Problem 42 - Elementary Algebra

Question

Applications involving variation. The sales tax on the purchase of a new car varies directly as the price of the car. If an \(\$ 18,000\) new car is purchased, then the sales tax is \(\$ 1,350\). How much sales tax is charged if the new car is priced at \(\$ 22,000 ?\)

Step-by-Step Solution

Answer

The sales tax on a \(22,000\) car is \(1,650\).

Step 1: Understanding Direct Variation

Direct variation means when one quantity increases, the other quantity increases at a constant rate. Here, the sales tax varies directly with the price of the car.

Step 2: Formulating the Variation Equation

Since the sales tax varies directly with the price, we can express this relationship as: \( T = kP \) where \( T \) is the sales tax, \( P \) is the price of the car, and \( k \) is the constant of variation.

Step 3: Finding the Constant of Variation

Substitute the known values into the variation equation to find \( k \). With a car price of \( 18,000 \) and tax of \( 1,350 \), we have:\[ 1,350 = k \times 18,000 \]Solving for \( k \), \[ k = \frac{1,350}{18,000} = 0.075 \]

Step 4: Applying the Variation Equation

Using \( k = 0.075 \), determine the sales tax for a new car priced at \( 22,000 \) using the equation:\[ T = 0.075 \times 22,000 \]

Step 5: Calculating the Sales Tax

Calculate the value:\[ T = 1,650 \]

Key Concepts

Variation Equation

In mathematics, a variation equation is an expression that defines a relationship between two variables in which one variable is a constant multiple of the other. Direct variation occurs when one variable increases, the other also increases proportionally. This concept is usually expressed in a simple equation of the form:

• \( T = kP \)

Here, \( T \) can represent the sales tax, \( P \) the price of the car, and \( k \) the constant of variation. This setup helps in easily predicting changes in one variable based on changes in the other. In our exercise, the sales tax "varies directly" as the car price, which means if the price of the car increases, so does the sales tax. This linear relationship makes it simpler for us to calculate the unknown values once we have the constant of variation in hand.

Constant of Variation

The constant of variation, denoted as \( k \), is a fixed number that relates two variables exhibiting a linear relationship, such as in direct variation. Once determined, \( k \) allows you to predict one variable based on the other. In this exercise, we already know that sales tax and car price are directly proportional.To find \( k \), use the equation that describes the variation, substituting known values for sales tax and price:

• \( 1,350 = k \times 18,000 \)

Solve for \( k \):

• \( k = \frac{1,350}{18,000} = 0.075 \)

Thus, the constant \( k = 0.075 \) tells us that the sales tax is \( 7.5\% \) of the car price. This relationship remains constant irrespective of the car price, as long as that percentage applies.

Sales Tax Calculation

Once you've determined the constant of variation, calculating the sales tax for any other car price becomes straightforward. This involves substituting the new car price into the variation equation. For example, if the new car price is \( \\(22,000 \), use the previously calculated \( k \) value:

• \( T = 0.075 \times 22,000 \)

Carrying out this calculation:

• Calculate the product: \( T = 1,650 \)

Thus, the sales tax for a car priced at \( \\)22,000 \) amounts to \( \$1,650 \). This approach underscores the importance of understanding direct variation and its application in real-world scenarios such as calculating sales tax, where we can see a direct relationship between price and tax.