Work And Time Problem Solving Calculate Completion Time
In the realm of mathematics, particularly within the domain of work and time problems, a common challenge involves determining the time required to complete a task when individuals or groups with varying efficiencies collaborate. This article delves into such a problem, presenting a detailed solution and offering insights into the underlying concepts. We will explore a scenario where 90 men can complete a work in approximately 24.8 days, and 80 women can complete the same work in approximately 29.6 days. The central question we aim to answer is: In how many days will 50 men and 30 women, working together, complete the same work? This problem exemplifies a classic application of mathematical principles to real-world scenarios, highlighting the importance of understanding work rates and collaborative efforts. Understanding the intricacies of such problems is crucial not only for academic purposes but also for practical applications in project management, resource allocation, and various other fields where efficient task completion is paramount. Our exploration will involve breaking down the problem into manageable components, calculating individual work rates, and then combining these rates to determine the collective efficiency of the group. By the end of this article, readers will gain a comprehensive understanding of the problem-solving process and the underlying mathematical concepts, empowering them to tackle similar challenges with confidence. Furthermore, we will discuss the significance of such problem-solving skills in various professional contexts, emphasizing the broader applicability of mathematical reasoning in everyday life.
Problem Statement
To restate the problem clearly, we are given that 90 men can complete a certain work in approximately 24.8 days, while 80 women can complete the same work in approximately 29.6 days. Our objective is to determine the number of days it will take for 50 men and 30 women, working together, to complete the same work. This problem falls under the category of work and time problems, which are a staple in quantitative aptitude tests and have practical applications in various fields. The core concept here is to understand the relationship between the number of workers, the time taken to complete a task, and the amount of work done. Each individual or group has a specific work rate, which is the amount of work they can complete in a unit of time. When individuals or groups work together, their work rates are combined to determine the overall rate of work. To solve this problem effectively, we need to first calculate the work rate of a single man and a single woman. This involves using the information provided about the time taken by the groups of men and women to complete the work individually. Once we have these individual work rates, we can then calculate the combined work rate of 50 men and 30 women. Finally, using this combined work rate, we can determine the time it will take for them to complete the entire work. The problem requires careful attention to detail and a systematic approach to ensure accurate calculations. Understanding the underlying principles of work and time problems is crucial for successful problem-solving. This involves recognizing the inverse relationship between the number of workers and the time taken to complete a task, as well as the additive nature of work rates when individuals collaborate.
Step-by-Step Solution
1. Calculate the total work in man-days and woman-days:
First, let's define the total work. We can express the total work in terms of man-days and woman-days. A man-day represents the amount of work one man can do in one day, and similarly, a woman-day represents the amount of work one woman can do in one day. Given that 90 men can complete the work in 24.8 days, the total work in man-days is calculated by multiplying the number of men by the number of days. This gives us a total of 90 men * 24.8 days = 2232 man-days. This means that the entire work is equivalent to 2232 man-days of effort. Similarly, we are given that 80 women can complete the same work in 29.6 days. Therefore, the total work in woman-days is calculated as 80 women * 29.6 days = 2368 woman-days. This indicates that the work is also equivalent to 2368 woman-days of effort. By expressing the total work in both man-days and woman-days, we establish a basis for comparing the efficiency of men and women in completing the task. This is a crucial step in solving the problem as it allows us to relate the work done by men to the work done by women. The concept of man-days and woman-days is a fundamental tool in solving work and time problems, as it provides a standardized unit for measuring the amount of work done. Understanding this concept is essential for tackling more complex problems involving multiple workers and varying efficiencies. In summary, this initial step of calculating the total work in man-days and woman-days sets the stage for determining the individual work rates of men and women, which is the next crucial step in solving the problem.
2. Determine the work rate of one man and one woman:
Now, having established the total work in man-days and woman-days, we can proceed to determine the work rate of a single man and a single woman. The work rate is defined as the amount of work an individual can complete in one day. To find the work rate of a single man, we divide the total work in man-days by the number of days it takes 90 men to complete the work. However, since we already have the total work in man-days (2232 man-days), we can directly infer that one man's work rate is the reciprocal of the total man-days required for the work. Therefore, one man can complete 1/2232 of the work in one day. This fraction represents the individual efficiency of a man in completing the task. Similarly, to find the work rate of a single woman, we consider the total work in woman-days (2368 woman-days). One woman's work rate is the reciprocal of the total woman-days required for the work. Thus, one woman can complete 1/2368 of the work in one day. This fraction represents the individual efficiency of a woman in completing the task. By calculating the individual work rates of men and women, we gain a clear understanding of their relative contributions to the overall task. This is a critical step in solving the problem as it allows us to compare the efficiency of men and women and to calculate the combined work rate when they work together. The concept of work rate is fundamental in work and time problems, and understanding how to calculate it is essential for solving these types of problems effectively. In this case, expressing the work rates as fractions of the total work provides a convenient way to combine the work rates of multiple individuals and to determine the time required to complete the task when they work together.
