Problem 15 - Prealgebra

Question

Clara purchased a pair of jeans for \(\$ 26,\) a T-shirt for \(\$ 12,\) and a pair of socks for \(\$ 4 .\) What is the total cost of the items without tax? Explain how the Commutative Property of Addition can be used to find the total mentally.

Step-by-Step Solution

Answer

The total cost of the items is \( \$42 \). The Commutative Property allows us to rearrange and mentally simplify the calculation.

Step 1: Identify the Costs

List out the cost of each item: the jeans cost \( \\(26 \), the T-shirt costs \( \\)12 \), and the socks cost \( \$4 \).

Step 2: Apply the Commutative Property of Addition

The Commutative Property of Addition states that numbers can be added in any order. Use this property to rearrange the sum: \( \\(26 + \\)12 + \\(4 = \\)4 + \\(26 + \\)12 \).

Step 3: Calculate the Rearranged Sum

First, add \( \\(4 + \\)26 \) to get \( \\(30 \). Then, add \( \\)30 + \\(12 \) to get \( \\)42 \).

Step 4: Conclude the Calculation

By rearranging the sum as \( \\(4 + \\)26 + \\(12 \), you have calculated the total cost more easily and found it to be \( \\)42 \).

Key Concepts

Addition

When calculating the total cost of items, addition is the fundamental operation to use. Adding numbers essentially means finding the combined value of two or more numbers. In our example, Clara wants to determine the total cost of her shopping. She has purchased jeans for \(26, a T-shirt for \)12, and socks for $4. To find the total cost, you perform the addition of these three numbers: - Add the cost of jeans (\(26\))- Add the cost of the T-shirt (\(12\))- Add the cost of the socks (\(4\))This looks like: \[ 26 + 12 + 4 \] The Commutative Property of Addition allows you to add this in any order. This property makes solving problems easier, especially when using mental math techniques.

Mental Math

Mental math is a valuable skill that helps you perform calculations in your head quickly and accurately. It is especially useful when you don’t have a calculator handy and want to solve a problem efficiently. To use mental math effectively, you can apply strategies like reordering numbers, which is where the Commutative Property of Addition comes into play. In Clara's case, instead of adding the numbers in the order given, rearranging them might make the mental calculation simpler. Doing - \(4 + 26 = 30\)- Then adding the remaining \(12\) gives: \(30 + 12 = 42\) Arranging numbers to add first those that make round numbers (like 30) simplifies mental calculations. Thus, the total cost is found to be 42 quickly without needing pen and paper.

Cost Calculation

Cost calculation involves adding up expenses to find out the total amount spent. It's a necessary skill for budgeting, shopping, and financial planning. In the given exercise, Clara is calculating the total cost of her purchases. This includes: - Jeans for $26 - A T-shirt for $12 - Socks for $4 By totaling these amounts, you determine the entire expense, which helps Clara understand her budget. The step-by-step solution shows how using the Commutative Property can simplify this process by allowing her to add numbers in a convenient order for quicker calculation. Such skills ensure you're always aware of exactly how much you spend, helping avoid overspending. Calculating costs effectively is a responsibility that ensures smart and informed financial choices every day.