Inverse Proportion in Work and Time: Solving the 8 People to 4 People Work Time Problem

When dealing with problems involving work and time, it's essential to understand the concept of inverse proportion. Essentially, if more people are working on a task, it takes less time to complete the work. Conversely, fewer people will take more time to complete the same task. This relationship can be quantified using a simple formula. Let's explore how we can apply this principle to a specific scenario.

Problem Statement

The question at hand is: If 8 people can do a work in 10 hours, how long will it take if this work is done by 4 people?

Solution Using Inverse Proportion

To solve this problem, we use the concept of inverse proportion, which states that the time taken to complete a task is inversely proportional to the number of people working on it. This means that:

Time1 × Number of People1 Time2 × Number of People2

Let's denote the given information:

Time1 10 hours (initial time)

Number of People1 8 (initial number of people)

Number of People2 4 (new number of people, which we need to find the time for)

Time2 ? (the time for 4 people to complete the work, which we need to find)

Plugging the given values into the formula:

10 hours × 8 people Time2 × 4 people

Now, we solve for Time2 by rearranging the equation:

Time2 (10 hours × 8 people) / 4 people

Time2 20 hours

Therefore, it will take 4 people 20 hours to complete the same work that 8 people can do in 10 hours.

Alternative Approaches to Understand the Same Problem

Approach 1: Work in Hours

Let's break it down further using the logic that 80 hours of work (8 people × 10 hours) must be completed by 4 people. Since the total work remains the same, the time calculation can be simplified as:

8 people × 10 hours 80 hours of work content

80 hours divided by 4 people:

20 hours of work for each person.

Approach 2: Man, Day, Hour, and Work Equations

Another way to solve this problem is by using the equation:

Man × Day × Hour / Work Constant

In the first case:

Man 8

H 10 hours

In the second case:

Man 4

Day and Work are not given and can be ignored for this calculation.

Substituting the values into the equation:

8 × 10 4 × H

Solving for H:

H 20 hours

Conclusion

In solving the work and time problem, we see the power of inverse proportion. By understanding and applying this relationship, we can efficiently calculate the time needed for different numbers of workers to complete the same task.

The key takeaway is that when the number of people working on a task decreases, the time to complete the work increases proportionally. This principle is widely applicable in fields such as project management, workforce planning, and productivity studies.

Keywords: inverse proportion, work time problem, manpower efficiency