Problem 43 - Introductory Algebra for College Students

Question

Including \(6 \%\) sales tax, a car sold for \(\$ 23,850 .\) Find the price of the car before the tax was added.

Step-by-Step Solution

Answer

The price of the car before the tax was added is \$22,500

Step 1: Understand the problem

The problem involves finding the initial price of the car before a 6% sales tax was added i.e. going from 106% of the price back to 100%.

Step 2: Set up an equation

Let 'x' be the price of the car before tax. After the tax of 6% is added to 'x', the price becomes 106% of 'x' or $23,850. Represent this as an equation: \(0.106x = 23850\)

Step 3: Solve the equation

To solve for 'x', divide both sides of the equation by 0.106: \(x = \frac{23850}{0.106}\)

Step 4: Calculate the result

Perform the division to get 'x' which will be the price of the car before tax was added.

Key Concepts

Percentage Problems

Percentage problems often involve calculations that show how quantities change with respect to a certain percentage. In this case, understanding that sales tax is an additional cost calculated as a percentage of the pre-tax price helps in solving the problem.
When dealing with percentage problems:

• Identify the whole, which is the original amount or 100%. In our problem, it's the car’s price before tax.
• Add the percentage or subtract it, depending on whether it's a markup or a markdown. Here, it's a 6% tax.
• Translate this into a problem statement: You are given 106% of the price and need to find 100%.

This understanding makes setting up equations based on percentages straightforward. Grasping how percentages operate means comprehending the effects of percentage changes. It lets you slice and dice different numerical problems, using percentages as a tool for scaling values.

Algebraic Equations

Algebraic equations are crucial in transforming word problems into solvable mathematical expressions. The key is identifying what you are solving for and how other pieces relate.
In this exercise, setting up an algebraic equation involves:

• Recognizing that you need an equation to express the relationship between the pre-tax and post-tax prices.
• Using 'x' to represent the unknown pre-tax price of the car.
• Stating that the post-tax price (\(23850\)) is 106% of the price (\(0.106x\)).

This boils down the problem into a clear mathematical expression. Equations allow us to encode complex relationships simply and to solve for our target values efficiently. They convert abstract information into a more familiar form, numbers, and operations, making them easier to handle mathematically.

Solving for Unknowns

The ultimate goal in many percentage and algebra problems is finding the unknown value. Here, to find the pre-tax price of the car, you need to solve for 'x' in the equation.

• Start with the equation derived from understanding the percentage relationship: \(0.106x = 23850\)
• Think of it as isolating 'x', where you divide both sides by \(0.106\) to solve for 'x': \(x = \frac{23850}{0.106}\)
• Carry out the arithmetic operation. This division yields the value of 'x', which is the original car price without the tax.

Solving for unknowns involves navigating backwards from something given to uncover the initial quantity. It involves simple arithmetic manipulations once the equation is set, making it a methodical path to answer or value retrieval. Mastery of this ensures you can tackle a wide variety of algebraic and percentage problems confidently.