Problem 116 - Intermediate Algebra

Question

Checking Accounts. To find the total dollar amount of the checks entered in the register below, we could add the check amounts in the order in which they are written: \(\mathrm{S} 39+\$ 75+\$ 34+\$ 25+\$ 111+\$ 16 .\) Write an expression with the amounts reordered and grouped in such a way that the addition is easier. Then find the sum. What properties of real numbers did you use? (TABLE CANNOT COPY)

Step-by-Step Solution

Answer

The sum is $300. The Commutative and Associative Properties were used.

Step 1: Identify the amounts

The amounts of checks to be added are \(39, 75, 34, 25, 111,\) and \(16\).

Step 2: Reorder the amounts

To make the addition easier, reorder the amounts by pairing numbers that sum up to a multiple of 10. Pair \((39, 111), (75, 25), (34, 16)\).

Step 3: Group the sums of each pair

Calculate the sum of each pair: \(39 + 111 = 150\), \(75 + 25 = 100\), and \(34 + 16 = 50\).

Step 4: Add the grouped sums

Add the sums of each group: \(150 + 100 + 50 = 300\).

Step 5: Identify properties used

This reordering and grouping use the Commutative Property (which allows numbers to be added in any order) and the Associative Property (which allows grouping of numbers to be changed).

Key Concepts

Associative Property

When working with addition, the associative property is your best friend for simplifying and rearranging expressions. This mathematical property states that changing the grouping of numbers does not affect the sum. Essentially, it reassures us that we can decide how to group numbers when adding them. For example, consider the expression \((a + b) + c\). According to the associative property, this is equivalent to \(a + (b + c)\). Notice how the parentheses change but the result remains the same.

The key idea is that the total sum will not change, which gives us flexibility to perform additions that are more convenient or intuitive. In the original exercise, we grouped numbers like \( (39 + 111) + (75 + 25) + (34 + 16)\). Adopting this strategy helps in computing numbers briskly and accurately.

• Associative property works with addition and multiplication.
• Helps in rewriting expressions for simplified calculations.
• Encourages grouping numbers for computational ease.

Real Numbers

Real numbers constitute a broad category of numbers used every day, including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. This includes any number you can think of, such as 5, -3, 1.25, and \( \pi \).

The versatile nature of real numbers allows us to perform various arithmetic operations comfortably. Real numbers can be represented on a number line, making them very relatable to real-world applications.

In the context of our exercise, the checks were recorded in dollar amounts, which are real numbers. Whether you’re adding whole numbers or decimals, the properties of real numbers—like the commutative and associative properties—remain consistent, enabling smooth calculations.

• Real numbers encompass all possible magnitudes from the smallest fraction to the largest integer.
• Arithmetic operations can be applied without restrictions.
• Useful for accurate representation of quantitative real-world measurements.

Addition Strategies

Addition can sometimes seem daunting, especially with larger or numerous numbers. However, there are simple strategies to make it easier. One effective approach involves pairing numbers that make mental arithmetic easier, usually focusing on sums ending in zeroes or whole tens. In our exercise, this strategy allowed us to simplify \(39 + 111\), \(75 + 25\), and \(34 + 16\), grouping them to sum to 150, 100, and 50, respectively.

This method stems from the properties of commutativity and associativity, enabling us to rearrange and regroup numbers with no change to the overall result. Using these properties, you can quickly spot numbers that, when combined, give effortlessly simple sums:

• Identify pairs of numbers that make round figures.
• Apply properties to rearrange for simpler calculations.
• Take advantage of mental math tricks for faster computation.

By using these strategies, addition becomes less cumbersome and more intuitive, allowing for faster calculations with fewer errors.