Problem 28 - Fundamentals of Mathematics

Question

For the following problems, show that the pairs of quantities yield the same sum. $$(731+256)+171 \text { and } 731+(256+171)$$

Step-by-Step Solution

Answer

The sums are the same due to the associative property of addition, both equaling 1158.

Step 1: Understand the problem

The exercise is asking us to show that two expressions, \[(731 + 256) + 171\] and \[731 + (256 + 171)\], result in the same sum. This is a demonstration involving the associative property of addition.

Step 2: Apply the associative property to the first expression

The first expression is \[(731 + 256) + 171\]. Using the associative property of addition, which states that the way in which numbers are grouped does not change their sum, we can rearrange this expression as \[731 + (256 + 171)\]. This shows that the two expressions are equivalent by associative property, even without calculating their sum.

Step 3: Calculate the sum of each expression to verify

Although we have already shown they are equivalent through the associative property, let's calculate each separately to confirm.First, calculate \[(731 + 256) + 171\]: \[731 + 256 = 987\]. Then, \[987 + 171 = 1158\].Now, calculate \[731 + (256 + 171)\]: \[256 + 171 = 427\]. Then, \[731 + 427 = 1158\].Both sums are indeed 1158, confirming they yield the same result.

Key Concepts

Addition

Addition is one of the basic operations in mathematics. It involves combining two or more numbers to get a total or sum. In this exercise, we're adding three numbers: 731, 256, and 171. Although these numbers come together differently in the two expressions, the final sum should be the same. Why is that? Addition allows for flexibility in how numbers are grouped. This flexibility is essential when verifying sums, especially when dealing with larger or more complex numbers.

• Start by adding the first pair of numbers to form an intermediate sum.
• Next, add this result to the third number.

Both approaches, direct addition or grouping and adding, should deliver the same total sum: 1158.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, and others. In this context, our expressions both involve addition but are structured differently:

• Expression 1: \((731 + 256) + 171\)
• Expression 2: \(731 + (256 + 171)\)

Despite these differences in order, mathematical expressions can use properties such as the associative property to demonstrate equivalence. Understanding the structure of these expressions helps reveal how properties work in practice. Both expressions share the same core components, allowing the use of associative property to affirm their equivalence. This understanding simplifies problem-solving by highlighting that different expressions can lead to the same result.

Sum Verification

Sum verification is an important step when dealing with mathematical expressions. It confirms that two different ways of grouping numbers under addition give the same result. Once the expressions are rearranged, we calculate each sum to verify equivalence. While the associative property theoretically shows equivalence without computation, actual calculations ensure accuracy and understanding:

• Expression 1: Calculate \((731 + 256) + 171\)
• Result: 731 + 256 = 987, then 987 + 171 = 1158
• Expression 2: Calculate \(731 + (256 + 171)\)
• Result: 256 + 171 = 427, then 731 + 427 = 1158

Both sums match, affirming that the associative property holds true. The verification step supports our theoretical understanding with practical proof, reinforcing the method's reliability.