Problem 29 - Algebra for College Students

Question

Solve each equation. Judy bought a coat at a \(20 \%\) discount sale for \(\$ 72\). What was the original price of the coat?

Step-by-Step Solution

Answer

The original price of the coat was $90.

Step 1: Understand the Problem

We are given the sale price of a coat which was bought at a 20% discount. We need to find the original price before the discount was applied.

Step 2: Express the Relationship Mathematically

The original price of the coat, when reduced by 20%, results in the sale price of $72. Let's denote the original price as \( P \). A 20% discount implies that Judy paid 80% of the original price. Therefore, \[ 0.8P = 72 \]

Step 3: Solve for the Original Price

To find the original price \( P \), we need to solve the equation \( 0.8P = 72 \). Divide both sides by 0.8 to isolate \( P \): \[ P = \frac{72}{0.8} \] Simplify the division: \[ P = 90 \]

Step 4: Verify the Solution

We found \( P = 90 \). Let's verify: Calculating 20% of \( 90 \), we get \( 0.2 \times 90 = 18 \). After applying the discount, the price becomes \( 90 - 18 = 72 \), which matches the given sale price.

Key Concepts

Percentage Problems

Understanding percentage problems is crucial in many real-life situations. A percentage represents a part per hundred, serving as a method to compare quantities or express increases and decreases. In this exercise, the original price of the coat is reduced by a percentage, leading to the final sale price. To express this mathematically, we apply the concept:

• Identify the original whole value. Here, it's the original price of the coat, denoted as \( P \).
• The given percentage, 20%, represents the discount applied to \( P \).
• Calculate what remains after the discount: if 20% is taken off, 80% of the original remains, leading to the equation \( 0.8P = 72 \).

These steps help you set up the right equation to find the original price by using the percentage given. With practice, identifying the right percentage in a problem becomes more intuitive.

Discount Calculations

Discount calculations are a straightforward but important aspect of percentage problems. They involve finding how much price is reduced when a discount is applied. For this calculation, remember:

• The discount percentage tells you what fraction of the original price is subtracted. In this case, 20% of the original price is the discount.
• Subtract the discount from the original price to find what you'll actually pay.
• In the example, if the original price is \( P \), the discount amount is \( 0.2P \).
• Thus, Judy paid \( P - 0.2P = 0.8P = 72 \), where 0.8 represents the remaining percentage of the price after the discount.

By understanding this process, you can easily calculate how any discount will affect the original price, whether in a store or when solving algebraic percentage problems in a classroom.

Mathematical Verification

Once you've calculated a solution, verifying its accuracy is a crucial step to ensure your answer is correct. It's a method of checking whether the calculations hold true when reversed. For instance, after finding the original price \( P \), we verify:

• Calculate 20% of the original price \( P = 90 \). Thus, \( 0.2 \times 90 = 18 \).
• Subtract the discount from the original price: \( 90 - 18 = 72 \).
• Compare this result with the given sale price.

If the numbers match, your solution is confirmed. Verification strengthens your mathematical process, highlighting the reliability of your problem-solving method. Incorporating verification checks helps build confidence in solving not just percentage problems, but a wide range of mathematical equations.