Problem 30 - Algebra for College Students

Question

Solve each equation. Jim bought a pair of slacks at a \(25 \%\) discount sale for \(\$ 24\). What was the original price of the slacks?

Step-by-Step Solution

Answer

The original price was $32.

Step 1: Identify the given information

We know that Jim bought a pair of slacks for $24 after a 25% discount. The price he paid is 75% of the original price because he received a 25% discount. We need to find the original price.

Step 2: Set up the equation

Let's denote the original price of the slacks as \( x \). Since Jim paid 75% of the original price, we can write the equation as: \( 0.75x = 24 \).

Step 3: Solve the equation for x

To find the original price \( x \), divide both sides of the equation by 0.75: \[ x = \frac{24}{0.75} \] Calculate the right side to find \( x \).

Step 4: Calculate the original price

Calculate \( \frac{24}{0.75} \) to find \( x \). This results in \( x = 32 \). Therefore, the original price of the slacks was $32.

Key Concepts

Understanding Percentage Discount

A percentage discount is a reduction from the original price, expressed as a percentage. It's a common concept in shopping and sales.
When a product is advertised with a percentage discount, it means that the price is decreased by that specific percentage.

• For example, a 25% discount means the price is reduced by 25% of its original value.
• To convert a percentage into a decimal, divide by 100. So, 25% becomes 0.25.

To find out how much you will pay, subtract the discount percentage from 100% to get the percentage you actually pay.
For Jim's slacks, the calculation looked like this:

• Original price percentage = 100% - 25% = 75%.
• Converted to decimal: 75% = 0.75.

You use this percentage as a multiplier with the original price to find the sale price.

Solving Equations in Word Problems

Solving equations involves finding the value that makes a statement true. In word problems, like the one with Jim’s slacks, the trick is to translate the problem into an equation.
Here's a step-by-step guide to help you:

• Identify the unknown quantity, which in this case is the original price, denoted by the variable x.
• Understand the relationship given in the problem: Jim paid 75% of the original price.
• Set up the equation to reflect this: 0.75x = 24.

Equations are balanced scales; whatever you do to one side, you must do to the other to maintain balance. By dividing both sides by 0.75, you isolate x:

• \[ x = \frac{24}{0.75} \]

This calculation helps find the value of x, the original price of the slacks.

Calculating the Original Price

Calculating the original price requires reversing the percentage calculation to find the price before the discount.
In the example, the sale price is \(24, which was 75% of the original price.
To find the original price, you perform the following steps:

• Use the equation: \[ x = \frac{24}{0.75} \]
• Divide 24 by 0.75 to solve for x.
• This yields \( x = 32 \).

Thus, the original price of the slacks was \)32. This method of dividing by the percentage paid (in decimal form) can be used whenever you want to find an original value that has undergone a percentage change.