Introduction

Understanding how to manipulate numbers based on given sums and differences can be a valuable tool in mathematics. This article will explore the concept of finding two numbers whose sum and difference have specific values, using both long and short methods to solve such problems.

Identifying Number Pairs with Specific Sum and Difference

Consider two numbers with a sum of 12 and a difference of 7. This combination presents a unique challenge because it's impossible to find two numbers that fit both criteria. However, the process of solving such a problem provides insight into the relationship between numbers and their properties.

Let's start with the equations:

Sum Equation: x y 12 Difference Equation: x - y 7

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

x y 7

Substitute this into the sum equation:

(y 7) y 12

Combine like terms:

2y 7 12

Solve for y:

2y 5

y 5/2

Now, substitute y back into the difference equation:

x 5/2 7

x 19/2

Exploring General Properties

There are some general properties to note:

If the sum of two numbers is 12, they are either both even or both odd. If the difference between two numbers is 7, they are one even and one odd. There are no two numbers that will both add to 12 and subtract to 7.

Example: Finding Number Pairs with Given Sum and Difference

Now, let's consider a concrete example where the sum of two numbers is 12 and their difference is 8. Using the equations:

x y 12 x - y 8

Solve for x in terms of y:

x y 8

Substitute this into the sum equation:

(y 8) y 12

Solve for y:

2y 8 12

2y 4

y 2

Now, substitute y back into the difference equation:

x 2 8

x 10

Therefore, the two numbers are 10 and 2.

Alternative Solving Methods

For enthusiasts or students who prefer visual learning, here is a step-by-step visual method to solve the problem:

Draw a long bar representing Value A. Draw a shorter bar representing Value B. Write the sum (12) at the end of the combined bar. Write the difference (8) at the end of the bar representing the extra value. Calculate the remainder (12 - 8 4). Divide the remainder by 2 to find the equal value (4 / 2 2). Subtract the equal value from 8 to find Value A (8 - 2 10). Therefore, the two numbers are 10 and 2.

Conclusion

Understanding how to solve for number pairs based on given sums and differences is a fundamental skill in mathematics. The methods covered in this article provide both theoretical and practical approaches to solving such problems, enhancing problem-solving skills and deepening mathematical understanding.