Solving A Time And Work Problem Determining When A Left The Project
In the realm of quantitative aptitude, time and work problems often present intriguing scenarios that test our ability to analyze and solve real-world situations. These problems typically involve individuals or groups working together to complete a task, with varying efficiencies and time constraints. Understanding the fundamental concepts of work rate, combined work, and individual contributions is crucial for tackling these challenges effectively. In this article, we will delve into a classic time and work problem, meticulously dissecting the given information and employing logical reasoning to arrive at the solution. This comprehensive exploration will not only provide a step-by-step guide to solving the specific problem at hand but also enhance your understanding of the underlying principles applicable to a wide range of similar scenarios. By grasping these core concepts, you will be well-equipped to confidently tackle future time and work problems, regardless of their complexity.
The problem we will be addressing presents a scenario involving two individuals, A and B, working together on a project. Each individual possesses a distinct work rate, meaning they can complete the task at different speeds. Specifically, A can complete a piece of work in 45 days, while B can complete the same work in 40 days. The problem introduces a collaborative element, where A and B begin working together on the project. However, the situation takes an interesting turn when A decides to leave the project after a certain number of days. This departure leaves B to complete the remaining work alone. B diligently continues working and finishes the remaining portion of the work in 23 days. The central question we aim to answer is: How many days did A work on the project before leaving? This problem requires careful consideration of individual work rates, combined work, and the impact of A's departure on the overall project timeline. To solve this puzzle, we will employ a systematic approach, breaking down the problem into smaller, manageable steps and utilizing logical deductions to arrive at the final answer. Understanding the interplay between individual contributions and the overall progress of the work is essential for successfully navigating this challenge.
To effectively tackle this problem, the first crucial step is to determine the individual work rates of A and B. The work rate of an individual represents the fraction of work they can complete in a single day. This concept forms the foundation for calculating combined work and the time required to complete the entire project. We are given that A can complete the entire work in 45 days. This implies that A's work rate is 1/45 of the work per day. In other words, A completes one forty-fifth of the total work each day. Similarly, B can complete the same work in 40 days. Therefore, B's work rate is 1/40 of the work per day. B completes one fortieth of the total work each day. These individual work rates are essential building blocks for understanding how A and B contribute to the project when working together. By quantifying their individual contributions, we can begin to analyze the combined work and the impact of A's departure on the project timeline. Understanding work rates allows us to establish a clear mathematical framework for analyzing the problem and ultimately determining the number of days A worked before leaving. This foundational understanding is crucial for solving the problem efficiently and accurately.
Now that we have established the individual work rates of A and B, we can determine their combined work rate. When individuals work together on a project, their work rates are additive. This means that the combined work rate is simply the sum of their individual work rates. In our problem, A's work rate is 1/45 of the work per day, and B's work rate is 1/40 of the work per day. To find their combined work rate, we add these fractions: 1/45 + 1/40. To add these fractions, we need to find a common denominator. The least common multiple of 45 and 40 is 360. Therefore, we can rewrite the fractions with a common denominator of 360: (1/45) * (8/8) = 8/360 and (1/40) * (9/9) = 9/360. Now we can add the fractions: 8/360 + 9/360 = 17/360. This means that when working together, A and B complete 17/360 of the work each day. This combined work rate is a crucial piece of information for determining how much work A and B completed before A's departure. By understanding their collective progress, we can calculate the remaining work that B had to complete alone and ultimately solve for the number of days A worked on the project. The concept of combined work is a fundamental principle in time and work problems, allowing us to analyze the efficiency of collaborative efforts.
Let's denote the number of days A and B worked together as 'x'. During these 'x' days, they worked at their combined work rate of 17/360 of the work per day. Therefore, the total work completed by A and B together in 'x' days is (17/360) * x. This expression represents the fraction of the total work that was completed before A left the project. It is crucial to understand that this work was accomplished through the combined efforts of A and B, working at their respective efficiencies. The remaining work, which B completed alone, can be calculated by subtracting the work done together from the total work (which we can represent as 1, signifying the completion of the entire task). This step sets the stage for calculating the amount of work B completed individually and, subsequently, the number of days A worked before leaving. By focusing on the work completed during the collaborative phase, we can isolate the impact of A's departure and establish a clear path towards solving for the unknown variable 'x'. This careful dissection of the problem into distinct phases allows for a more organized and accurate solution.
