Problem 32 - Algebra for College Students

Question

Solve each equation. Find the discount sale price of a \(\$ 72\) item that is on sale for \(35 \%\) off.

Step-by-Step Solution

Answer

The sale price is $46.80.

Step 1: Understand the Problem

We need to find the sale price of an item that originally costs $72, given that it is on sale at a 35% discount. The formula for finding the discount is to multiply the original price by the discount rate.

Step 2: Calculate the Discount Amount

Calculate 35% of the original price. The discount amount can be found by using the formula: \[\text{Discount amount} = \text{Original price} \times \text{Discount rate}\] Substitute the known values: \[\text{Discount amount} = 72 \times 0.35 = 25.2\].

Step 3: Determine the Sale Price

Subtract the discount amount from the original price to get the sale price. Use the formula: \[\text{Sale price} = \text{Original price} - \text{Discount amount}\] Substitute the values: \[\text{Sale price} = 72 - 25.2 = 46.8\]

Key Concepts

Percentage Discounts

Percentage discounts are a common way to reduce the cost of an item. This type of discount is an amount taken off the original price, based on a certain percentage. For instance, if an item with an original price of $100 has a 20% discount, the price is reduced by 20% of $100.

To calculate any percentage discount, you simply multiply the original price by the discount percentage expressed as a decimal. Here’s a simple guide to doing this:

• Convert the discount percentage to a decimal by dividing by 100. For example, 35% becomes 0.35.
• Multiply the original price by this decimal to find the discount amount.
• Subtract the discount amount from the original price to get the sale price.

This method ensures you understand exactly how much you're saving with any percentage discount.

Solving Equations

Solving equations involves finding the value of an unknown number that makes the equation true. In the case of calculating the sale price after a discount, the equation helps us determine how much money we're saving and how much we'll need to pay.

Here’s how you can solve these equations:

• First, identify the equation you need to solve. For the discount problem, it involves determining the discount amount: \( \text{Discount amount} = \text{Original price} \times \text{Discount rate} \).
• Plug in the known values into the equation, replace the variables with the values you know, and solve the equation.
  
• Another equation to solve is for the sale price: \( \text{Sale price} = \text{Original price} - \text{Discount amount} \).

By repeatedly using these basic equations, you can become adept at solving similar real-world problems.

Algebraic Operations

Algebraic operations are the basic mathematical procedures used to rearrange and solve equations. In discount calculations, we use these operations to calculate both the discount amount and the final sale price.

• Addition and subtraction: After finding the discount amount, subtract it from the original price to find the sale price.
• Multiplication: Used to calculate the discount amount by multiplying the original price by the discount rate.

These operations form the foundation of algebraic manipulation and allow us to systematically approach and solve problems. Understanding them simplifies complex problems into more manageable steps.

By mastering these simple operations, you're better equipped to handle not just mathematics but also everyday decisions involving discounts and purchases.