Question

Sydney buys her textbooks online and spends \(\$ 1147\). The sales tax in her town is \(5 \frac{3}{4} \%\), but the bookseller does not charge tax on Internet sales. Before she pays for shipping, how much has she saved by not paying sales tax? (Round to the nearest cent.) A. \(\$ 6.60\) B. \(\$ 57.35\) C. \(\$ 65.95\) D. \(\$ 68.82\)

Short answer

Option C. \(\$65.95\)

Step-by-step solution

Convert the sales tax percentage to a decimal

The sales tax is given as a mixed number: \(5 \frac{3}{4} \%\). First, convert it to an improper fraction: \(5 \frac{3}{4} = \frac{20}{4} + \frac{3}{4} = \frac{23}{4}\). Now, convert this fraction to a decimal by dividing: \(\frac{23}{4} = 5.75\). Finally, convert the percentage to a decimal by dividing by 100: \(5.75 \% = 0.0575\).

Calculate the amount saved by not paying sales tax

To find out how much Sydney saved by not paying the sales tax, multiply the total amount spent by the decimal form of the sales tax: \(1147 \times 0.0575\).

Perform the multiplication

Multiply \(1147 \times 0.0575 = 65.9525\).

Round to the nearest cent

Since we need to round to the nearest cent, look at the third decimal place: \(65.95\underline{2}5\). The third decimal place is 2, which is less than 5, so we round down to 65.95.

Choose the correct answer

The rounded answer is \(65.95\). Therefore, the amount saved by not paying sales tax is \(C. \$65.95\).

Key concepts

percentage to decimal conversion

When you're dealing with percentages, it's often necessary to convert them to decimals to make calculations easier. For example, let's take Sydney's problem with her sales tax. The sales tax is given as a mixed number: \(5 \frac{3}{4}\)%.

The mixed number can be converted to an improper fraction first. To do this, multiply the whole number (5) by the fraction's denominator (4) and then add the numerator (3). This gives \(\frac{20}{4} + \frac{3}{4} = \frac{23}{4}\).

Next, divide the numerator (23) by the denominator (4) to get the decimal: \(\frac{23}{4}=5.75\). Finally, since it's a percentage, divide this result by 100 to convert it to a decimal: \(5.75\% = 0.0575\).

This step is crucial for making further calculations, like figuring out the tax savings.

multiplication of decimals

Once you have your percentage converted to a decimal, you can then use this decimal to find out specific amounts, like how much Sydney saved on her textbooks. The sales tax as a decimal was \(0.0575\).

To find the amount saved, multiply the total amount spent by the sales tax decimal. Here, Sydney spent \$1147. So, you calculate that by: \(1147 \times 0.0575\).

When multiplying decimals:

• Ignore the decimal points at first and multiply the numbers as if they were whole numbers.
• After the multiplication, count the total number of decimal places in the factors combined.
• Place the decimal point in the result, ensuring it has the same number of decimal places you counted.

By performing the operation, we get \(1147 \times 0.0575 = 65.9525\).

rounding to the nearest cent

After you've multiplied and arrived at a result, sometimes you need to round to a specific place value. In this case, we need to round to the nearest cent, which is the hundredths place in a decimal.

Look at the third decimal place of your number, which is \(65.9525\). The digit here is 2.

• If this digit is 5 or higher, you would round up, increasing the hundredths place by 1.
• If this digit is less than 5, you would round down, leaving the hundredths place as it is.

Here, 2 is less than 5, so you round down, and thus the amount is rounded to \$65.95.

This step is important to provide precise monetary values.

mixed number conversion

Dealing with mixed numbers is another essential math skill. A mixed number consists of a whole number and a fraction. In Sydney's problem, the sales tax was given as \(5 \frac{3}{4}\)%.

To use this in calculations, you first need to convert it to an improper fraction.

• Multiply the whole number part (5) by the denominator of the fractional part (4).
• Add the numerator (3) to this result to get the new numerator.

So, \(5\cdot4+3=23\).

The fraction form of the mixed number is thus \(\frac{23}{4}\). From here, you can proceed to convert it to a decimal as previously discussed.

Mastering mixed number conversions simplifies various calculations and enhances your mathematical proficiency.