Problem 66 - Introductory and Intermediate Algebra for College Students 4th

Question

Suppose that the local sales tax rate is \(7 \%\) and you buy a graphing calculator for \(\$ 96\) a. How much tax is due? b. What is the calculator's total cost?

Step-by-Step Solution

Answer

The sales tax due on the calculator is \$6.72, making the total cost \$102.72.

Step 1: Convert the Tax Rate from Percentage to Decimal

First, convert the tax rate from percentage to decimal. This can be done by dividing the tax rate by 100. So, \(7 \% = \frac{7}{100} = 0.07\).

Step 2: Calculate the Amount of Tax Due

Now multiply the cost of the calculator by the decimal value of the tax rate. This will give the tax amount due. So, \( \$96 * 0.07 = \$6.72\).

Step 3: Calculate the Total Cost

The total cost of the calculator is simply the sum of its original price and the tax amount due. So, \(\$96 + \$6.72 = \$102.72\).

Key Concepts

Algebraic Problem-Solving

Algebra is often about finding the unknown or putting real-life variables into equations and then solving them. In the given exercise, we tackle a real-world situation of calculating sales tax for a purchase. This problem requires us to use basic algebraic skills, such as converting percentages and performing simple arithmetic operations.

Understanding the relationship between different mathematical concepts is crucial. For example, connecting the tax rate (a percentage) with the cost of an item (a price in dollars), and then using multiplication to find the resulting tax amount is a typical problem-solving scenario. Break down the problem into smaller, manageable steps – convert the rate, multiply to get the tax, and add to find the total cost. By approaching algebraic problems systematically, you make them less intimidating and easier to solve.

Converting Percentages to Decimals

Calculating tax or interest rates often requires you to convert percentages to decimals. This is because percentages are a way of expressing a number as a fraction of 100. To transform a percentage into a decimal, divide it by 100. This conversion is essential in algebra and financial calculations because it allows you to work with numbers in a more straightforward arithmetic format.

For instance, a sales tax rate of \(7\%\) becomes \(0.07\) as a decimal. Remember, moving the decimal point two places to the left is the quick trick to convert a percentage to a decimal. This process is not only crucial for solving the exercise at hand but is also a widely applicable skill, useful in various contexts ranging from science to economics.

Calculating Total Cost

When you are buying something, the total cost is not always the price tag you see. Sales tax, tips, or discounts can change your final payout. Understanding how to calculate the total cost is not only essential for algebra problems but for everyday financial literacy as well.

The total cost is the sum of the original price and any additional charges like tax. To calculate it, first find each of these amounts, then add them together. Using our example, you multiply the item's price by the sales tax rate expressed as a decimal to get the tax amount, and then you add it to the original price to find out what you'll actually pay at the register – in this case, \(\$96 + \$6.72 = \$102.72\). Learning to calculate the total cost teaches you to anticipate expenses effectively, which is a useful skill for budgeting and financial planning.