Question

You have 125.79 dollars to spend for a CD player. The sales tax is \(6 \% .\) What is your price limit for the CD player? (Review 3.6)

Short answer

The price limit for the CD player before applying the sales tax is approximately $118.67.

Step-by-step solution

Understanding the Sales Tax

When you buy a product, you typically pay a sales tax which is a percentage of the selling price. In this case, the sales tax is \(6\%\). This means that for every dollar spent on the product, \(\$0.06\) is added as sales tax. If \(P\) represents the price of the CD player before tax, then the total cost including tax would be \(P+0.06P=1.06P\).

Setting up the equation

You know the total amount of money available for the purchase, which is \$125.79. This amount includes the price of the CD player and the sales tax. So, we can set up the equation \(1.06P = 125.79\) where \(P\) is the price limit for the CD player.

Solving the equation

In order to find the price \(P\), you can divide both sides of the equation by 1.06. This gives \(P = 125.79 \div 1.06\).

Calculating the price limit

Using a calculator, one can divide 125.79 by 1.06 to calculate \(P\).

Key concepts

Understanding Sales Tax Percentage

Calculating sales tax is crucial when budgeting for purchases, as this affects the ultimate amount you pay. The sales tax is determined by a specified percentage rate applied to the price of a good or service. For instance, if an item is priced at \( \(100 \) and the sales tax rate is \( 6\text{%} \), the sales tax is \( \)100 \times 0.06 = \(6 \). Therefore, the total cost becomes \( \)100 + \(6 = \)106 \).

This percentage can vary depending on the locality and type of item. To calculate the total cost of an item including the tax, you can use the algebraic expression \( P + (P \times \text{tax rate}) \) or simply \( P \times (1 + \text{tax rate}) \), where \( P \) represents the original price before the tax. Applying this to a real-world scenario ensures that you don’t overspend and helps in maintaining a budget.

Deciphering Algebraic Equations

Algebraic equations are mathematical statements that show the balance between two expressions. They can help you figure out unknown values, such as the pre-tax price of an item, by using the given information. In the context of calculating sales tax, the equation takes the form of \( P \times (1 + \text{tax rate}) = \text{Total cost} \).

To isolate \( P \), divide both sides by \( (1 + \text{tax rate}) \), yielding \( P = \frac{\text{Total cost}}{(1 + \text{tax rate})} \). Using this formula is incredibly helpful for financial planning and making informed purchasing decisions. By mastering algebraic equations, you not only solve math problems but also cultivate strong analytical and problem-solving skills.

Calculating Price Before Tax

Knowing how to find the price before tax is advantageous for your pocket. To get this price, you decompose the final amount paid back into its base price and tax elements. Given a total amount that includes tax, work backward by dividing by \( (1 + \text{tax rate}) \).

For a clearer perspective, if your total budget is \( $125.79 \) including a \( 6\text{%} \) sales tax, the calculation would be: \[ P = \frac{125.79}{1.06} \]. This provides the price without tax. It's essential in ensuring you stay within budget before reaching the checkout. Learning to calculate the pre-tax price equips you with a transparent understanding of where your money goes and helps you to identify potential savings.