Solving Work Rate Problems: Men, Women, and Children

Introduction

When dealing with complex work rate problems, understanding the contributions of different individuals (such as men, women, and children) is crucial. In this article, we will delve into a specific problem where the work rates of men, women, and children are calculated and applied. This article aims to provide a comprehensive guide on solving such problems, ensuring clarity and ease of understanding for both beginners and advanced readers.

Problem Statement

A scenario is presented where 3 men and 5 women can complete a certain task in 8 days, while 2 men and 7 children can do the same task in 12 days. The question posed is: How many women can equivalently complete the same task in one day as 21 children?

Solution Approach

The solution will follow a step-by-step approach, defining variables, setting up equations, and solving for the desired work rates. The process will be meticulously detailed to ensure readers can follow and understand each step.

Step 1: Define Work Rates

Let m represent the work rate of one man, w represent the work rate of one woman, and c represent the work rate of one child in work units per day.

Step 2: Set Up Equations

Based on the problem statement:

3m 5w 1/8 (since 3 men and 5 women can complete the work in 8 days) 2m 7c 1/12 (since 2 men and 7 children can complete the work in 12 days) 21c 1 (since 21 children can complete the work in one day)

Step 3: Equate the Two Expressions for Work

Multiplying both sides of the first and second equations by the respective denominators to clear the fractions:

24m 40w 1
24m 84c 1

Subtracting the first equation from the second:

40w - 84c 0
w (84/40)c
w (21/10)c

Step 4: Find the Work Done by 21 Children

From the third equation, if 21 children can complete the work in one day, then the work done by 21 children in a day is 1 unit.

Step 5: Determine How Many Women Match This Work

Let x represent the number of women who can do as much work in a day as 21 children.

xw 21c

Substituting w (21/10)c:

x * (21/10) * c 21c

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

x * (21/10) 21

Multiplying both sides by 10/21:

x 10

Conclusion

In conclusion, 10 women can do as much work in a day as 21 children.

For a deeper understanding of these concepts, consider delving into additional similar problems and practice questions available online. This method can be applied to solve various complex situations involving work rates, making it a versatile skill to master.