Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Work Exercises \(1-6\) without pencil and paper. Do not use a calculator. Suppose that a computer that originally sold for \(x\) dollars has been discounted \(30 \% .\) Which one of the following expressions does not represent the sale price of the computer? A. \(x-0.30 x\) B. \(0.70 x\) D. \(x-0.30\) C. \(\frac{7}{10} x\)

Step-by-Step Solution

Answer

Option D (\( x - 0.30 \)) does not represent the sale price.

Step 1: Understand the Expression for Sale Price

The computer is discounted by 30%, which means we subtract 30% of its price from the original price. This is represented by the expression: \( x - 0.30x \). This expression calculates the sale price directly.

Step 2: Simplify the Expression

To simplify the expression \( x - 0.30x \), factor out the common term \( x \):\[x - 0.30x = (1 - 0.30)x = 0.70x\]This means the sale price is 70% of the original price.

Step 3: Check Other Options

We will check if each of the given options represents the simplified expression (i.e., \( 0.70x \)) or not:- Option A: \( x - 0.30x \) simplifies to \( 0.70x \).- Option B: \( 0.70x \) directly matches the sale price expression.- Option C: \( \frac{7}{10}x \) is equivalent to \( 0.70x \) since \( \frac{7}{10} = 0.70 \).- Option D: \( x - 0.30 \) does not match because it subtracts a fixed amount, not a percentage of \(x\).

Step 4: Identify the Mismatched Option

From the analysis in Step 3, Option D (\( x - 0.30 \)) is the mismatched expression because it does not represent 70% of \( x \). Instead, it subtracts a constant value of 0.30, which is incorrect according to the conditions of the problem.

Key Concepts

Percentage Reduction

Percentage reduction is a common concept used in real-life scenarios, especially in sales and discounts.
Understanding this will help you decide the final price of a product after a specific discount is applied.

When you see a discount like 30%, it means you are reducing the price by 30% of its original value.
For instance, if a product costs \(x\) dollars originally, a 30% discount is calculated as follows:

• Find 30% of \(x\) which is \(0.30 \times x\).
• Subtract that amount from the original price \(x\).
• The formula: \(x - 0.30x = 0.70x\) represents the sale price after the discount.

This means you are effectively paying 70% of the original price.
Remember, the percentage reduction helps you find out how much you save as well as the new reduced price.

Expression Simplification

Simplification of expressions is a crucial mathematical process.
It involves reducing the complexity of an expression to make calculations easier.

In our scenario, to find the sale price of the computer, we use the expression \(x - 0.30x\).
Here is how simplification works:

• Factor out the common variable: \(x\).
• You get \((1 - 0.30)x\).
• Simplifying \(1 - 0.30\) to get \(0.70\), so the expression becomes \(0.70x\).

This shows that simplification is valuable because it allows for easier comparison and understanding of different expressions representing the same value.
Reducing expressions to their simplest form also helps in identifying incorrect or misleading expressions easily.

Mathematical Reasoning

Mathematical reasoning is the logical thought process you use to solve problems.
It helps you systematically identify correct solutions and spot errors.

When working with expressions in mathematics, reasoning is key.
In the exercise, different expressions represent the sale price, but not all are correct.

• For example, options such as \(x - 0.30x\) and \(0.70x\) logically represent the sale price.
• However, option \(x - 0.30\) is incorrect because it subtracts a static value, not 30% of \(x\).
• Checking logic against each potential solution shows how reasoning aids your understanding.

Good mathematical reasoning enables you to apply concepts accurately in problem-solving situations.
It helps you verify the truth and relevance of each solution option when making decisions.