Solving Work Rate Problems: A Comprehensive Guide

When dealing with work rate problems in mathematics, such as the one where a certain number of men can cut a certain number of trees in a given time, it is essential to understand how to approach and solve these problems. These types of problems are fundamental in fields like engineering, construction, and human resource management. This guide will walk through a detailed, step-by-step solution to the problem: 'If 20 men can cut 40 trees in 5 hours and 4 men leave the job, how many trees will be cut in 12 hours.'

Understanding the Problem

The given problem involves a direct relationship between the number of men, the number of trees, and the time taken. The key is to maintain this relationship while adjusting for the changes in the number of people and time.

Solving the Problem

Let's break down the problem using different approaches.

Approach 1

The first approach involves calculating the total man hours and then using this to determine the number of trees that can be cut in the new conditions.

Step 1: Calculate the total man hours for 60 trees.

40 men can cut 60 trees in 8 hours, which amounts to 40 * 8 320 man hours.

Step 2: Adjust for the reduced number of men.

When 4 men leave, 32 men are left. Using the total man hours, we can calculate how many trees 32 men can cut in 12 hours.

32 * 12 384 man hours.

Step 3: Calculate the number of trees for the adjusted man hours.

No. of trees 384 / 320 * 60 72 trees

Conclusion: In the new conditions, 32 men can cut 72 trees in 12 hours.

Approach 2

Another approach involves setting up equations to reflect the given conditions and solving them.

Step 1: Set up the equation.

(30 / (20 * 4)) (t / (16 * 6))

Step 2: Solve for t.

t (30 * 16 * 6) / (20 * 4)

t (6 * 16 * 6) / (4 * 4)

t 6 * 6

Conclusion: t 36 trees.

Approach 3

This method involves calculating the total work hours for the initial conditions and then adjusting for the new conditions.

Step 1: Calculate the total work hours.

Total work hours for 40 trees 20 * 5 100 hours. After 4 men leave, 16 men are left. The new work hours are 16 * 12 192 hours.

Step 2: Calculate the number of trees.

No. of trees 192 * 40 / 100 76.8 trees.

Conclusion: The adjusted conditions allow for cutting 76.8 trees, but since we can't cut a fraction of a tree, we round to 76 or 77 trees.

Approach 4

This approach simplifies the problem by using proportional reasoning.

Step 1: Set up the proportionality.

40 men … 8 hours … 60 trees

32 men … 12 hours … ? trees

Step 2: Solve using the cross-multiplication method.

32/40 * 12/8 * 60 72 trees

Conclusion: In 12 hours, 32 men can cut 72 trees.

Conclusion

The correct answer, through all these methods, is 72 trees. Understanding and applying these methods can help you solve similar work rate problems effectively.

Keywords: work rate, man hours, tree cutting, productivity