Solving Work Rate Problems: Men and Women’s Efficiency in Completed Work

When dealing with work rate problems involving a combination of different workers, understanding the individual work rates and how they interact is essential. This article will guide you through solving a specific problem using a step-by-step approach. By the end, you will be able to tackle similar problems with confidence.

Understanding the Problem Context

A common type of work rate problem involves determining how long it will take a given number of men and women to complete a work task based on their combined work rates. In this article, we will solve the following problem: If 6 men and 8 women can do a work in 20 days, how long can 3 men and 4 women do the same work?

Step-by-Step Solution

Step 1: Determine the Total Work Done

Let's denote the work done by 1 man in 1 day as M and the work done by 1 woman in 1 day as W. Given that 6 men and 8 women can complete the work in 20 days, we can express the total work T as:

T (6M 8W) * 20

Step 2: Calculate the Work Rate of 6 Men and 8 Women

This simplifies to:

T 120M 160W

Step 3: Determine the Work Rate of 3 Men and 4 Women

The work rate of 3 men and 4 women in one day can be expressed as:

3M 4W

Step 4: Relate the Two Work Rates

To find the time it takes for 3 men and 4 women to finish the work, we need to relate the work rates. We can express the work rate of 3 men and 4 women in terms of the total work T:

D T / (3M 4W)

Step 5: Find the Ratios of Work Rates

To find the time taken D by 3 men and 4 women to complete the same work T:

D (120M 160W) / (3M 4W)

Step 6: Calculate the Ratio of Work Rates

Since the work rate of 3 men and 4 women is half of the work rate of 6 men and 8 women:

3M 4W 1/2 (6M 8W)

Step 7: Substitute in the Total Work Equation

From the earlier equation, we know:

T 120M 160W

Substituting, we find:

3M 4W 1/2 (120M 160W) 60M 80W

Step 8: Calculate the Time Taken by 3 Men and 4 Women

Now to find the time D:

D T / (3M 4W) (120M 160W) / (60M 80W)

Step 9: Simplify the Equation

Now we can simplify:

D (120M 160W) / (60M 80W) 2

Conclusion

Therefore, the time taken by 3 men and 4 women to complete the work is 40 days.

Alternative Solutions

Job Requires 8 x 20 160 Women Days.

Men Work at a Rate of 8/6 or 4/3 or 1.333 Women Days Per Day.

Working Together Each Day 3 Women Days 6 x 1.333 Women Days Are Completed 11 Women Days.

160 / 11 14.54545 Days

Men Work at the Rate of 1/620 1/120 Job Per Person Per Day.

Women Work at the Rate of 1/820 1/160 Job Per Person Per Day.

Working Together Their Rates Add.

6 Men and 3 Women Will Work at the Rate of 6/120 3/180 3/60 1/60 4/60 1/15 Job Per Day.

It Will Take 15 Days to Do the Job.

Conclusion and Summary

Through this problem, we have demonstrated a systematic approach to solving work rate problems. By understanding the individual work rates and how they combine, you can solve similar problems efficiently. The key is to break down the problem into manageable steps and systematically apply the principles of work rate.

Keywords

Work rate, productivity, time and work problems