Problem 19 - Algebra A Combined Function

Question

Solve.Find the original price of a pair of shoes if the sale price is \(\$ 78\) after a \(25 \%\) discount.

Step-by-Step Solution

Answer

The original price of the shoes was $104.

Step 1: Understand the Problem

We need to find the original price of the shoes before the discount was applied. The sale price is given as $78 after a 25% discount.

Step 2: Setup the Equation

Let the original price be denoted as \( x \). The shoes were sold with a 25% discount, which means the sale price is 75% of the original price. This can be expressed by the equation: \( 0.75x = 78 \).

Step 3: Solve for x

To find \( x \), divide both sides of the equation by 0.75. Therefore, \( x = \frac{78}{0.75} \).

Step 4: Calculate the Original Price

Perform the division to find \( x \). This gives us \( x = 104 \).

Step 5: Verify the Calculation

Check the work by ensuring a 25% discount on $104 results in a sale price of $78. Calculate 25% of $104, which is $26. Subtracting $26 from $104 gives $78, verifying the solution is correct.

Key Concepts

Discount Calculations

Discount calculations are a common way to determine how much you will save when buying an item on sale. In this context, understanding the percentage discount is crucial.
When a store offers a discount, it’s usually given as a percentage of the original price. To calculate the discount:

• Identify the percentage of the discount.
• Convert that percentage to a decimal by dividing by 100.
• Multiply the original price by this decimal to find the discount value.

In the case of the shoes, a 25% discount means you will save 25% of the original price. For these calculations, if you already know the discounted price like in this example, you can work backward to find the original price. Knowing how to calculate discounts will not only help you in math exercises but will also assist in making smarter financial decisions while shopping.

Original Price Determination

Determining the original price from a discounted price might seem tricky, but it's quite simple once you understand the process. When you have the sale price and the discount rate, you're set to solve for the original price.Here's how you can determine it:

• Set the sale price equal to the fraction of the original price that remains after the discount. In our example, the sale price is the result of a 75% value of the original price.
• Use the formula: \[ 0.75 \times x = 78 \]where 0.75 is the percentage remaining after a 25% discount, and 78 is the sale price.
• Solve the equation for the original price, which is most often denoted by the variable \( x \).

Thus, the original price of the shoes before applying the 25% discount is found using such logical steps often used in reverse calculations of percentages.

Equation Solving

Solving equations is an essential skill in mathematics that lets you find unknown values. When you are determining the original price after a discount, such as in this scenario, you will often create and solve an equation.When solving for the original price:

• Set up the equation based on the given discount details. In this scenario, it's 0.75 times the original price equals the sale price \( 78 \).
• Rearrange the equation to isolate the unknown variable. Here, we divide both sides by 0.75 to find \( x \).
• Perform the division: \[ x = \frac{78}{0.75} \]
• Calculate this division to find the value of \( x \), the original price.

This approach not only helps in solving discount-related problems but also strengthens overall problem-solving skills. Practicing these steps helps understand how equations model real-world scenarios and how they can be solved efficiently.