Problem 37 - Intermediate Algebra

Question

Use linear functions. "All Items \(20 \%\) Off Marked Price" is a sign at a local golf course. Create a function and then use it to determine how much one has to pay for each of the following marked items: a \(\$ 9.50\) hat, a \(\$ 15\) umbrella, a \(\$ 75\) pair of golf shoes, a \(\$ 12.50\) golf glove, a \(\$ 750\) set of golf clubs.

Step-by-Step Solution

Answer

Hat: $7.60, Umbrella: $12, Shoes: $60, Glove: $10, Clubs: $600.

Step 1: Define the Function

We need a function that represents the discounted price after applying a 20% discount. The original price can be represented as \( x \). The discount amount is 20% of \( x \), which is \( 0.20x \). Therefore, the discounted price is given by \( f(x) = x - 0.20x \), simplifying to \( f(x) = 0.80x \).

Step 2: Calculate Price of the Hat

Substitute \( x = 9.50 \) into the function to find the discounted price of the hat. \[ f(9.50) = 0.80 \times 9.50 = 7.60 \] So, the hat costs \( \$7.60 \).

Step 3: Calculate Price of the Umbrella

Substitute \( x = 15 \) into the function to find the discounted price of the umbrella. \[ f(15) = 0.80 \times 15 = 12 \] So, the umbrella costs \( \$12 \).

Step 4: Calculate Price of the Golf Shoes

Substitute \( x = 75 \) into the function to find the discounted price of the golf shoes. \[ f(75) = 0.80 \times 75 = 60 \] So, the golf shoes cost \( \$60 \).

Step 5: Calculate Price of the Golf Glove

Substitute \( x = 12.50 \) into the function to find the discounted price of the golf glove. \[ f(12.50) = 0.80 \times 12.50 = 10 \] So, the golf glove costs \( \$10 \).

Step 6: Calculate Price of the Golf Clubs

Substitute \( x = 750 \) into the function to find the discounted price of the golf clubs. \[ f(750) = 0.80 \times 750 = 600 \] So, the golf clubs cost \( \$600 \).

Key Concepts

Discount Calculation

When shopping, you often encounter discounts on the original price of items. A discount is a reduction applied to the original price, making the item cheaper for the buyer. Here, we'll focus on a specific case of a **20% discount** at a golf course.

For any item with a marked price, calculating the discount involves a few simple steps:

• Determine the percentage of the discount. For our example, it's 20%.
• Convert this percentage into a decimal by dividing by 100, which for 20% becomes 0.20.
• Multiply the original price by this decimal to find the discount amount.
• Subtract this discount amount from the original price to find the final price.

By applying these steps, you can quickly calculate how much you will pay after the discount. This is a fundamental skill for anyone looking to make wise financial decisions.

Function Notation

Function notation provides a simple way to write mathematical functions, using symbols to represent relationships between quantities. In our example, the function describes the relationship between the original and discounted prices of items.

A function is usually denoted by the symbol **f** followed by parentheses containing a variable, like this: **f(x)**. Here, **x** represents the input (original price), and **f(x)** is the output (discounted price).

For our problem, the notation helps us easily express the operation needed to find the discounted price. We defined a function as:

• Since a 20% reduction means paying 80% of the original price, our function becomes: \[ f(x) = x - 0.20x \]
• This simplifies to: \[ f(x) = 0.80x \]

This equation means to get the discounted price, you simply multiply the original price by 0.80. Function notation, thus, not only simplifies complex mathematical calculations but also makes them more readable.

Percentage Reduction

Understanding percentage reduction is crucial in calculating discounts, which is a popular application of linear functions. A percentage reduction tells us what fraction of the original amount is deducted.

Here's how it works in our example:

• A 20% discount means that 20% of the original price is removed.
• It is calculated by multiplying the original price by the reduction percentage divided by 100, i.e., original price \( \times \) 0.20.
• This is subtracted from the original price to find the reduced price, essentially paying 80%.
• So, using our established formula \( f(x) = 0.80x \) easily shows this step.

Percentage reduction is not only relevant for discounts but also applies to understanding increases and decreases in various contexts like tax calculations, markups, and sales increases. Grasping this concept will enhance your mathematical reasoning and practical problem-solving skills.