Solving Equations with Whole Numbers - A Practical Example

Mathematical equations can be challenging, but they also offer insight into how numbers work together in real-world scenarios. This article will walk through a specific problem where we need to divide 184 into two parts. The challenge is further complicated by the requirement that one-third of one part should exceed one-seventh of the other part by 8. Let's break this down step-by-step.

Problem Statement

The problem can be stated as follows:

Divide 184 into two parts such that one-third of one part may exceed one-seventh of the other part by 8.

Solving the Problem

Let's denote the two parts as x and y.

Step 1: Define the Equations

Based on the problem, we can define the following equations:

The sum of the two parts:

x y 184

One-third of one part exceeds one-seventh of the other part by 8:

x/3 - y/7 8

Step 2: Express y in terms of x

From the first equation, we can express y in terms of x as:

y 184 - x

Step 3: Substitute y into the second equation

Substituting y 184 - x into the second equation, we get:

x/3 - (184 - x)/7 8

Step 4: Expand and Simplify the Equation

Expanding and simplifying the equation, we get:

x/3 - 184/7 x/7 8

Combining the terms involving x:

(1/3 1/7)x - 184/7 8

Since 1/3 1/7 10/21, the equation becomes:

10/21x - 184/7 8

Step 5: Find a Common Denominator

The common denominator of 21 and 7 is 21. Thus:

10/21x - 184*3/7*3 8

Which simplifies to:

10/21x - 552/21 168/21

Step 6: Eliminate the Fraction

Multiplying the entire equation by 21 to eliminate the fraction:

1 - 552 168

Step 7: Solve for x

Adding 552 to both sides of the equation:

1 720

Dividing by 10:

x 72

Step 8: Find y

Substituting x 72 back into the equation for y:

y 184 - 72 112

Conclusion

The two parts are:

x 72 y 112

Thus, the two parts are 72 and 112.

About the Explorations

This example uses basic algebra and demonstrates how to solve real-world problems through step-by-step equations. The problem-solving process highlights the importance of defining the right equations, eliminating common denominators, and solving linear equations. Understanding these concepts is crucial for anyone dealing with mathematical problems in various fields, including economics, engineering, and daily life.

By practicing similar problems, you can improve your algebraic skills and confidence in handling more complex equations. If you have any questions or need further clarification, feel free to reach out or explore more related problems on this topic.