work, understanding the efficiency and individual contributions of workers is crucial. This article delves into a specific problem where two individuals, A and B, have different work completion times. A starts the work alone, works for a certain number of days, and then leaves. The challenge lies in determining how many days B will take to complete the remaining work. This article provides a step-by-step solution and explanation to help you grasp the concepts involved.

Let's consider the problem at hand: A can complete a work in 7 days, and B can complete the same work in 18 days. A works alone for 5 days and then leaves. The question is: How many days will B take to complete the remaining work? The options provided are:

1. 5 \frac{3}{7}
2. 5 \frac{2}{7}
3. 5 \frac{1}{7}

To solve this, we need to break down the problem into smaller, manageable steps and apply the principles of work and time.

Before we dive into the solution, it's essential to understand the fundamental concepts of work and time. In such problems, we often consider the total work as a unit (e.g., 1) and express the work done by individuals in terms of fractions.

• Work Rate: The amount of work an individual can complete in one day is their work rate. If a person can complete a work in n days, their work rate is \frac{1}{n} per day.
• Total Work: We usually consider the total work to be 1 unit. This makes it easier to calculate fractions of work done.
• Combined Work: When individuals work together, their work rates are added to find the combined work rate.
• Remaining Work: After someone has worked for a certain number of days, the remaining work is the total work minus the work already completed.

1. Determine Individual Work Rates

First, we need to find out how much work A and B can do in one day individually.

• A can complete the work in 7 days, so A's work rate is \frac{1}{7} per day.
• B can complete the work in 18 days, so B's work rate is \frac{1}{18} per day.

Understanding the individual work rates is crucial because it allows us to quantify how much each person contributes to the completion of the task. These rates form the basis for calculating the amount of work done in a given period and the remaining work.

2. Calculate Work Done by A in 5 Days

Next, we calculate how much work A completes in the 5 days he works alone. To do this, we multiply A's work rate by the number of days he works:

• Work done by A in 5 days = A's work rate × Number of days
• Work done by A in 5 days = \frac{1}{7} × 5 = \frac{5}{7}

This step is pivotal as it quantifies A's contribution before leaving. By determining the fraction of work completed by A, we can then assess the remaining workload for B.

3. Calculate Remaining Work

Now, we need to find out how much work is left after A has worked for 5 days. We subtract the work done by A from the total work:

• Remaining work = Total work - Work done by A
• Remaining work = 1 - \frac{5}{7} = \frac{2}{7}

Calculating the remaining work is essential as it sets the stage for determining how long it will take B to finish the job. This fraction represents the portion of the task that B must complete.

4. Calculate Days for B to Complete Remaining Work

Finally, we calculate how many days B will take to complete the remaining work. To do this, we divide the remaining work by B's work rate:

• Days for B to complete remaining work = \frac{Remaining work}{B's work rate}
• Days for B to complete remaining work = \frac{\frac{2}{7}}{\frac{1}{18}}
• Days for B to complete remaining work = \frac{2}{7} × 18
• Days for B to complete remaining work = \frac{36}{7}
• Days for B to complete remaining work = 5 \frac{1}{7}

Therefore, B will take 5 \frac{1}{7} days to complete the remaining work.

This calculation provides the final answer to our problem. By dividing the remaining work by B's work rate, we determine the time required for B to complete the task. This step highlights the importance of understanding individual work rates and their impact on overall project completion time.

In conclusion, B will take 5 \frac{1}{7} days to complete the remaining work. This problem illustrates how to approach work and time problems by breaking them down into manageable steps. Understanding individual work rates, calculating work done, and determining remaining work are essential skills in solving such problems. The correct option is 3) 5 \frac{1}{7}.

Mastering these techniques allows for the effective resolution of complex problems related to time and work, which often appear in various competitive exams and real-life scenarios. The ability to dissect the problem, apply the correct formulas, and sequentially calculate the solution is crucial for success. By understanding the underlying concepts and practicing similar problems, individuals can enhance their problem-solving skills and improve their mathematical proficiency.

To further enhance your problem-solving skills in time and work scenarios, consider these additional tips:

• Visualize the Problem: Sometimes, drawing a diagram or visualizing the work being done can help you understand the problem better.
• Use a Table: Organizing the information in a table can make it easier to keep track of work rates, time, and work done.
• Practice Regularly: The more you practice, the more comfortable you will become with these types of problems. Try solving a variety of questions to cover different scenarios.
• Understand the Units: Make sure you are consistent with your units (e.g., days, hours). If the problem involves different units, convert them to a common unit before solving.
• Check Your Answer: After solving the problem, take a moment to check if your answer makes sense in the context of the problem.

By employing these strategies, you can tackle even the most challenging time and work problems with confidence and precision. Remember, the key to success lies in understanding the fundamental principles, practicing regularly, and developing a systematic approach to problem-solving. This ensures that you are well-prepared to handle similar questions effectively and efficiently.

When solving time and work problems, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and improve your accuracy. Here are some of the common mistakes to watch out for:

• Forgetting to Account for Work Done: A frequent error is not calculating the work done by the first person before the second person starts. Always subtract the initial work from the total work.
• Incorrectly Adding Work Rates: When people work together, their work rates should be added. Make sure you are adding fractions correctly, finding common denominators when necessary.
• Using the Wrong Formula: Using the wrong formula or method can lead to incorrect results. Ensure you understand the basic formulas for work, time, and rate.
• Misunderstanding the Question: Always read the question carefully to understand exactly what is being asked. Misinterpreting the question can lead to solving for the wrong variable.
• Unit Inconsistency: If the problem involves different units (e.g., days and hours), failing to convert them to a common unit can result in an incorrect answer.

By being mindful of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy in solving time and work problems. Remember, careful attention to detail and a systematic approach are crucial for success. Regularly reviewing your work and identifying areas where you commonly make errors can also help you refine your problem-solving skills.

The concepts of time and work are not just confined to the realm of mathematics textbooks; they have numerous real-world applications. Understanding these applications can help you appreciate the practical significance of this topic and improve your problem-solving abilities in everyday situations. Here are some examples of how time and work problems are relevant in real life:

• Project Management: In project management, estimating the time required to complete tasks and allocating resources efficiently are crucial. Time and work principles help in scheduling and resource planning.
• Construction: Construction projects involve various tasks with different time requirements. Estimating the time needed to complete a project and assigning workers to tasks can be optimized using these concepts.
• Manufacturing: In manufacturing, determining the production rate and the time required to produce a certain number of items is essential for efficient operations. Time and work calculations help in production planning.
• Software Development: Software development projects involve multiple developers working on different modules. Estimating the time required for each module and managing the team’s workload are important aspects where these principles can be applied.
• Daily Life: Even in daily life, we encounter situations where time and work concepts are useful. For example, planning a household chore schedule or estimating how long it will take to complete a set of tasks involves similar calculations.

By recognizing these real-world applications, you can better appreciate the importance of mastering time and work problems. This understanding can also motivate you to improve your skills in this area and apply these principles in various practical scenarios. The ability to analyze and solve such problems effectively is a valuable asset in both professional and personal contexts.

By understanding these key concepts, practicing regularly, and avoiding common mistakes, you can master time and work problems and excel in your mathematical endeavors.