Question

You have \(\$ 35.72\) to spend for a sweatshirt. The sales tax is \(7 \% .\) What is your price limit for the sweatshirt?

Short answer

The price limit for the sweatshirt before tax is approximately \$ 33.39.

Step-by-step solution

Understand the question

The given amount \(\$ 35.72\) includes both the price of the sweatshirt and the sales tax of \(7 \% .\) Now we need to find the original price of the sweatshirt before tax. The original price plus \(7 \%\) of the original price should equal \$ 35.72.

Set up the equation

Let's denote the price of the sweatshirt before tax as \(x .\) Then we can write down the equation as \(1.07x = 35.72 .\) The \(1.07\) comes from \(1 + 0.07,\) representing the original price plus the sales tax.

Solve for x

To isolate \(x ,\) divide both sides of the equation by \(1.07 .\) Therefore \(x = 35.72 / 1.07.\)

Calculate the price

Finally, calculate \(x\) to find the pre-tax price of the sweatshirt.

Key concepts

Percent Problems

Understanding percent problems is essential in many real-world situations, such as calculating discounts, interest rates, and, as in our example, sales tax. The term 'percent' literally means 'per hundred' and is a way to express a number as a fraction of 100.

In the context of sales tax, the given percentage represents the amount of tax that must be paid for every 100 currency units of the item's price. So, a 7% sales tax implies that for every \(100 worth of items purchased, an additional \)7 is paid as tax. This tax is then added to the original price to determine the total cost. To compute the taxes, or reverse-calculate from the total cost to the pre-tax price, one must understand the fundamental relationship between percentages, their decimal equivalents, and how they apply to quantities in question.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions, which typically consist of variables and constants. Solving an algebraic equation means finding the value(s) of the variable(s) that make the equation true.

In the exercise, we form the equation based on the given sales tax problem. If the pre-tax price of the sweatshirt is represented by the variable \(x\), and knowing that sales tax adds an additional 7% to this value, we express the total as \(1.07x\). Here, the factor 1.07 combines 100% of the item’s price (the whole item, denoted by 1) plus the 7% sales tax (0.07). The equation \(1.07x = 35.72\) ties together the pre-tax price, the tax rate, and the total amount spent, encapsulating the entire purchasing scenario in a single algebraic expression.

Solving for Variable

When you're faced with an equation like \(1.07x = 35.72\), the goal is to find the value of \(x\), the variable that represents the unknown quantity we are interested in—here, it's the pre-tax price of the sweatshirt. To isolate \(x\), we perform operations that will 'undo' the multiplication by 1.07.

This process is called 'solving for a variable'. It involves methods such as addition, subtraction, multiplication, division, or a combination of these operations to get the variable on its own on one side of the equation. In this instance, dividing both sides of the equation by 1.07 will give us the pre-tax price of the sweatshirt, effectively solving for the variable \(x\). This kind of manipulation is at the heart of algebra and is a powerful tool for solving a wide array of problems.