Question

A pair of gloves retails for \(\$ 17.89\). If the sales tax is \(8 \frac{1}{2} \%\), what is the total cost of the gloves? A. \(\$ 19.32\) B. \(\$ 19.41\) C. \(\$ 26.39\) D. \(\$ 32.20\)

Short answer

B. \( \$ 19.41 \)

Step-by-step solution

- Convert Percent to Decimal

First, convert the sales tax percentage to a decimal for easier calculation. The sales tax is given as \(8 \frac{1}{2} \%\). To convert this to a decimal, divide by 100: \[ 8 \frac{1}{2} \% = 8.5\% = \frac{8.5}{100} = 0.085 \]

- Calculate the Sales Tax Amount

Next, calculate the amount of sales tax on the pair of gloves. Multiply the retail price by the decimal form of the sales tax: \[\text{Sales Tax} = \text{Retail Price} \times \text{Sales Tax Rate} = 17.89 \times 0.085 = 1.52065 \]

- Calculate the Total Cost

Add the sales tax amount to the retail price to find the total cost: \[\text{Total Cost} = \text{Retail Price} + \text{Sales Tax} = 17.89 + 1.52065 = 19.41065 \]

- Round to the Nearest Cent

Finally, round the total cost to the nearest cent. Since the third decimal place is 0, round down: \[19.41065 \approx 19.41\]

- Select the Correct Answer

Match the rounded total cost of \(19.41\) with the provided options. The correct answer is \textbf{(B) \$19.41}.

Key concepts

percent to decimal conversion

Converting a percentage to a decimal is an essential skill in calculating sales tax. Start by understanding that percentages are out of 100. For the given problem, we have a sales tax of \(8 \frac{1}{2}\%\). To convert this to a decimal:

1. Change the mixed number to an improper fraction: \(8 \frac{1}{2}\% = 8.5\%\).
2. Divide by 100: \(\frac{8.5}{100} = 0.085\).

This means that \(8.5\%\) as a decimal is \(0.085\).

multiplying decimals

Multiplying decimals involves a precise process. Let's see how it applies to our problem:

To find the amount of sales tax on the gloves, we'll multiply the retail price by the sales tax rate in decimal form. This is:

\( \text{Sales Tax} = \text{Retail Price} \times \text{Sales Tax Rate} \).
In our example: \(\$17.89 \times 0.085\).

Steps to multiply decimals:

• Ignore the decimals and multiply the numbers as whole numbers: \(1789 \times 85 = 152065\).
• Count the total number of decimal places in the factors (here, 2 in 17.89 and 3 in 0.085, totaling 5 decimal places).
• Place the decimal in the product so that it has the same number of decimal places: \(1.52065\).

The calculated sales tax is \(\$1.52065\).

rounding numbers

Rounding numbers is about simplifying them to a desired precision. Here, we need to round the total cost to the nearest cent, which means two decimal places.

Follow these steps:

• Identify the digit in the third decimal place.
• If this digit is 5 or greater, round up the second decimal place by one.
• If this digit is less than 5, leave the second decimal place as it is.

For example, our calculated total cost before rounding is \(19.41065\). The third decimal place is 0, so we round down:
\(19.41065 \approx 19.41\).
The rounded total is \(\$19.41\).

total cost calculation

Calculating the total cost involves adding the retail price to the sales tax amount. Let's walk through this final step:

1. Start with the retail price: \(\$17.89\).
2. Add the calculated sales tax amount: \(\$1.52065\).

This gives us a preliminary total cost: \( \text{Total Cost} = 17.89 + 1.52065 = 19.41065 \).

We then round this result to the nearest cent to get:
\( \$19.41 \).
This is the final total cost of the gloves, including sales tax. So the correct answer is \( \text{(B)} \$19.41 \).