How Many Days Will 24 Men Take to Finish the Job?

Understanding the relationship between the number of workers and the number of days required to complete a task is essential in project management. The concept of man-days can be very useful in determining such relationships. Let’s dive into a specific example to clarify how this works.

Problem Statement

The original problem states that 12 men can finish a job in 24 days. The task is to determine how long it will take for 24 men to complete the same job. To solve this, we will use the concept of man-days, which is the product of the number of workers and the number of days they work.

Total Man-Days Calculation

First, let's calculate the total man-days required to complete the work:

$$ 12 text{ men} times 24 text{ days} 288 text{ man-days} $$

This total amount of work is fixed, regardless of the number of workers available. Therefore, the total man-days needed to complete the job is 288.

Calculating the Days for 24 Men

Now, let's find out how many days it will take for 24 men to complete the same job. We will use the following formula:

$$ text{Number of days} times text{Number of men} text{Total man-days} $$

Let (d) be the number of days it takes for 24 men to finish the job. Therefore:

$$ 24 text{ men} times d text{ days} 288 text{ man-days} $$

Solving for (d):

$$ d frac{288}{24} 12 text{ days} $$

Therefore, it will take 12 days for 24 men to finish the job.

Algebraic Solution

We can also solve this problem using an algebraic approach. Given that 12 men can finish the work in 24 days, let:

$$ M_1 12 text{ men, } D_1 24 text{ days} $$

Let (M_2) be 24 men and (D_2) be the number of days it takes 24 men to complete the work. We have the equation:

$$ M_1 times D_1 M_2 times D_2 $$

Substituting the known values:

$$ 12 times 24 24 times D_2 $$

Solving for (D_2):

$$ D_2 frac{12 times 24}{24} 12 text{ days} $$

Again, it will take 12 days for 24 men to complete the job.

Additional Examples and Concepts

Let's explore some additional scenarios and related concepts:

Indirect Variation

This problem is an example of indirect variation. In indirect variation, the product of the two quantities remains constant. For instance, if 16 men work for 12 days, the total man-days are:

$$ 16 text{ men} times 12 text{ days} 192 text{ man-days} $$

If we now have 24 men, we need to determine the number of days ((D_2)) required to achieve the same total man-days:

$$ 192 text{ man-days} 24 text{ men} times D_2 $$

Solving for (D_2):

$$ D_2 frac{192}{24} 8 text{ days} $$

Therefore, 24 men can complete the job in 8 days.

Indirect Variation with Proportions

Another way to look at this is through proportions. If 8 men take 12 days to do a certain job, we can set up the following proportion to find out how many days 24 men will take:

$$ 8 text{ men} times 12 text{ days} 24 text{ men} times X text{ days} $$

Solving for (X):

$$ X frac{8 times 12}{24} 4 text{ days} $$

Therefore, 24 men will take 4 days to complete the job.

Unit Work Concept

One can also approach this problem by finding the amount of work done per man per day. If 16 men can complete a job in 12 days, then:

$$ text{Total work} 16 text{ men} times 12 text{ days} 192 text{ units of work} $$

Now, if we have 24 men, the number of days needed to complete the same amount of work is:

$$ text{Number of days} frac{192 text{ units of work}}{24 text{ men}} 8 text{ days} $$

This confirms that 24 men will complete the job in 8 days.

Conclusion

Understanding the relationship between the number of workers and the number of days is crucial in project management. The concept of man-days and the principles of indirect variation are valuable tools in determining the time required to complete a job given a specific number of workers. The examples provided here illustrate how these principles can be applied in practical scenarios to optimize work schedules and resource allocation.