Question

A car dealership offers \(45 \%\) off all its cars during its annual sale. essica pays \(\$ 7,150\) for her discounted vehicle. What was the original rice of the car? Round your answer to the nearest dollar. A) \(\$ 3,933\) B) \(\$ 10,368\) C) \(\$ 13,000\) D) \(\$ 15,889\)

Short answer

C) \(\$ 13,000\)

Step-by-step solution

Understand the problem

We are given that Jessica pays $7,150 for her discounted car. The dealership offers a 45% discount. We need to find the original price (before discount) of the car.

Create an equation relating the original price and the discounted price

Let x represent the original price of the car. Since Jessica pays 45% less for her car, she pays 100% - 45% = 55% of the original price. We can write this relationship as an equation: \(0.55x = 7150\)

Solve for the original price

To find the original price, we need to solve for x. To do this, we can divide both sides of the equation by 0.55: \(x = \frac{7150}{0.55}\) Now, we can calculate the value of x: \(x = 13000\)

Round the answer and compare it to the given options

The original price of the car is $13,000. This value matches option C, so the correct answer is: C) \(\$ 13,000\)

Key concepts

Percentage Calculation

Understanding how to work with percentages is a fundamental skill in math that helps in a variety of applications, such as calculating discounts, interest rates, and much more. When you talk about percentage, you are referring to a part out of 100. To calculate a percentage of a number, you multiply the number by the percentage in its decimal form. For example, if you want to find 45% of a number, you convert 45% to a decimal by dividing by 100, which gives 0.45, and then multiply by the number. In Jessica's case, the car dealership offered a 45% discount. This means she paid only 55% of the original price (100% - 45%). The discounted price she paid was $7,150, which is 55% of the original price. By setting up the equation \(0.55x = 7150\), you can find what the full 100% (original price) should be.

Discount Problems

Discount problems are practical applications of percentage calculations. They are often encountered during sales, where you are given a percentage off the regular price. To solve these problems, it's crucial to understand how the discount percentage relates to the final price you pay.Here’s a straightforward approach:

• Determine the percentage you will pay after the discount. In Jessica's situation, she paid 55% of the original price because 100% minus the 45% discount is 55%.
• Use the percentage equation to find the original price before the discount. The method we used for Jessica involves converting the percentage into decimal form (i.e., 0.55) and forming the equation \(0.55x = 7150\).

When faced with discount problems, always remember to check that the percentage paid plus the discount equals 100%. It's an easy way to verify you're on the right track.

Algebraic Equations

Algebra forms the basis for solving equations like the one in the car discount problem, where you solve for an unknown variable (in this case, the original price \(x\)). An equation is a mathematical statement that asserts the equality of two expressions. To solve an algebraic equation, follow these steps:

• Identify the unknown variable, typically represented by letters such as \(x\).
• Set up the equation based on the problem statement. For Jessica, this meant setting \(0.55x = 7150\), where \(x\) represents the original car price.
• Isolate the variable by performing inverse operations. Here, you divide both sides by 0.55 to solve for \(x\).
• Verify your answer by substituting it back into the original context to ensure it makes sense. Knowing \(x = 13000\) helps us validate that this was indeed option C.

Whether you're dealing with percentages or other mathematical problems, mastering algebraic methods allows you to find solutions efficiently.