Problem 102 - Elementary Algebra

Question

Kim bought a pair of shoes on sale for \(\$ 40.50 .\) The sale price was \(45 \%\) of the original price. What was the original price of the shoes?

Step-by-Step Solution

Answer

The original price of the shoes was 90.

Step 1: Understand the Relationship Between Sale Price and Original Price

The sale price represents 45% of the original price. This means that the sale price is 45% of some unknown original price. We can represent the original price as 'P'.

Step 2: Set Up the Equation

Since the sale price ( 40.50) is 45% of the original price (P), we can write the equation as: 40.50 = 0.45P

Step 3: Solve for the Original Price

To find the original price, 'P', divide both sides of the equation by 0.45: P = 40.50 / 0.45.

Step 4: Calculate the Original Price

Perform the division to solve for 'P': P = 90.

Key Concepts

Percentage Calculations

Understanding how to calculate percentages is fundamental in various mathematical applications, including solving word problems like the one in this exercise. Percentages are a way of expressing numbers as a fraction of 100. In this scenario, Kim bought a pair of shoes for 45% off the original price. This means that the sale price of 40.50 USD represents 45% of the original price. To establish this relation, we use the formula for percentage calculation:

• 45% can be written as a decimal: 0.45

So, if we let 'P' represent the original price, then 0.45P equals 40.50 USD. Breaking down the problem this way simplifies the process of finding the original price.

Solving Equations

In algebra, equations are used to find an unknown value. Here, we have the equation 40.50 = 0.45P. Our goal is to isolate 'P' to determine the original price of the shoes. The following steps outline the basic procedure for solving simple linear equations:

• Identify the equation: 40.50 = 0.45P
• Isolate the variable on one side of the equation. In this case, we divide both sides of the equation by 0.45: \(\frac{40.50}{0.45} = P\)

After performing the division, we find that P equals 90. Thus, the original price of the shoes was 90 USD.

Problem-Solving Steps

Approaching word problems methodically ensures a higher likelihood of solving them correctly. Here's a detailed plan on tackling algebra word problems, which includes the steps used in this exercise:

• Read the problem carefully to understand what is being asked and identify the key pieces of information.
• Convert the word problem into a mathematical equation. In our example, we translated '45% of the original price' to 0.45P.
• Solve the equation step-by-step. Isolate the variable by performing the necessary arithmetic operations, such as division or multiplication.
• Double-check your calculations and ensure the solution makes sense contextually within the problem.
• To verify, substitute the solution back into the context of the problem to see if it satisfies the original conditions. For instance, multiplying 90 by 0.45 should give us the sale price of 40.50.

Following these systematic problem-solving steps helps in better understanding and solving algebra word problems efficiently.