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Chapter 1: Problem 33

Jeff knows that his neighbor Sarah paid \(\$ 40,230,\) including sales tax, for a new Buick Park Avenue. If the sales tax rate is \(8 \%\), then what is the cost of the car before the tax?

Short Answer

Expert verified
The cost of the car before tax is \( 37,250 \).

Step by step solution

01

Understand the problem

Jeff needs to find out the cost of the car before the sales tax was added. He knows the total amount paid and the sales tax rate.
02

Identify the variables

Let the cost of the car before tax be denoted as \( C \). The given total cost including tax is \( 40,230 \) and the sales tax rate is \( 8\text{ %} \).
03

Set up the equation

Since the total cost is the cost of the car plus the sales tax, the relationship can be written as: \[ C + 0.08C = 40,230. \]
04

Combine like terms

Combine the terms involving \( C \) on the left side of the equation: \[ 1.08C = 40,230 \]
05

Solve for \( C \)

To find \( C \), divide both sides of the equation by \( 1.08 \): \[ C = \frac{40,230}{1.08} \]
06

Perform the division

Calculate \( \frac{40,230}{1.08} = 37,250 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sales Tax Calculation
When you buy a product, sales tax is an additional percentage of the product's price. This tax is calculated based on the item's total value. To find the sales tax alone, you take a percentage of the initial cost.
For example, with a sales tax rate of 8%, you would multiply the item's price by 0.08 to determine the tax.

Understanding this helps you see how the total price inflates due to tax.
In our exercise, this knowledge is used to determine the original cost of the car before tax was applied. Knowing the final price including tax allows us to work backward using algebraic methods.
Algebraic Equations
Algebra is essential for solving real-life problems like determining pre-tax prices.
In our exercise, we used the following equation to represent the problem:
\[ C + 0.08C = 40,230 \]
Here, \(C\) represents the car's original price. We know that the total price, which includes the tax (8%), equals $40,230.
By combining like terms, we simplify the equation: \[1.08C = 40,230 \]

This step is crucial because it consolidates all the variable terms into one, making it easier to solve. Finally, to isolate \(C\), divide both sides by 1.08:
\[ C = \frac{40,230}{1.08} \]

Performing the division gives us the original price of the car before tax.
Problem-Solving Steps
When tackling word problems, it's important to follow specific steps. This approach ensures clarity and accuracy:
  • Understand the Problem

    • Read the problem carefully. Identify what is given and what needs to be found.

\(\)
  • Identify the Variables

    • Determine what each variable represents in the problem. For example, \(C \) is the car's cost before tax.


  • Set Up the Equation

    • Translate the problem into a mathematical equation. In this case: \(C + 0.08C = 40,230\).


  • Combine Like Terms

    • Simplify the equation by combining terms with the same variable.


  • Solve for the Variable

    • Isolate the variable by performing algebraic operations. Here, divide both sides by 1.08.


  • Perform the Calculations

    • Carry out the necessary computations to find the value of the variable.


By following these precise steps, you can systematically solve similar word problems effortlessly.

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Most popular questions from this chapter

Equal Slopes Show that if \(y=m x+b_{1}\) and \(y=m x+b_{2}\) are equations of lines with equal slopes, but \(b_{1} \neq b_{2}\), then they have no point in common. Assume that they have a point in common and see that this assumption leads to a contradiction.

Cost Accounting An accountant has been told to distribute a bonus to the employees that is \(15 \%\) of the company's net income. since the bonus is an expense to the accountant, it must be subtracted from the income to determine the net income. If the company has an income of \(\$ 140,000\) before the bonus, then the accountant must solve $$ B=0.15(140,000-B) $$ to find the bonus \(B\), Find \(B\).

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