After A's departure, B took on the responsibility of completing the remaining portion of the work alone. We are given that B completed this remaining work in 23 days. Since B's individual work rate is 1/40 of the work per day, the total work completed by B in 23 days is (1/40) * 23, which simplifies to 23/40. This fraction represents the portion of the work that B completed independently after A left the project. To further clarify, this work is equivalent to the remaining work after A and B worked together for 'x' days. Therefore, we can establish a relationship between the work done together, the work done by B alone, and the total work. This relationship will be crucial in setting up an equation to solve for the unknown variable 'x', which represents the number of days A and B worked together. By carefully analyzing B's solo effort, we can bridge the gap between the collaborative phase and the completion of the project, ultimately leading us to the solution. This step highlights the importance of considering individual contributions and the impact of changes in the workforce on the overall project timeline.
Now, we have all the necessary pieces to formulate an equation that will allow us to solve for 'x', the number of days A and B worked together. We know that the total work completed is the sum of the work done by A and B together and the work done by B alone. We can represent the total work as 1, signifying the completion of the entire task. The work done by A and B together is (17/360) * x, and the work done by B alone is 23/40. Therefore, we can write the equation as follows:
(17/360) * x + 23/40 = 1
This equation elegantly captures the relationship between the collaborative work, the individual work, and the overall project completion. It represents a crucial step in solving the problem, as it translates the verbal information into a mathematical expression that can be manipulated to isolate the unknown variable 'x'. To solve this equation, we will need to perform algebraic operations, such as finding a common denominator and isolating 'x' on one side of the equation. The ability to formulate such equations from word problems is a fundamental skill in quantitative aptitude and problem-solving. By carefully translating the given information into a mathematical representation, we can leverage the power of algebra to arrive at the solution. This step underscores the importance of bridging the gap between verbal and mathematical reasoning.
To solve the equation (17/360) * x + 23/40 = 1, we first need to isolate the term containing 'x'. We can do this by subtracting 23/40 from both sides of the equation:
(17/360) * x = 1 - 23/40
To subtract the fractions on the right side, we need a common denominator. The least common multiple of 1 and 40 is 40. Therefore, we can rewrite 1 as 40/40:
(17/360) * x = 40/40 - 23/40
(17/360) * x = 17/40
Now, to isolate 'x', we can multiply both sides of the equation by the reciprocal of 17/360, which is 360/17:
x = (17/40) * (360/17)
The 17 in the numerator and denominator cancels out:
x = 360/40
Simplifying the fraction, we get:
x = 9
Therefore, A and B worked together for 9 days before A left the project. This is the solution to the problem. By systematically solving the equation, we have successfully determined the number of days A contributed to the collaborative effort. This step demonstrates the power of algebraic manipulation in solving real-world problems. The ability to accurately solve equations is a crucial skill in various fields, including mathematics, science, and engineering. This problem highlights the practical application of algebraic principles in solving everyday scenarios.
In conclusion, by carefully analyzing the given information, breaking down the problem into smaller, manageable steps, and applying logical reasoning, we have successfully determined that A worked on the project for 9 days before leaving. This problem exemplifies the importance of understanding fundamental concepts such as work rate, combined work, and individual contributions in solving time and work problems. The systematic approach employed in this solution can be applied to a wide range of similar scenarios, providing a valuable framework for tackling future challenges. Furthermore, this problem highlights the practical application of mathematical principles in real-world situations. By mastering these problem-solving techniques, you can enhance your analytical skills and confidently approach quantitative aptitude challenges. Remember, the key to success lies in a clear understanding of the underlying concepts, a methodical approach to problem-solving, and the ability to translate verbal information into mathematical expressions. With practice and perseverance, you can conquer even the most complex time and work problems.
Keywords: time and work, work rate, combined work, problem-solving, quantitative aptitude
