Percent, Sales Tax, and Discounts: Study Notes for Liberal Arts Math

Percent, Sales Tax, and Discounts

Basics of Percent

Percents are a fundamental concept in mathematics, representing numbers as parts of 100. The term percent literally means "per hundred." Understanding how to convert between fractions, decimals, and percents is essential for solving real-world problems involving proportions.

• Expressing a Fraction as a Percent: Divide the numerator by the denominator, multiply the result by 100, and add a percent sign.

• Expressing a Decimal as a Percent: Move the decimal point two places to the right and attach a percent sign.

• Expressing a Percent as a Decimal: Move the decimal point two places to the left and remove the percent sign.

Example: Expressing a Fraction as a Percent

Suppose you have the fraction \( \frac{5}{8} \). To express it as a percent:

• Step 1: Divide 5 by 8 to get 0.625.

• Step 2: Multiply 0.625 by 100 to get 62.5.

• Step 3: Add the percent sign: 62.5%.

Example: Expressing a Decimal as a Percent

To convert 0.47 to a percent:

• Move the decimal two places right: 47.

• Add the percent sign: 47%.

Example: Expressing a Percent as a Decimal

• 19% becomes 0.19.

• 180% becomes 1.80 or 1.8.

Percent, Sales Tax, and Discounts

Many real-world applications of percent involve calculating sales tax and discounts. The key formula is:

$\text{Percent of a quantity} = \text{Percent (as decimal)} \times \text{Quantity}$

The word "of" in mathematics often implies multiplication. This formula is used to determine sales tax collected by states, counties, and cities on sales items.

Example: Calculating Sales Tax

If the local sales tax rate is 7.5% and you purchase a bicycle for $894:

• Tax paid: $\text{Tax} = 0.075 \times 894 = 67.05$

• Total cost: $894 + 67.05 = 961.05$

Percent and Sales Price

Businesses often reduce prices to attract customers or clear inventory. The discount rate is a percent of the original price.

Example: Calculating Discount and Sale Price

A gaming laptop with an original price of $1460 is on sale at 15% off:

• Discount amount: $\text{Discount} = 0.15 \times 1460 = 219$

• Sale price: $1460 - 219 = 1241$

Percent and Change

When a quantity changes, its percent increase or decrease can be calculated as follows:

• Find the fraction representing the change: $\frac{\text{Change}}{\text{Original Value}}$

• Express this fraction as a percent by multiplying by 100.

Example: Percent Increase and Decrease

Suppose world population was 6 billion in 2000. If projections show a population of 30 billion in 2150 (high projection):

• Percent increase: $\frac{30 - 6}{6} = 4$ (or 400%)

If projections show a population of 2 billion in 2150 (low projection):

• Percent decrease: $\frac{6 - 2}{6} = \frac{4}{6} \approx 0.667$ (or 66.7%)

Abuses of Percent

Percents can be misused or misunderstood, especially when reporting changes. For example, reporting a percent increase without proper calculation can lead to misleading statements.

Example: Percents of Percents

If a TV show audience increases from 4% to 13%, the percent increase is:

• Percent increase: $\frac{13 - 4}{4} = 2.25$ (or 225%)

Reporting this as "more than 300%" is incorrect.

Summary Table: Percent Conversions

Conversion	Steps	Example
Fraction to Percent	Divide, multiply by 100, add %	\( \frac{3}{4} = 0.75 \times 100 = 75\% \)
Decimal to Percent	Move decimal right two places, add %	0.25 = 25%
Percent to Decimal	Move decimal left two places, remove %	80% = 0.80