Problem 27 - Fundamentals of Mathematics

Question

For the following problems, show that the pairs of quantities yield the same sum. $$(114+226)+108 \text { and } 114+(226+108)$$

Step-by-Step Solution

Answer

Both expressions evaluate to 448, showing they yield the same sum.

Step 1: Write the Original Expressions

The exercise provides two expressions:1. \((114+226)+108 \)2. \(114+(226+108)\)

Step 2: Evaluate the First Expression

Let's solve the first expression: \((114 + 226) + 108\)First, calculate the sum inside the parentheses: \(114 + 226 = 340\)Now, add 108 to the result: \(340 + 108 = 448\)

Step 3: Evaluate the Second Expression

Now, solve the second expression:\(114 + (226 + 108)\)First, calculate the sum inside the parentheses:\(226 + 108 = 334\)Now, add 114 to the result:\(114 + 334 = 448\)

Step 4: Verify Equality

Both expressions have been evaluated:1. \((114+226)+108 = 448\)2. \(114+(226+108) = 448\)Since both expressions result in the same sum, we can confirm that the calculations are correct.

Key Concepts

Understanding Mathematical Proofs

When solving mathematical problems, proofs play a crucial role. They are logical arguments that help us verify the truth of a statement. Let's dive into why proofs matter:

• They build a strong foundation to demonstrate that what we're solving adheres to mathematical principles.
• In this exercise, we use proofs to confirm that two different-looking expressions actually deliver the same result, emphasizing the idea of the **Associative Property of Addition**.

The exercise demonstrates a straightforward proof involving addition, showing that the two sums are equal by solving both expressions independently and confirming they have the same result. This step-by-step validation provides a deeper understanding of how mathematical identities are consistent across different scenarios.

Addition as a Fundamental Operation

Addition is one of the basic arithmetic operations you encounter early in learning mathematics. It's the process of combining two or more quantities to find a total amount. Here’s why it’s fundamental:

• It is the foundation for more complex operations like subtraction, multiplication, and division.
• Addition is involved in everyday calculations, such as counting objects or combining numerical values.

A key point in understanding addition is knowing its properties, such as the associative property, which tells us that numbers can be added in any order, and the sum will be the same. This property supports the exercise where both expressions \((114+226)+108\) and \(114+(226+108)\) yielded identical results, regardless of how the numbers were grouped.

Exploring Arithmetic Operations

Arithmetic operations are the basic building blocks of mathematics. They include addition, subtraction, multiplication, and division. Each operation has unique properties and rules:

• **Addition**: Combining values to get a total. Associative and commutative properties are fundamental here.
• **Subtraction**: Removing a quantity from another; it’s essentially the inverse of addition.
• **Multiplication**: Repeated addition of a number; possesses similar properties like associativity and commutativity.
• **Division**: Splitting a number into equal parts; considered the inverse of multiplication.

These operations are crucial for solving mathematical problems and developing problem-solving skills. They are interconnected and often used together to tackle complex calculations. Understanding their properties simplifies tasks, such as the exercise at hand, which focuses on how arithmetic operations hold true under the associative property.