Problem 53 - College Algebra

Question

After a \(12 \%\) price reduction, a car sold for $$ 17,600 .$ What was the car's price before the reduction?

Step-by-Step Solution

Answer

The original price of the car was about $20,000.

Step 1: Translation into a mathematical equation

Firstly, this problem can be interpreted as a simple algebraic equation. If the original price is represented as \( x \), the problem states that the original price minus \(12 \%\) of the original price is equal to $17,600. This can be written as \(x - 0.12x = 17600\).

Step 2: Simplify the equation

Next, combine like terms on the left-hand side of the equation. The original price of \( x \) minus \( 12 \% \) of \( x \), or \( 0.12x \), equals \( 0.88x \). So the equation simplifies to \(0.88x = 17600\).

Step 3: Solve for x

Finally, solve for \( x \) by dividing both sides of the equation by \( 0.88 \). This results in \(x = 17600 / 0.88\).

Key Concepts

Price Reduction

When we say there's a price reduction, it means that the original price of an item is decreased by a specific amount or percentage. In our context, a price reduction of \( 12\% \) means the car's original price is reduced by \( 12\% \) of its value. This new price after the reduction is also known as the selling price.
Here's a simple breakdown:

• The car's original price: the amount before any reduction.
• The reduction percentage: the fraction of the original price that gets deducted.
• The selling price: the final amount after the reduction.

You can calculate the reduction amount by multiplying the original price by the percentage (in decimal form). Understanding price reductions is crucial for budgeting and getting the best deals when shopping.

Percentages in Algebra

Percentages are a way to express any number as a part of a hundred. In algebra, percentages are commonly involved in various types of problems, especially involving increases and decreases like price discounts or interest rates.
You should convert percentages into decimals to solve algebraic equations. For example, \( 12\% \) becomes \( 0.12 \). This conversion is essential for simplifying equations and performing calculations accurately. Here's how you can use them:

• Convert the percentage to a decimal by dividing by 100.
• Use the decimal in algebraic expressions to find percentage-based amounts.

When solving problems, always ensure the conversion is correctly done to avoid errors. Mastering this skill is very helpful in understanding and solving real-world problems involving percentages.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. A linear equation is an equation between two variables that gives a straight line when plotted on a graph. In this case, our task is to determine the original price of the car after a given percentage reduction.
The equation we have is \( 0.88x = 17600 \), where \( x \) represents the original price. Here’s how to tackle it:

• Identify the parts of your equation: the variable, coefficients, and constant.
• Isolate the variable \( x \) by performing operations such as addition, subtraction, multiplication, or division.
• In this example, divide both sides by \( 0.88 \) to solve for \( x \).

This approach allows you to solve for unknowns quickly. Understanding how to manipulate these equations is crucial in algebra and everyday problem-solving tasks.