Solving Work-Rate Problems: The Case of Men and Women Completing a Job

Work-rate problems are common in mathematics and related fields, often requiring a detailed understanding of how different individuals, teams, or machines can complete a task. In this article, we explore a problem that examines the productivity of men and women working together. Specifically, we'll dive into how 9 men and 15 women can complete a work in 16 days, and then determine how long 3 men and 7 women would take to complete the same job.

Understanding the Problem

First, let's define the work being done by each individual. We'll use the following variables:

tM - Work done by 1 man in 1 day tW - Work done by 1 woman in 1 day

Given that 9 men and 15 women can complete the work in 16 days, we need to determine how much work they do together in one day and then extend this to 3 men and 7 women.

Step-by-Step Solution

Step 1: Calculate the total work done by 9 men and 15 women in 16 days.

The total work done by 9 men and 15 women in one day is:

Work per day 9M 15W

Over 16 days, the total work done can be expressed as:

Total Work 16 × (9M 15W) 144M 240W

Step 2: Calculate the work done by 3 men and 7 women in one day.

The total work done by 3 men and 7 women in one day is:

Work per day 3M 7W

Step 3: Compare the Work Capacities

To find out how many days it takes for 3 men and 7 women to complete the same amount of work, we set up the following ratio:

Ratio frac{3M 7W}{9M 15W}

Let's express W in terms of M. Assume that W frac{1}{2}M. This common assumption simplifies our calculations.

The ratio then becomes:

frac{3M 7(frac{1}{2}M)}{9M 15(frac{1}{2}M)} frac{3M 3.5M}{9M 7.5M} frac{6.5M}{16.5M} frac{13}{33}

Step 4: Calculate the Total Time Taken

Using the total work calculated before, we find the number of days taken by 3 men and 7 women to complete the work:

D × (3M 7W) 144M 240W

Substituting W frac{1}{2}M gives:

D × (3M 7 times frac{1}{2}M) 144M 240 times frac{1}{2}M

D × (3M 3.5M) 144M 120M

D × 6.5M 264M

Dividing both sides by M (assuming M ≠ 0):

D × 6.5 264

D frac{264}{6.5} approx 40.615

For practical purposes, we can round D to 41 days.

Conclusion: It takes approximately 41 days for 3 men and 7 women to complete the job.

Key Takeaways:

tUnderstanding work rates is crucial for determining the time required for different teams or groups to complete a task. tThe key to solving work-rate problems is to express all work in the same unit (days, hours, etc.) and then set up the appropriate ratios. tMaking common assumptions, such as W frac{1}{2}M, can simplify the solution process.

Keyword Strategies and Related Terms

Keyword 1: Work-Rate Problems - These problems involve determining how long it takes to complete a task given the working rate of individuals or groups.

Keyword 2: Labor Productivity - This term is often associated with the efficiency of labor in completing tasks.

Keyword 3: Time and Work Problems - This includes a variety of problems related to determining the time required for a task based on different work rates.

Keyword 4: Job Completion Time - This term is useful for emphasizing the time aspect of work-rate problems.

Conclusion: By understanding and solving work-rate problems, you can better manage and optimize labor resources and project timelines.