• Unit 7: Percentages

We use percentages every day. For example, how about when your aunt's coffee shop advertised pumpkin spice lattes for 50% off – you and your aunt knew this meant half off, and so did the individuals in the ensuing rush of new customers. In this unit, we explore how to compute percentages. What does this mean, and how do we think about percentages more generally?

In this unit, we answer this question and lots of others that involve percentage computations and applications. For example, we will convert between percents and fractions or decimals and learn about percentage increases and decreases. We will also explore how to calculate some real-world uses, such as restaurant tips and sale prices.

Completing this unit should take you approximately 2 hours.

• 7.1: Describing the Meaning of Percent

What is a percentage or a percent? One clue is in the name itself: per-cent. We frequently use the word per to compare or divide quantities, as in rates and ratios, and the leftover word cent means centum or one hundred in Latin. The term percent refers to a quantity expressed as a fraction (per) whose denominator is 100 (cent).

For example, we can write $0.5 = \frac{1}{2} = \frac{50}{100} = 50 \%$. We pronounce the percentage symbol, %, percent. Isn't it telling that it looks a lot like our division symbol $\div$?

We see percentages all the time in the real world – frequently in sales at stores. During a sale, a store might mark their shirts 50% down so that a $20 shirt now costs$10. A week later, the store may announce customers can take an additional 20% off the sale price. How do you determine the new discounted price of the shirt? (Try it!)

As another example, we can say that $15/100$ is 15% or that $0.25$ is 25%.

• 7.2: Converting Between Decimals, Fractions, and Percents

As we discussed above, a percentage is just a new name for our dear old friend the fraction. Of course, we can also write these friendly rationals in decimal notation. All of this tells us we can convert between all three forms of expression.

• 7.3: Determine the Percent Given Two Numbers

It can be useful to express ratios or fraction information in terms of percentages. For example, let's say I have a class of 23 students, and 11 of them earned an A on a test. What percent of the class earned an A?

It is also useful to know how to reverse this process. When given percentage information, we may want to express it as fraction information. For example, if a pair of pants that normally costs \$32 is on sale for 20% off, what is the new, actual price? Can you figure this out?