A Mathematical Approach: Understanding Tree Cutting Productivity

When dealing with problems related to labor productivity and time, mathematical reasoning can provide clear and concise solutions. One such intriguing question is: If 10 people can cut 60 trees in 8 hours, how many trees can 8 people cut in 12 hours? This article will break down the logic and methodology behind solving this problem using real-world application, ensuring that the solution is both accurate and easy to understand.

Determining the Rate of Work

To solve this problem, we start by determining the rate at which one individual can cut trees. Given that 10 people collectively cut 60 trees in 8 hours, we can calculate the rate as follows:

Step 1: Calculate the rate for 10 people

The rate of cutting trees per hour for 10 people can be found by dividing the total number of trees by the total number of hours. This can be expressed as:

[ text{Rate for 10 people} frac{60 text{ trees}}{8 text{ hours}} 7.5 text{ trees per hour} ]

Step 2: Calculate the rate for 1 person

Next, we need to determine the rate for one individual by dividing the rate of 10 people by the number of people. This can be expressed as:

[ text{Rate for 1 person} frac{7.5 text{ trees per hour}}{10} 0.75 text{ trees per hour} ]

Step 3: Calculate the rate for 8 people

With the rate for one person, we can now calculate the rate for 8 people by multiplying the individual rate by the number of people. This is as follows:

[ text{Rate for 8 people} 8 times 0.75 text{ trees per hour} 6 text{ trees per hour} ]

Step 4: Calculate the total number of trees cut

Finally, to find out the total number of trees cut by 8 people in 12 hours, we multiply the rate for 8 people by the total number of hours. This can be expressed as:

[ text{Total trees} 6 text{ trees per hour} times 12 text{ hours} 72 text{ trees} ]

Therefore, 8 people can cut 72 trees in 12 hours.

Alternative Methods and Further Considerations

There are alternative methods to solve this problem, such as using proportional reasoning. For instance, the following calculations demonstrate an alternative approach:

[ frac{60}{10 times 8} frac{x}{8 times 12} ]

Solving for ( x ), we find:

[ x 72 text{ trees} ]

Another approach can be visualized as follows:

[ 60 text{ trees} div (10 text{ people} times 8 text{ hours}) 1.33 text{ people-hours per tree} ]

[ 8 text{ people} times 12 text{ hours} 96 text{ people-hours} ]

[ text{Total trees} 96 text{ people-hours} div 1.33 text{ people-hours per tree} 72 text{ trees} ]

Environmental Implications and Ethical Considerations

While mathematical solutions to such problems are accurate, it’s important to consider the environmental and ethical implications of tree cutting. Do the people cutting the trees have the necessary permits and approvals from their local authority? Encouraging such activities without due considerations can lead to ecological harm and legal issues. It is crucial to ensure that any tree cutting activities are conducted responsibly, with proper authorization and adherence to ecologically sound practices.

Through a meticulous analysis, we can see that the number of trees that can be cut by 8 people in 12 hours is 72. However, this finding should be coupled with mindful consideration of the environmental and ethical responsibilities of tree cutting activities. This ensures a sustainable and responsible solution that preserves natural resources for future generations.