Problem 40 - Introductory Algebra for College Students

Question

After a \(30 \%\) reduction, you purchase a DVD player for \(\$ 98 .\) What was the price before the reduction?

Step-by-Step Solution

Answer

The original price of the DVD player before the reduction was approximately $140.

Step 1: Understand the question

The ultimate price of the DVD player after a 30% discount is $98. Therefore, the $98 represents the remaining 70% of the original price (100% - 30%). The task is to find out the original price which represents the 100%.

Step 2: Formulate an equation

Given that 70% now equates to $98, we can represent this relationship with the equation \(0.70x = 98\), where \(x\) is the original price.

Step 3: Solve for \(x\)

Solving for \(x\) will give us \(x = \frac{98}{0.70}\).

Key Concepts

Percentage Calculation

Understanding how to calculate percentages is a foundational skill in mathematics that can be applied to a variety of real-world situations, like discounts during shopping. In percentage problems, you are usually given a part and the whole, or you need to find one of these when the other and the percentage are given.

In the context of our DVD player problem, the '\(98' after a 30% reduction represents 70% of the original price because 100% (the full original price) minus the 30% discount leaves us with 70%. To find the original price, we need to calculate what amount '100%' would represent if '\)98' is '70%'. This is a common application of percentage calculation in finding the original price before discount or figuring out the final amount after interest rates are applied in financial contexts.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions. They usually contain one or more variables representing unknown values that we need to solve for.

In our DVD player example, the original price of the DVD player is the unknown we represent with variable 'x'. The equation we create, \(0.70x = 98\), is a simple algebraic equation that illustrates the relationship between the unknown original price and the known discounted price. Here, '0.70x' symbolizes 70% of the original price. This kind of problem is common in algebra, where manipulation of equations is used to isolate and solve for the unknown variable.

Word Problem Solving

Solving word problems involves translating a real-world scenario into a mathematical one. It's a multi-step process that requires comprehension of the problem's context, identification of the relevant numerical information, and formulation of an equation or set of equations to solve it.

Improvement in solving word problems is achieved by practicing problem comprehension, focusing on the logic of the relationships involved, and systematically translating those relationships into mathematical representations. With the DVD player discount problem, we first understood the scenario and the meaning of the 30% discount. Then we selected the relevant details to construct an equation that models the situation. To effectively improve in this area, it is beneficial to practice various word problems across different contexts to build strong analytical and mathematical reasoning skills.