Solving Work Rate Problems: A Comprehensive Guide

In this article, we will delve into a classic problem of work rate, solving it step-by-step and breaking down the concepts involved. This approach is essential for understanding how to manage manpower and time efficiently, a crucial skill for managers and team leaders in various industries.

Introduction to Work Rate Problems

Work rate problems often involve determining the time it takes for a certain number of workers to complete a task. These problems are common in various fields such as construction, manufacturing, and project management. The core of these problems is understanding the relationship between the number of workers, the time spent, and the amount of work accomplished.

Problem Statement

The problem involves 4 women and 6 men completing a work in 8 days, while 3 women and 7 men complete the same work in 10 days. We need to determine the number of days 10 men will take to complete the work.

Step-by-Step Solution

Let's denote the work rate of one woman as (w) and the work rate of one man as (m). The work undertaken by a group of workers is the product of the number of workers and their work rate.

Step 1: Define Work Rates

Let:

(w): the amount of work done by one woman in one day. (m): the amount of work done by one man in one day.

Step 2: Set Up Equations

Using the given information:

4 women and 6 men can complete the work in 8 days: 3 women and 7 men can complete the work in 10 days:

This gives us two equations:

left(4w 6mright) times 8 1 left(3w 7mright) times 10 1

These can be simplified to:

4w 6m frac{1}{8} 3w 7m frac{1}{10}

Step 3: Solve the Equations

Multiply the first equation by 5 and the second by 4:

20w 30m frac{5}{8} 12w 28m frac{2}{5}

Multiply the first equation by 3 to make the coefficients of (w) equal:

60w 90m frac{15}{8}

Multiply the second equation by 5:

60w 140m 2

Subtract the first modified equation from the second:

50m 2 - frac{15}{8}

Convert 2 to eighths:

50m frac{16}{8} - frac{15}{8} 50m frac{1}{8} m frac{1}{400}

Substitute (m) back to find (w)

Substitute (m) into the first simplified equation:

4w 6 cdot frac{1}{400} frac{1}{8} 4w frac{6}{400} frac{1}{8} 4w frac{3}{200} frac{1}{8} frac{1}{8} frac{25}{200} 4w frac{25}{200} - frac{3}{200} frac{22}{200} frac{11}{100} w frac{11}{400}

Calculate Work Done by 10 Men

The work rate of 10 men is:

10m 10 cdot frac{1}{400} frac{10}{400} frac{1}{40}

Therefore, the time (T) taken by 10 men to complete the work is:

T frac{1}{frac{1}{40}} 40 text{ days}

Conclusion

10 men will complete the work in 40 days. This solution demonstrates the importance of breaking down complex problems into manageable parts and using algebraic methods to find the desired solution.

By following these steps, you can efficiently solve a wide range of work rate problems, which is a valuable skill in project management and industry settings.