Problem 78 - Algebra A Combined Function

Question

Solve. A woman received a statement of her charge account at Old Navy. She spent \(\$ 93\) on purchases last month. She returned an \(\$ 18\) top because she didn't like the color. She also returned a \(\$ 26\) nightshirt because it was damaged. What does she actually owe on her account?

Step-by-Step Solution

Answer

The woman owes $49 on her account.

Step 1: Calculate Total Purchases

The woman initially charged $93 to her account. This amount reflects the total purchases made during the month.

Step 2: Calculate Returns

The woman returned two items: an \(18 top and a \)26 nightshirt. We need to calculate the total amount of these returns: \[ 18 + 26 = 44 \]

Step 3: Subtract Returns from Total Purchases

Subtract the total returns from the initial amount spent to find out what the woman actually owes: \[ 93 - 44 = 49 \]

Step 4: Final Step: State the Final Amount Owed

After accounting for the returns, the woman owes $49 on her charge account.

Key Concepts

Understanding Addition in Basic Arithmetic

Addition is a mathematical operation that combines two or more quantities into a single sum. It is one of the four basic operations of arithmetic alongside subtraction, multiplication, and division. In our everyday lives, addition is used for various purposes, such as in calculating total costs, determining distances, or simply keeping track of numbers.

• Addition is often described as "putting together" or "combining" numbers.
• When adding numbers, each number is called an 'addend', and the result is known as the 'sum'.
• Addition is both associative and commutative, meaning that the order of numbers doesn't affect the sum. In mathematical terms, this is such that \(a + b = b + a\) and \((a + b) + c = a + (b + c)\).

In the given problem, addition is used to find the total of amounts returned to Old Navy.When the woman returned a top and a nightshirt, we used addition to find the sum total of these items. The calculation, where we added \( 18+ 26 = 44 \), is a straightforward application of this principle.

The Role of Subtraction in Resolving Financial Balances

Subtraction is another fundamental arithmetic operation that involves taking one quantity away from another. It is crucial for resolving balances, comparing amounts, and determining differences. Subtraction asks, "How much is left?" or "What remains?"

• In subtraction, the number from which others are taken is called the 'minuend', the number that is subtracted is the 'subtrahend', and the result is called the 'difference'.
• Unlike addition, subtraction isn't commutative; changing the order of numbers will affect the outcome. For example, \(5 - 3 = 2\) differs from \(3 - 5 = -2\).

This concept directly applies to the problem at hand where the woman's returns reduce the amount she originally owed.After adding up the total returns from her purchases, subtraction is used to find out her final balance.By subtracting the total returns \( \)44\( \) from her original purchases \( \)93\( \),we calculate what she rightfully owes; resulting in \( \)49\( \).

Effective Problem-Solving with Arithmetic Operations

Problem-solving in arithmetic involves the application of mathematical concepts to resolve real-world scenarios. It encourages logical thinking and is fundamental to understanding mathematics. The core of problem-solving lies in identifying the problem, devising a plan, and then carrying out the arithmetic operations to reach a solution. Here are some key points:

• Firstly, clearly understand the problem. Recognize what is known and what needs to be found.
• Break down the problem into manageable steps. Determine which arithmetic operations are necessary.
• Calculate carefully and review each step to ensure accuracy.
• Finally, interpret the result in the context of the problem to confirm that it satisfies the initial query.

In the given exercise, the problem-solving process involved:1. Reading the charge account issue.2. Using addition to calculate total returns.3. Applying subtraction to determine the correct amount owed after returns were processed.Thus, problem-solving with addition and subtraction led to accurately assessing the woman's account balance and concluding that she owes \( \)49\( \). This approach underscores the importance of strategy and operations in tackling everyday math challenges.