Solving Work and Time Problems: A Comprehensive Guide

When dealing with work and time problems, it is crucial to understand the relationship between the amount of work, the number of workers, and the duration of work. This is particularly important for SEO optimization as problems like these often appear in various standardized tests and real-world applications. In this article, we present a detailed solution to a common problem: determining the number of men needed to complete a certain amount of work within a specified period.

Understanding the Problem

The problem at hand involves a scenario where 3/5 of the work is completed by 12 men in 10 days. We need to find out how many men will be required to complete the entire work in 20 days. This problem can be solved using the concept of man-days, which is a fundamental unit in work and time problems.

Step-by-Step Solution

Step 1: Determine the Total Amount of Work

Since 3/5 of the work is done in 10 days, the remaining 2/5 of the work must be completed in the next 10 days. This step helps us understand the distribution of work over the given time frame.

Step 2: Find the Amount of Work Done Per Day

First, we need to find how much work is done per day. Since 3/5 of the work is done in 10 days, the work done per day is:

[ text{Work done per day} frac{3/5 cdot text{Total Work}}{10} frac{3/50 cdot text{Total Work}}{1} ]

Step 3: Find the Work Done by 12 Men in One Day

The work done by 12 men in one day is equal to the work done per day:

[ text{Work done by 12 men in 1 day} frac{3/50 cdot text{Total Work}}{1} ]

Step 4: Find the Work Done by One Man in One Day

To find the work done by one man in one day, we divide the work done by 12 men in one day by 12:

[ text{Work done by 1 man in 1 day} frac{text{Work done by 12 men in 1 day}}{12} frac{3/50 cdot text{Total Work}}{12} frac{1/200 cdot text{Total Work}}{1} ]

Step 5: Find the Number of Men Required to Complete the Whole Work in 20 Days

Now, to complete the whole work in 20 days, we need to determine how many men are required. The total work needed to be completed is the remaining 2/5 of the work. Using the work done by one man in one day, we find the required number of men:

[ text{Number of men required} frac{2/5 cdot text{Total Work}}{1/200 cdot text{Total Work} / 20} 40 text{ men} ]

Therefore, 40 men will be required to complete the whole work in 20 days.

Alternative Approaches

There are several alternative methods to solve this problem. One such approach is through the use of man-days. Here, we calculate the total amount of work in man-days and then use it to find the required number of men for the given duration.

Man-Days Calculation:

Since 3/5 of the work is done by 12 men in 10 days, the total work done is 12 men x 10 days 120 man-days. The total work is 5/3 times the work done, so the total work is 200 man-days. To complete this work in 20 days, we need:

[ text{Number of men required} frac{200 text{ man-days}}{20 text{ days}} 10 text{ men} ]

Therefore, 10 men working for 20 days would be required to complete the work.

Conclusion

Solving work and time problems requires a clear understanding of the basic relationship between work, time, and the number of workers. By using man-days and logical steps, we can efficiently solve these problems and optimize our solutions for better performance in standardized tests and real-world applications.