Solving a Complex Work Rate Problem: 4 Men and 6 Women vs 3 Men and 7 Women

In this article, we'll explore a complex work rate problem involving men and women completing a job. We'll break down the problem into a series of logical steps, solving for the individual work rates of men and women, and then use these rates to determine the time required for a different number of men and women to complete the same job. This type of problem is relevant for understanding work rate calculations in real-world scenarios, such as project management and scheduling.

Introduction to the Problem

The problem states: '4 men and 6 women can complete a job in 8 days while 3 men and 7 women can complete it in 10 days. We need to determine in how many days 10 women and 8 men can complete the job.

Step 1: Setting Up the Equations

Let's define:- m: Work rate of one man (jobs per day)- w: Work rate of one woman (jobs per day)

From the given information, we can set up the following equations:

4 men and 6 women complete the job in 8 days: 4m 6w 1/8 3 men and 7 women complete the job in 10 days: 3m 7w 1/10

Step 2: Solving the System of Equations

Multiplying the first equation by 5 and the second by 4, we get:

20m 30w 5/8 12m 28w 1/2.5 2/5

By subtracting the second equation from the first, we eliminate the fraction and solve for w:

8m 1/5 (after common denominator and simplification) w 1/400

Step 3: Finding the Work Rate of Men

Substituting w 1/400 back into the first equation:

4m 6(1/400) 1/8 4m 3/200 1/8 4m (25/200) - (3/200) 22/200 m 11/400

Step 4: Determining the Total Work Rate

The total work rate for 10 women and 8 men is calculated as:

Work rate of 10 women: 10w 10 * 1/400 1/40 Work rate of 8 men: 8m 8 * 11/400 88/400 11/50 Total work rate 1/40 11/50 (common denominator 200) Total work rate 5/200 44/200 49/200

Step 5: Calculating the Time to Complete the Job

The time required to complete 1 job is the reciprocal of the total work rate:

T 1 / (49/200) 200/49 ≈ 4.08 days

Conclusion

Thus, the time it will take for 10 women and 8 men to complete the job is approximately 4.08 days.

In summary, this problem demonstrates how to solve complex work rate problems by setting up and solving a system of linear equations. Understanding these techniques is crucial for accurate project and resource allocation in various fields.