Problem 115 - College Algebra

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% of its original price. It's selling at 36% of its original price.

Step 1: Factor Out

First, factor out \( (x-0.4x) \) from each term in the expression \( (x-0.4x)-0.4(x-0.4x) \). Doing so will give us the expression \( (x-0.4x)(1-0.4) \).

Step 2: Simplify the Expression

Next, simplify the expression \( (x-0.4x)(1-0.4) \). This will result in the expression \( 0.6x \cdot 0.6 = 0.36x \). We can interpret this to mean that the computer is selling at 36% of its original price.

Step 3: Answer the Questions

Based on the simplification in Step 2, we can tell that with a 40% reduction followed by another 40% reduction, the computer is not selling at 20% of its original price. Instead, it's selling at 36% of its original price.

Key Concepts

Algebraic Simplification

Algebraic simplification is a mathematical process used to make an algebraic expression easier to understand and work with. It involves combining like terms, reducing fractions, and applying distributive laws to eliminate parentheses and combine terms whenever possible.

In the context of our exercise, simplification starts by recognizing that the expressions inside the parentheses are alike and can be combined. The term 0.4x represents a 40% reduction of the original price x, so the expression inside the parentheses simplifies to x - 0.4x, which is the same as 0.6x. This represents the price after the first reduction.

Applying simplification techniques to our problem is crucial to find the final sale price after successive discounts. Always remember that simplification is like tidying up your room: once everything is in order, you have a clear space to work and understand what you have.

Factorization

Factorization is a method used in algebra that entails breaking down numbers or expressions into a product of their factors. Factors are numbers or expressions that multiply together to give the original number or expression.

In our given exercise, factorization is utilized to streamline the calculation of the successive discounts on the computer's price. Initially, the expression looks a bit crowded with multiple instances of reductions by 40%. By factoring out the common term (x - 0.4x), we turn the given equation into a product of two factors: (x - 0.4x) and (1 - 0.4). This step significantly eases the task of further simplifications.

Percentage Calculation

Percentage calculation involves determining the part of a whole in terms of 100. It is a fundamental concept in mathematics that is commonly used in financial analysis, data interpretation, and real-life discount scenarios like our exercise.

To calculate the new price after each 40% reduction, we need to understand that a 40% reduction means the remaining price is 60% of the original (100% - 40% = 60%). After the first reduction, we get 0.6x. The second 40% reduction is applied to the new price, so we multiply 0.6x by 0.6 again. The result, 0.36x, represents 36% of the original price, which implies that a double reduction of 40% does not equate to a single reduction of 80%, but rather a smaller total reduction.

Understanding percentage calculation is critical in properly analyzing discounts, interest rates, and growth rates, among other contexts, to avoid misunderstandings and incorrect conclusions.