Problem 65 - Introductory and Intermediate Algebra for College Students 4th

Question

Suppose that the local sales tax rate is \(6 \%\) and you buy a car for \(\$ 16,800\) a. How much tax is due? b. What is the car's total cost?

Step-by-Step Solution

Answer

a. The tax due is \$1,008. b. The total cost of the car is \$17,808

Step 1: Calculate the tax due

To calculate the amount of tax, multiply the sales price of the car by the sales tax rate. Therefore, tax due can be calculated using the formula: Tax Due = Price of Car * (Sales Tax/100), which is Tax Due = $16,800 * (6/100).

Step 2: Calculate the total cost of the car

The total cost of the car is the sum of the original price of the car and the sales tax. Therefore the formula to calculate the total cost is: Total Cost = Price of Car + Tax Due. After calculating the tax due from step one, add that amount to the initial price of the car to get the total cost.

Step 3: Evaluation

Evaluate the expressions in step 1 and 2 to get the actual numeric values for tax due and total cost respectively.

Key Concepts

Mathematical Expressions

Understanding mathematical expressions is fundamental when dealing with algebraic problems. In our given example, the expression to calculate tax due is \(Tax Due = Price of Car \times \frac{Sales\ Tax}{100}\). Here, the mathematical expression encompasses variables, numerical values, and operations. The price of the car is a specific value (\(\$16,800\)) while the sales tax rate is a percentage (\(6\%\)). In algebra, a percentage is expressed as a fraction of 100, making mathematical expressions like this one straightforward to evaluate. By converting the percentage into a decimal (\(0.06\)) and multiplying it by the price, we can find the amount of tax due.

Expressions simplify real-world transactions into algebraic terms, making it possible to perform calculations and predict outcomes. It's crucial to understand each component of an expression—as they represent real values and relationships—so that when you're challenged to insert different numbers or rates, you can confidently adjust the expression and solve for the unknowns.

Percentages in Algebra

Percentage problems are very common in algebra, especially in real-world applications such as calculating sales tax, discounts, or interest rates. In the context of our exercise, the percentage represents the sales tax rate. To work with percentages in algebra, we convert them into a more manageable form by dividing by 100. Therefore, a tax rate of \(6\%\) becomes \(0.06\) in our formula: \(Tax Due = Price of Car \times 0.06\).

One key point to remember is that percentages represent a part of a whole, with the whole being defined as 100%. This concept allows us to easily scale numbers up or down. For example, finding \(6\%\) of any number is the same as multiplying that number by \(\frac{6}{100}\) or \(0.06\). It's a simple, effective way to use algebra to calculate various parts of numerical quantities, and it's essential for students to be comfortable with this process to handle day-to-day calculations involving percentages.

Real-world Applications of Algebra

Algebra is not just a subject confined to the classrooms; it has numerous applications in everyday life. Our example of calculating sales tax on a car purchase is a perfect illustration. By using algebraic formulas, we can figure out financial implications of purchases, taxes, and other transactions. The total car cost formula \(Total Cost = Price of Car + Tax Due\) is a practical use of algebra that helps consumers understand how much they will be spending in total.

Whether it's budgeting for household expenses, analyzing business profits, or computing loan interest, algebra serves as a powerful tool to solve problems and make informed decisions. The beauty of algebra lies in its ability to turn real situations into equations and expressions that can be methodically solved. Grasping these algebraic concepts can empower students not just to excel academically but also to navigate complex financial landscapes with confidence.