Problem 6 - Precalculus Student Solutions Manual 5th

Question

Sale Price Suppose that a computer that originally sold for \(x\) dollars has been discounted \(60 \%\). Which one of the following expressions does not represent its sale price? A. \(x-0.60 x\) B. \(0.40 x\) C. \(\frac{4}{10} x\) D. \(x-0.60\)

Step-by-Step Solution

Answer

D. \(x-0.60\)

Step 1: Identify the Original Price

Let the original price of the computer be represented by the variable \(x\).

Step 2: Calculate the Discounted Amount

The computer is discounted by \(60\%\). To find the discounted amount, calculate \(0.60x\).

Step 3: Determine the Sale Price Expression

The sale price is the original price minus the discounted amount. Mathematically, this can be expressed as \(x - 0.60x\).

Step 4: Simplify the Sale Price Expression

Simplify the expression \(x - 0.60x\): \[ x - 0.60x = (1 - 0.60)x = 0.40x\]

Step 5: Compare Given Options

The expressions given are: A. \(x-0.60 x\)B. \(0.40 x\)C. \(\frac{4}{10} x\)D. \(x-0.60\)Options A, B, and C all represent the same sale price. Option D is different because it does not properly account for the percentage discount.

Step 6: Final Step: Identify the Incorrect Expression

Hence, the expression \(x - 0.60\) does not represent the sale price.

Key Concepts

Percentage Discount

Discounts are often given as a percentage of the original price. To understand how a percentage discount works, let's start with the basics. A percentage indicates a part of a hundred. So, if you have a 60% discount, it means 60 out of every 100 units of the original price are taken off.

For example, if the original price of a product is \(100 and the discount is 60%, you calculate the discount by multiplying the original price by 0.60 (since 60% = 60/100 = 0.60). The discounted amount, in this case, is \)100 * 0.60 = \(60. Therefore, the sale price would be the original price minus the discount, which is \)100 - \(60 = \)40.

To generalize, if the original price is represented by the variable \( x \), and the discount percentage is 60%, the discount amount is \( 0.60x \).

Simplifying Expressions

Simplifying expressions helps make complex mathematical problems easier to handle. In our exercise, after calculating the discounted amount \( 0.60x \), we subtract this from the original price \( x \) to find the sale price.

In mathematical terms, this is written as:

• Original Price: \( x \)
• Discounted Amount: \( 0.60x \)
• Sale Price Expression: \( x - 0.60x \)

To simplify, we combine like terms. When you subtract \( 0.60x \) from \( x \), you get \( 1x - 0.60x \). This simplifies to \( 0.40x \). This is easily understandable as holding on to 40% of the original value after discounting 60%.

Simplifying expressions is useful as it gives a clearer understanding and often allows easier calculations.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators that represent a value. In the context of sale price calculations, expressions allow us to derive useful information from given data.

Consider the various expressions provided in the original exercise. We are given:

• \( x - 0.60 x \)
• \( 0.40 x \)
• \( \frac{4}{10} x \)
• \( x - 0.60 \)

First, understand each expression:

\( x - 0.60 x \) means the original price minus 60% of the original price.

\( 0.40 x \) means directly taking 40% of the original price.

\( \frac{4}{10} x \) is the same as \( 0.40 x \), since \( \frac{4}{10} \) = 0.40.

\( x - 0.60 \) incorrectly subtracts a constant 0.60, not accounting for the percentage of the price.

Options A, B, and C all represent the same sale price, calculated correctly. However, option D does not. Recognizing the structure of mathematical expressions helps identify correct representations of concepts like sale price.