Introduction

Positive integers form the backbone of many mathematical problems, whether they are used in basic arithmetic, accounting, or more complex mathematical applications. A particular class of these problems involves finding positive integers based on their difference and ratio. This article explores how to solve such problems with a step-by-step approach, using an example to illustrate the process.

Understanding the Problem

Consider the following scenario: two positive integers have a difference of 30 and their ratio is 2:5. The task is to find these integers and determine their total sum. Let's delve into the solution step by step.

Step 1: Identify Variables

Let's denote the two positive integers as x and y (where x y).

Step 2: Formulate Equations

From the problem, we can form two key equations:

The difference between the two integers is 30: The ratio of the two integers is 2:5.

Mathematically, these can be represented as:

y - x 30 x:y 2:5, which can be expressed as 2y 5x.

Step 3: Solve the Equations

From the second equation, we can express y in terms of x and then substitute it into the first equation:

From 2y 5x, we can derive y 5/2x. Substitute y 5/2x into the first equation y - x 30 to get:

5/2x - x 30

Simplifying this, we get:

(5x - 2x)/2 30

3x/2 30

3x 60

x 20

Now that we have x 20, we can find y using the ratio equation y 5/2x:

y 5/2 × 20 50

Step 4: Find the Total Sum

The final step is to find the total sum of these integers:

x y 20 50 70

Conclusion

In conclusion, two positive integers, with a difference of 30 and a ratio of 2:5, are 20 and 50, respectively. Their total sum is 70. This method can be applied to solve similar problems involving positive integers, their differences, and ratios.

Further Reading and Resources

For a deeper understanding of problem-solving techniques in mathematics and practical applications, you can explore additional resources such as algebra textbooks or online tutorials on solving equations and ratios.