Solving Work Rate Problems: A Comprehensive Guide

Work rate problems are a common type of mathematical challenge that often appear in examinations and real-world scenarios. One classic example relates to determining the number of men required to complete a task within a given time frame. In this article, we will solve the problem of cutting trees and explore the underlying principles of work rate and proportional reasoning.

Understanding Work Rate Problems

Work rate problems involve determining how long a task will take if the amount of work or the number of workers changes. These problems often require you to apply proportional reasoning, especially when the relationship between different variables is linear.

Problem Statement

Consider the following problem: If five men can cut 23 trees in 5 hours, how many men would be needed to cut 40 trees in the same 5-hour period?

Proportional Reasoning in Action

To solve this problem, we can use proportional reasoning. Let's denote the number of men needed to cut 40 trees in 5 hours as 'm'. We can set up the proportion:

(frac{5 text{ men} cdot 5 text{ hours}}{23 text{ trees}} frac{m text{ men} cdot 5 text{ hours}}{40 text{ trees}})

Solving for 'm', we find:

(m frac{5 text{ men} cdot 5 text{ hours} cdot 40 text{ trees}}{5 text{ hours} cdot 23 text{ trees}})

(m frac{200}{23} text{ men} approx 8.698)

Since a fraction of a man is not practical, we need to round up to the nearest whole number. Therefore, we would need at least 9 men to cut 40 trees in 5 hours.

Alternative Approach: Direct Calculation

Alternatively, we can use direct calculation to determine the man-hours required to cut one tree. The man-hours required to cut one tree can be calculated as follows:

(frac{5 text{ men} cdot 5 text{ hours}}{23 text{ trees}} frac{25 text{ man-hours}}{23 text{ trees}})

If 23 trees require 25 man-hours, then 40 trees would require:

(40 text{ trees} cdot frac{25 text{ man-hours}}{23 text{ trees}} frac{1000}{23} text{ man-hours})

Dividing the total man-hours by the number of hours in the task:

(frac{frac{1000}{23} text{ man-hours}}{5 text{ hours}} frac{200}{23} text{ men} approx 8.698 text{ men})

Rounding up, we again need 9 men.

Strategic Team Sizing

For safety and efficiency, it is often advisable to divide the workforce into two teams. Therefore, we would engage two teams of five men each, totaling 10 men. This ensures that we can get the job done even if one team is unavailable or requires additional assistance.

Comparative Analysis: Man-Hours Per Tree

Further analysis shows that cutting 1 tree requires:

(frac{25 text{ man-hours}}{23 text{ trees}} approx frac{25}{23} text{ man-hours per tree})

Therefore, cutting 40 trees requires:

(40 text{ trees} times frac{25}{23} text{ man-hours per tree} approx 36.8 text{ man-hours})

To complete this in 5 hours, the number of men required is:

(frac{36.8 text{ man-hours}}{5 text{ hours}} approx 7.36 text{ men})

Given the need for additional safety margins, we would need around 8 men to cut 40 trees in 5 hours. However, this still results in more efficient use of time, as only 4.6 hours would be required.

By understanding these principles, you can effectively solve work rate problems and make informed decisions in various scenarios involving team dynamics and task allocation.