Problem 6 - College Algebra and Calculus: An Applied Approach

Question

Write an algebraic expression for the verbal expression. Discount The sale price of an item that is discounted by \(20 \%\) of its list price \(L\)

Step-by-Step Solution

Answer

The algebraic expression is \(L - 0.20L\).

Step 1: Comprehend the Problem

Every percentage can be converted to a decimal by dividing it by 100. Hence, \(20\%\) discount can be represented as \(0.20\), meaning that 20% of the original price is subtracted from the original price itself to get the sale price after the discount.

Step 2: Identify the Operation

In this context, the operation involved is subtraction. A discount is calculated by subtracting a certain percentage of the original price from the original price itself. Hence, 20% of \(L\) needs to be subtracted from \(L\) to arrive at the sale price.

Step 3: Algebraic Expression

Now, 20% of \(L\) can be obtained by multiplying \(L\) with \(0.20\). So, this value is calculated as \(0.20L\). To get the sale price, this value needs to be subtracted from \(L\). Hence, the algebraic expression that represents this operation is \(L - 0.20L\).

Key Concepts

Understanding Percents

Percentages are a fundamental concept in mathematics, often used to express how large or small one quantity is in relation to another. In simple terms, a percent is a way to express a number as a fraction of 100. So, 20% is equivalent to 20 out of 100, or 0.20 in decimal form.
Understanding percents is crucial in everyday situations, such as calculating discounts, interest rates, or statistics.
When you see a percentage, you can convert it to a decimal by dividing by 100. For example:

• To convert 20% to a decimal, divide 20 by 100 which gives 0.20.
• For 50%, you would divide 50 by 100 to get 0.50.

Using decimals makes it easier to perform mathematical operations, like finding parts of a number, which is vital for calculating discounts, taxes, and more.

Subtraction in Algebra

Subtraction is a basic mathematical operation, representing the process of taking one quantity away from another. When applied in algebra, subtraction involves working with variables and can be used to solve various types of problems, especially when finding differences or values after reductions.
In the scenario of a discount, subtraction comes into play when you need to remove a certain portion of the original amount.
For instance, finding a sale price after a 20% discount involves subtracting 20% of the list price from the list price itself.

Algebraic Subtraction Formula

For our example with a list price of \(L\), the subtraction of 20% becomes \(L - 0.20L\).
This represents the original price (\(L\)) minus the part that is taken away (\(0.20L\)). This allows us to easily calculate the discounted price using algebra.

Solving Word Problems in Algebra

Word problems in algebra are exercises where you use language-based descriptions to derive mathematical solutions. They are a fantastic way to practice translating everyday situations into mathematical operations.
To solve word problems, it's important to:

• First, carefully read the problem to identify what information is given and what needs to be found.
• Next, determine which mathematical operations and formulas will lead to the solution.
• Then, set up an algebraic expression or equation to represent the scenario.

In the case of discount calculation, the word problem might describe the reduction as a percentage of the list price.
By translating this into an algebraic expression like \(L - 0.20L\), you apply both subtraction and the conversion of percent to decimal to find the answer.

Key Steps for Success

Transforming the verbal description into equations or expressions is the heart of solving word problems. Practice and attention to detail improve one's ability to dissect the language of the problem and construct a logical mathematical path to the solution.