Problem 115 - College Algebra Essentials

Question

Your computer store is having an incredible sale. The price on one model is reduced by \(40 \% .\) Then the sale price is reduced by another \(40 \% .\) If \(x\) is the computer's original price, the sale price can be modeled by $$(x-0.4 x)-0.4(x-0.4 x)$$ a. Factor out \((x-0.4 x)\) from each term. Then simplify the resulting expression. b. Use the simplified expression from part (a) to answer these questions. With a \(40 \%\) reduction followed by a \(40 \%\) reduction, is the computer selling at \(20 \%\) of its original price? If not, at what percentage of the original price is it selling?

Step-by-Step Solution

Answer

No, the computer is not selling at \(20%\); it's selling at \(36%\) of the original price.

Step 1: Factoring

Factoring involves identifying a common element in both terms of the expression and factoring it out. In this case, the expression is \((x-0.4 x)-0.4(x-0.4 x)\). The term \((x-0.4 x)\) is common to both terms, so we can write the expression as: \( [(x-0.4 x)*(1-0.4)]\).

Step 2: Simplifying Expressions

To simplify it further, reduce the terms in the parentheses: \((x-0.4x)\) simplifies to \(0.6x\), and \((1-0.4)\) simplifies to \(0.6\). This gives the final expression: \(0.6x * 0.6\).

Step 3: Calculating the Final Price

Now we perform the multiplication which will give us \(0.36x\). This is the final sale price of the computer, represented as a fraction of the original price.

Step 4: Converting to Percentage

In the context of this problem, the \(x\) represents 100% (the original price) and the coefficient in front of it is the percentage that is the sale price. So the final sale price is \(36%\) of the original price. This means that the computer is not selling at \(20%\), but rather at \(36%\).

Key Concepts

Factoring

Factoring is a fundamental concept in algebra that helps simplify expressions and solve equations. In simple terms, factoring is the process of breaking down an equation into products of simpler expressions. It’s like finding ingredients in a complex recipe.In our exercise, we factor the expression \[ (x - 0.4x) - 0.4(x - 0.4x) \] by recognizing the common element \( (x - 0.4x) \).

• Identify the repeated part: \( (x - 0.4x) \).
• Factor it out from both terms: \[ [(x - 0.4x) \, * \, (1 - 0.4)] \].

By factoring, we simplify the expression, making it easier to see how each component contributes to the final result. Factoring is the first step towards simplifying any algebraic expression, making problem-solving more straightforward.

Simplifying Expressions

After factoring, the next step is simplifying expressions. Simplification is all about making an expression as simple as possible by performing operations within the equation. It helps us understand and manipulate equations more easily.In the exercise, after factoring out \((x-0.4x)\), we simplify each part:

• \((x-0.4x)\) simplifies to \(0.6x\) because subtracting \(0.4x\) from \(x\) is the same as taking \(60\%\) of \(x\).
• \((1-0.4)\) simplifies to \(0.6\), indicating another \(60\%\).

The simplified expression becomes \(0.6x * 0.6\), which is much easier to work with.Simplifying expressions often involves combining like terms, performing basic arithmetic, and reducing coefficients to their simplest form. It’s a crucial skill for any mathematician because it transforms complex expressions into manageable pieces.

Percentage Reduction

Understanding percentage reduction is key to solving problems related to discounts, taxes, and other changes in value.In the problem, the original price \(x\) of the computer undergoes a sequence of reductions:

• First, a \(40\%\) reduction means the item is now \(60\%\) of the original. This is calculated as \(x - 0.4x = 0.6x\).
• Then, another \(40\%\) reduction is applied to the new price \((0.6x)\), resulting in a final price that is \(36\%\) of \(x\).

This showcases a common misconception: Two successive \(40\%\) reductions are not equal to a single \(80\%\) reduction. Each percentage reduction is applied to the most recent price, not the original.

Mathematical Modeling

Mathematical modeling is the process of representing real-life scenarios using mathematical expressions or equations. It’s a bridge that connects abstract math concepts with everyday problems.In this exercise, the expression\[ (x - 0.4x) - 0.4(x - 0.4x) \]is a model of how the price of a computer is affected by two successive discounts.

• The variable \(x\) represents the original price of the computer.
• The expressions \(0.4x\) and \((x - 0.4x)\) model the effects of each \(40\%\) discount.

This type of modeling helps in predicting outcomes, planning budgets, and making informed decisions based on mathematical reasoning. Understanding how to construct such models is an essential skill, especially when dealing with financial and statistical data. It’s all about turning numbers into stories that show us the impact of changes. The exercise teaches us that mathematical models aren’t just numbers; they are powerful tools for understanding complex dynamics in a simplified way.