Problem 9 - Algebra 1

Question

Solve the percent problem. The price of a book without tax is \(\$ 5.99\) and the sales tax rate is \(6 \% .\) Find the amount of the tax by using an equation of the form \(a=p b\) and by using a proportion. How are the two methods similar?

Step-by-Step Solution

Answer

The amount of sales tax is \$0.3594. Both the percentage equation and the proportion are mathematically equivalent and represent the tax as a fraction of the original price.

Step 1: Define the variables in the equation

In the percentage equation a=p*b, 'a' represents the amount of tax, 'p' is the price of the book (pre-tax), and 'b' is the tax rate. Here, p=\$5.99 and b=6/100=0.06 (since tax rate is given in percentage, we will convert it into decimal).

Step 2: Use the equation to calculate sales tax

Substitute the given values into the equation to find 'a'. a = p*b = \$5.99 * 0.06 = \$0.3594. This is the amount of sales tax on the book.

Step 3: Define the variables in the proportion

In a proportion, if 'x' is the amount of tax applied to the price of the book, which is \$5.99, then the proportion comparing the price of the book to the tax can be written as x/\$5.99 = 6/100.

Step 4: Use the proportion to calculate sales tax

Solving the proportion for 'x' gives x = (6/100)*\$5.99 = \$0.3594. This is the amount of sales tax on the book calculated using proportion.

Step 5: Analyze the similarity between the two methods

Both the percentage equation and the proportion represent the same mathematical relationship. Both are used to express the sales tax as a fraction of the original price, and that fraction is the tax rate.

Key Concepts

Sales Tax Calculation

Sales tax calculation is an essential skill that helps you determine the additional cost of an item due to tax. When buying a product, you might see a listed price. But often, there's a sales tax added to this price.

• The listed price is the cost of the item before any tax is added.
• The sales tax is calculated by multiplying the listed price by the tax rate (expressed in decimal form).

To find out how much extra you need to pay because of the tax, simply multiply the price by the tax rate. If the sales tax rate is 6% for an item costing $5.99, you multiply $5.99 by 0.06.
This results in a sales tax amount of $0.3594. Understanding this helps you anticipate the total cost of your purchases.

Percentage Equation

The percentage equation is a useful tool for calculating parts of a whole, like sales tax. It uses the formula: \(a = p \times b\), where:

• \(a\) is the part you're finding (amount of tax).
• \(p\) is the whole or total (pre-tax price).
• \(b\) is the percentage in decimal form (tax rate divided by 100).

To solve our problem, plug in the values: the price before tax \(p = 5.99\) and the tax rate \(b = 0.06\). The equation becomes: \(a = 5.99 \times 0.06\).
Using this equation gives the tax as \$0.3594. This method is direct and efficient for solving many real-world problems.

Proportion Method

The proportion method offers another way to solve percent problems. It involves setting up a ratio to find the unknown value. In our example, we want to find the sales tax as part of the whole price.Set the proportion: \(\frac{x}{5.99} = \frac{6}{100}\), where:

• \(x\) is the amount of tax.
• 5.99 is the pre-tax price of the book.

Cross-multiplication helps find \(x\), resulting in \(x = \frac{6}{100} \times 5.99\).
This solves to \$0.3594, showing that proportions provide a flexible way to approach many kinds of mathematical problems, especially when dealing with percentages.

Mathematical Relationships

Mathematical relationships help us understand the connections between numbers and variables in problems. In both methods described, the relationship between the book's price, the tax rate, and the sales tax is carefully maintained.

• The percentage equation stands for one direct relationship where multiplication instantly connects all components.
• The proportion method showcases a more comparative approach, presenting the tax as a fraction of the whole.

Both methods reveal the underlying concept that the tax is a part or percentage of the item's pre-tax cost.
They reinforce the idea that different mathematical approaches can yield the same result by adhering to these foundational relationships.