Problem 109 - Intermediate Algebra

Question

As the cost of a purchase that is less than \(\$ 5\) increases, the amount of change received from a five-dollar bill decreases. Is this inverse variation? Explain.

Step-by-Step Solution

Answer

No, this is not inverse variation.

Step 1: Understand the Problem

We need to determine if the relationship between the cost of an item and the change received from a five-dollar bill exhibits inverse variation. Inverse variation implies that when one variable increases, the other decreases in a manner such that their product is constant.

Step 2: Define the Variables

Let's denote the cost of the purchase as \( x \) and the change received from a five-dollar bill as \( y \). The relationship can be expressed by the equation \( y = 5 - x \).

Step 3: Test for Inverse Variation

For inverse variation, the product of the two variables \( x \) and \( y \) should remain constant, i.e., \( x \cdot y = k \) where \( k \) is a constant. Checking this for \( x \) and \( y \):Substituting for \( y \), we get \( x \cdot (5 - x) = 5x - x^2 \). This expression does not simplify to a constant; hence, it does not fit the form of inverse variation.

Step 4: Conclude with the Analysis

Since the product \( 5x - x^2 \) is not constant, the relationship between cost \( x \) and change \( y \) is not an inverse variation. Instead, it is a direct subtraction from a constant value of 5.

Key Concepts

Cost and Change Relationship

In this exercise, we are exploring the relationship between the cost of a purchase and the change received from a five-dollar bill. When you buy an item that costs less than \(5, you receive change.
This relationship can be expressed as the equation:

• \( y = 5 - x \)

Here, \( x \) represents the cost of the purchase, and \( y \) is the change. As you can see, this relationship is not direct proportionality, because for an increase in \( x \), there is a decrease in \( y \). Importantly, this decrease doesn't affect overall value in a product form, and it's an instance of a linear relationship where you directly subtract the cost from a fixed constant—\)5 in this case.
Understanding this cost and change relationship is crucial because it shows how spending a part of a fixed sum reduces the remainder or the change you receive.

Direct Subtraction

In this problem, the cost of the purchase is directly subtracted from a fixed amount, which is a five-dollar bill.
This subtraction is straightforward:

• The equation for the change is \( y = 5 - x \).

This tells us that each additional dollar spent on the purchase directly reduces the change you get in an equal amount. Simply put, if you spend \( x \) dollars, you subtract \( x \) from 5 to find \( y \), the change. It's a clear case of how subtraction works in an everyday scenario, emphasizing that the more you spend, the less you get back. Hence, the relationship shown here is one of direct subtraction. It is essential to identify when such straightforward arithmetic operations are at play in practical contexts, rather than assuming more complex relationships such as inverse variation.

Constant Value Analysis

To test if a relationship is an inverse variation, we need to see if the product of the two variables is a constant.
Let's apply this check to our scenario:

• We substitute \( y = 5 - x \) into \( x \cdot y = k \), resulting in \( x(5 - x) = 5x - x^2 \).

This expression does not result in a constant because it changes depending on the value of \( x \). An inverse variation relationship means that multiplying two variables results in the same number, \( k \), for any combination of those variables. In our problem, as \( x \) increases, the expression \( 5x - x^2 \) changes, proving that it is not a constant. Constant value analysis helps confirm that the relationship doesn't qualify as an inverse variation, but instead is characterized by the simple deduction of a varying value from a fixed sum.