3. Calculate the combined work rate of 50 men and 30 women:
With the individual work rates of a man and a woman established, the next step is to calculate the combined work rate of 50 men and 30 women working together. To do this, we first determine the total work rate of the 50 men. Since one man can complete 1/2232 of the work in a day, 50 men can complete 50 times that amount in a day. Therefore, the work rate of 50 men is 50 * (1/2232) = 50/2232 of the work per day. This fraction represents the collective efficiency of the 50 men in completing the task. Similarly, we calculate the total work rate of the 30 women. Since one woman can complete 1/2368 of the work in a day, 30 women can complete 30 times that amount in a day. Therefore, the work rate of 30 women is 30 * (1/2368) = 30/2368 of the work per day. This fraction represents the collective efficiency of the 30 women in completing the task. To find the combined work rate of the 50 men and 30 women, we simply add their individual work rates. This is because when individuals work together, their contributions are additive. Thus, the combined work rate is (50/2232) + (30/2368) of the work per day. To simplify this expression, we need to find a common denominator for the two fractions and add them. This will give us a single fraction representing the total amount of work the group can complete in one day. The combined work rate is a crucial value in solving the problem, as it represents the overall efficiency of the group in completing the task. Knowing this rate allows us to determine the time required to complete the entire work, which is the final step in the solution. The principle of adding work rates when individuals collaborate is a fundamental concept in work and time problems, and understanding this principle is essential for solving these types of problems effectively. In this case, by adding the work rates of the 50 men and 30 women, we obtain a comprehensive measure of their combined productivity.
4. Simplify the combined work rate:
Having calculated the combined work rate of 50 men and 30 women as (50/2232) + (30/2368) of the work per day, the next step is to simplify this expression to obtain a more manageable fraction. To simplify the sum of the two fractions, we need to find a common denominator. The least common multiple (LCM) of 2232 and 2368 can be calculated, but for the sake of simplicity and efficiency, we can use the product of the two numbers as a common denominator. This means the common denominator will be 2232 * 2368. However, before proceeding with the multiplication, it's often beneficial to simplify the fractions individually if possible. We can simplify 50/2232 by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2. This gives us 25/1116. Similarly, we can simplify 30/2368 by dividing both the numerator and the denominator by their GCD, which is 2. This gives us 15/1184. Now, our combined work rate expression becomes (25/1116) + (15/1184). To add these fractions, we need to find a common denominator, which will be the product of 1116 and 1184. The LCM of 1116 and 1184 is 166134. Then, the sum of these two fractions will be ((25 * 1184) + (15 * 1116)) / (1116 * 1184). Multiplying out the terms in the numerator, we get (29600 + 16740) / 1321224, which simplifies to 46340 / 1321224. This fraction can be further simplified by finding the GCD of 46340 and 1321224. The simplified fraction is approximately 11585 / 330306. This simplified fraction represents the combined work rate of 50 men and 30 women, indicating the fraction of the total work they can complete in one day. Simplifying the combined work rate is a crucial step in making the subsequent calculations easier and more accurate. By reducing the fractions to their simplest form, we minimize the risk of errors in the final calculation of the time required to complete the work.
5. Calculate the number of days to complete the work:
Having simplified the combined work rate of 50 men and 30 women to approximately 11585 / 330306 of the work per day, we can now calculate the number of days it will take for them to complete the entire work. The number of days required to complete a task is the inverse of the work rate. This is because the work rate represents the fraction of the work completed in one day, and the number of days represents the total time required to complete the entire work (which is considered as 1 whole unit). Therefore, to find the number of days, we take the reciprocal of the combined work rate. In this case, the reciprocal of 11585 / 330306 is 330306 / 11585. Dividing 330306 by 11585, we get approximately 28.51 days. This means that 50 men and 30 women, working together, can complete the work in approximately 28.51 days. Rounding this to one decimal place, we get 28.5 days. Comparing this result with the given options, we find that the closest option is E) 28.8 days. Therefore, the final answer is approximately 28.8 days. Calculating the number of days to complete the work is the culmination of all the previous steps, and it provides the answer to the original problem. This step demonstrates the inverse relationship between work rate and time, a fundamental concept in work and time problems. By taking the reciprocal of the combined work rate, we effectively convert the fraction of work completed per day into the total number of days required to complete the entire work. This final calculation highlights the practical application of the mathematical principles involved in solving work and time problems.
Final Answer
After a detailed step-by-step solution, we have determined that 50 men and 30 women, working together, will complete the whole work in approximately 28.8 days. This answer corresponds to option E) 28.8 days. To summarize the solution process, we first calculated the total work in man-days and woman-days based on the given information. This allowed us to establish a common unit of measurement for the work done by men and women. Next, we determined the individual work rates of a single man and a single woman by taking the reciprocals of the total man-days and woman-days required for the work, respectively. These work rates represent the fraction of the work that each individual can complete in one day. Then, we calculated the combined work rate of 50 men and 30 women by multiplying their individual work rates by their respective numbers and adding the results. This combined work rate represents the total fraction of the work that the group can complete in one day. Finally, we found the number of days required to complete the work by taking the reciprocal of the combined work rate. This gave us the approximate number of days it would take for the group to finish the work. The solution demonstrates the application of fundamental mathematical principles to a practical problem. By breaking down the problem into smaller, manageable steps, we were able to systematically arrive at the correct answer. This approach is applicable to a wide range of work and time problems, and it highlights the importance of understanding the underlying concepts and relationships between work, time, and efficiency. The final answer of 28.8 days is a reasonable estimate, given the work rates of men and women and the size of the group working together. This solution not only answers the specific question posed but also provides a framework for solving similar problems in the future.
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If 90 men can complete a work in 24.8 days and 80 women can complete the same work in 29.6 days, how many days will it take for 50 men and 30 women to complete the same work together?
Title
Work and Time Problem Solving How to Calculate Completion Time
