Question

A beverage store charges a base price of \(x\) dollars for one keg of root beer. A sales tax of a certain percentage is applied to the base price, and an untaxed deposit for the keg is added. If the total amount, in dollars, paid at the time of purchase for one keg is given by the expression \(1.07 x+17\), then what is the sales tax, expressed as a percentage of the base price? A) \(0.07 \%\) B) \(1.07 \%\) C) \(7 \%\) D) \(17 \%\)

Short answer

The sales tax percentage is 7%. Answer: \(\textbf{(C) } 7\%\).

Step-by-step solution

Identify the sales tax and the deposit

From the expression \(1.07x + 17\), we can identify that 1.07x represents the base price after applying the sales tax and 17 represents the deposit.

Extract the sales tax factor

Since the expression \(1.07x\) represents the price after sales tax is applied, we can determine the sales tax factor by noticing that the base price \(x\) is multiplied by 1.07. This means the sales tax factor is \((1.07-1)\).

Convert the tax factor to percentage

The sales tax factor is the multiplicative effect of the sales tax on the base price, and it's expressed as decimal value (in this case, \((1.07 - 1)\)). To convert it to a percentage, we simply multiply the value by 100. Sales tax percentage = \((1.07 - 1) * 100\).

Calculate the sales tax percentage

Now that we have the expression for the sales tax percentage, we can calculate it by using the formula: Sales tax percentage = \((1.07 - 1) * 100 = 0.07 * 100 = 7\%\)

Choose the correct answer

Since the sales tax is calculated to be 7%, the correct answer is: C) \(7 \%\)

Key concepts

Algebraic Expressions

In algebra, expressions are formed using variables, constants, and operational symbols like addition, subtraction, multiplication, and division. Algebra provides a systematic approach to model real-world situations, allowing for simplification and manipulation of formulas.

In the given problem, the expression \(1.07x + 17\) represents the total cost of purchasing one keg of root beer. Here, \(x\) is a variable representing the base price of the keg, while 17 is a constant denoting the deposit for the keg.

Understanding how each part of the expression relates to real-world quantities is crucial. The term \(1.07x\) indicates the base price adjusted by a sales tax, and knowing how to interpret these components is key to solving for specific elements like the sales tax percentage.

Percentage Conversion

Converting between percentages and decimal numbers is an essential math skill, especially when dealing with financial transactions like calculating sales tax.

The basic operation to convert a decimal to a percentage is to multiply by 100. Conversely, to convert a percentage to a decimal, divide by 100.

In this scenario, the expression \(1.07x\) suggested a 7% sales tax. This required us to subtract 1 from 1.07, giving us 0.07. To express this as a percentage, we multiplied 0.07 by 100, resulting in a 7% sales tax.

These conversions are vital in calculating values accurately when analyzing costs and taxes in algebraic expressions.

Tax Factor

The tax factor is used to determine how much a price increases due to tax, expressed as a multiplier.

When a base price \(x\) is multiplied by a tax factor, it gives the total price inclusive of tax. In this context, \(1.07\) is the tax factor, indicating a 7% increase on the base price due to sales tax. The calculation can be summarized into:

• Subtract 1 from the tax factor to isolate the tax effect.
• In this case: \(1.07 - 1 = 0.07\).
• This means that 7% of the base price \(x\) is the tax.

The tax factor gives a straightforward way to include tax directly in algebraic expressions, making it a handy tool in financial mathematics.

Keg Pricing

Keg pricing involves understanding different elements that come into play when determining the sales price, such as base cost, taxes, and deposits.

In this exercise, the price of a keg is given by the expression \(1.07x + 17\), breaking down into two parts:

• \(1.07x\): This part includes the cost of the keg along with the sales tax.
• The constant 17 represents the untaxed deposit added to the price.

Understanding these components helps in analyzing how taxes and deposits affect the overall pricing of goods. Such breakdowns are essential for both consumers, who wish to know what they are paying for, and businesses, which need to correctly set prices and tax calculations.