Problem 67 - Introductory Algebra for College Students

Question

An exercise machine with an original price of \(\$ 860\) is on sale at \(12 \%\) off. a. What is the discount amount? b. What is the exercise machine's sale price?

Step-by-Step Solution

Answer

a. The discount amount is $\$103.20$ b. The sale price of the exercise machine is $\$756.80$.

Step 1: Determine the Discount Amount

To calculate the discount amount, multiply the original price ($860) by the discount rate (12%). In mathematical terms, this is expressed as \(Discount Amount = $860 \times 0.12\). This will yield the discount amount.

Step 2: Subtract the Discount Amount from the Original Price

After calculating the discount amount, subtract it from the original price to get the sale price. As a formula, this is given as \(Sale Price = Original Price - Discount Amount\) which in this case becomes \(Sale Price = $860 - Discount Amount\).

Step 3: Compute the values

By substituting the values into the two above formulas, we will get the values for the discount amount and the sale price.

Key Concepts

Percentage Calculation

Understanding percentage calculation is crucial when working with discounts and sale prices. When we talk about percentages, we refer to a part of a whole expressed as a fraction of 100.
For example, if something is marked with a "12% discount", it means you only need to pay 88% of the original price. To calculate how much 12% is in dollars, we multiply the original price by 12%, or 0.12 in decimal form.
This kind of calculation helps to easily determine the exact amount of a discount or any other proportionate values.

• To convert a percentage to a decimal, move the decimal point two places to the left. So, 12% becomes 0.12.
• Multiplying the decimal by the original price gives the discount amount.

Understanding these basics helps you quickly comprehend how much you're saving or what proportion of a total you're working with.

Discount and Sale Price

Discount and sale prices are two key terms often used when shopping. A discount is the amount taken off the original price, while the sale price is what you actually pay after the discount is applied.
Think of the original price as the unit cost, say \(\$860\) in this case. The 12% off means you're paying just 88% of that original price. In calculations:

• Discount Amount: Multiply the original price by the discount percentage (\(Discount Amount = 860 \times 0.12\).
• Sale Price: Subtract the discount amount from the original price (\(Sale Price = 860 - Discount Amount\)).

This simple calculation shows you how much you'll need to handover at the cash counter, after discounts.

Problem-Solving Steps

Solving problems involving discounts and sale prices often follows a straightforward method. This step-by-step process helps ensure that you don't miss a detail and calculate correctly. Start by determining what you need to find. For the exercise, we need to find both the discount and the sale price. Follow these steps:

• Step 1: Identify the original price (given as \(\$860\)) and the discount rate (12%). Convert the percentage into a decimal by dividing by 100.
• Step 2: Calculate the discount amount. Multiply the original price by the decimal equivalent of the discount rate. This gives you the portion of the price being deducted.
• Step 3: Find the sale price by subtracting the discount amount from the original price. The result is the amount that needs to be paid.

Practicing such step-by-step approaches can make tackling similar problems much easier and efficient.