Solving Work Problems with Mathematical Logic: A Laborer’s Task

Mathematical logic is a powerful tool that can be used to solve real-world problems, such as determining the original number of laborers required for a task based on changes in workforce and completion time. This article delves into such a scenario, using algebraic methods to find the solution.

Introduction to the Laborer Problem

A group of laborers undertake a piece of work that they initially promise to complete in 10 days. However, due to unforeseen circumstances, five of them become absent. Despite the decreased number of workers, the remaining laborers complete the task in 12 days. The question is, what is the original number of laborers in the group?

Determining the Original Number of Laborers

To solve this problem, we can use the concept of labor-days, which is the product of the number of laborers and the number of days they work. Let’s denote the original number of laborers in the group as (N).

Setting Up the Equation

The total amount of work can be expressed as:

(text{Total Work} N times 10 text{ (in labor-days)})

When five laborers are absent, the number of laborers left is (N - 5). These remaining laborers complete the work in 12 days, so the total work can also be expressed as:

(text{Total Work} (N - 5) times 12)

Since both expressions represent the same total work, we can set them equal to each other:

(N times 10 (N - 5) times 12)

Solving the Equation

Let’s solve for (N):

(N times 10 12N - 60)

Rearranging gives:

(10N - 12N -60)

Simplifying this leads to:

(-2N -60)

Dividing both sides by -2 gives:

(N 30)

Thus, the original number of laborers in the group is 30.

Verification of the Solution

To ensure the solution is correct, let’s verify it by checking the work content:

If there were 30 laborers and the work was completed in 10 days, the total work content is: 30 times 10 300 man-days. If 25 laborers completed the work in 12 days, the total work content is: 25 times 12 300 man-days.

The man-days tally, confirming that the original number of laborers is indeed 30.

Additional Solution Methods

Here are alternative methods to solve the same problem, showcasing different algebraic setups:

Method 1: Simplified Algebraic Approach

Let there be (x) number of workers in the group. The work content (x times 10 1) The remaining workers are (x - 5) and they work for 12 days to complete the same work, thus: (frac{1}{12} x - 5) Solving for (x): (1 12(x - 5) ) (1 12x - 60) (2x 60) (x 30)

Method 2: Direct Work Content Calculation

Let the original number of laborers be (N). The work content is 10N. With 5 workers absent, the remaining laborers complete the work in 12 days, so the work content is: (12(N - 5)) Solving for (N): (10N 12(N - 5) ) (10N 12N - 60) (2N 60) (N 30)

Conclusion

Through various algebraic methods, we have consistently arrived at the conclusion that the original number of laborers in the group is 30. This approach not only verifies the initial solution but also illustrates the power of using mathematical logic to solve real-world problems.

By understanding and applying these methods, you can tackle similar work problems efficiently and accurately. Whether you are in construction, manufacturing, or any field where labor management is crucial, mastering these concepts will be invaluable.