Question

The sale price of a shirt is \(\$ 17.25\) after a \(25 \%\) discount is taken. You can model the situation with an equation of the form \(a\) is \(p\) percent of \(b\). Is the base \(b\) the sale price or the regular price?

Short answer

'b' represents the regular price.

Step-by-step solution

Identify the variables in the problem

The variables from the equation are mapped as: 'a' is the sale price which is $17.25, 'p' is the percentage that is 25%, and 'b' is what we need to determine.

Understand the concept of discount

A discount is a reduction applied on the original price of an item. Therefore, the sale price is the result after the discount percent has been taken off from the original price. The original price is always greater than the sale price if the discount is given.

Apply the discount concept to the given scenario

Based on the understanding of discounts, it can be concluded that 'b' in the given scenario represents the regular price (before the discount), not the sale price, since it is from the regular price that the discount is computed. In other words, the sale price is 75% (100% - 25%) of the original price.

Key concepts

Percentage in Algebra

When we delve into the relationship between algebra and percentages, we effectively merge two powerful tools of mathematics to understand and solve practical problems such as finding discounts. In this context, percentages are used to describe a part of a whole, and algebra is the mechanism by which we express these relationships with variables and equations.

In our exercise, percentages are symbolized by the variable 'p' which stands for the discount rate (25%). Algebra comes into play as we establish an equation to interpret the situation. By setting variables like 'a' for sale price and 'b' for the unknown original price, algebra enables us to create a model that represents the problem. Thus, when we talk about percentage in algebra, we are essentially discussing how to use variables and equations to represent parts of a quantity—in this case, a discount on a price.

Modeling with Equations

Modeling with equations is akin to painting a picture with numbers and symbols; it allows us to illustrate real-world situations mathematically. By constructing an equation, we gain the power to analyze and solve problems in a systematic way.

In terms of our discount problem, the equation \(a = p\% \times b\) is the chosen model. This equation is a blueprint that guides us to uncover the original price before the discount was applied. Here, 'a' is the amount after the discount, 'p' symbolizes the percentage of the original price that remains after the discount, and 'b' represents the original price that we're calculating. The art of modeling with equations lies in choosing the right equation form and accurately assigning the variables to get to the solution.

Understanding Discounts

A discount is like a financial shortcut offered to consumers—it's the difference between the original price and the sale price. When you understand discounts, you unlock the ability to find out how much money you save, or conversely, how much you’d have to pay without that discount.

In our textbook problem, we grapple with a 25% discount which tells us that the sale price reflects only 75% of the whole, or the original price. It’s essential to recognize that the original price is the complete 100%, and the discount is subtracted from this total. By applying this understanding, we can backtrack from the sale price, known here as 'a', to find the pre-discount amount, labeled as 'b', which is the objective of our exercise.