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Percent Concepts

A candy bar that normally costs 75¢ is on sale for 60¢. What is the percent of the decrease?

A student took a test and got 23 out of 25 problems right. What percent is that?

Find the answer using two different methods: first division and decimal place scoot, then by altering the denominator to be 100.

I am 22 years old. My brother is 17. His age is what percentage of mine?

The biggest political donor between 1989 and 2010 was AT&T Inc., which donated $44,027,485. 44% went to the Democratic Party. How many dollars is that?

Instead of diving into percents suddenly, let's approach the topic slowly. First we'll think about how our society is used to using numerical scales to rate how nice things are.

Here is table to complete. We'll just make up answers.

Example 1

As a group, let's rate the following on a scale of:

Desert 1 to 5 1 to 10 1 to 100 Carrot Cake Banana Runts Dark Chocolate

In general there is no nice name in English for how something rates on a scale of 1 to 5, or a scale of 1 to 10. (There are some exceptions. The popularity of Auto Club travel guides means most people know a "4 star hotel" is rated on a scale of 1 to 5.)

There is a nice name in English for how something rates on a scale of 1 to 100. This used to be called *per cent*, meaning "per 100" since the word "cent" means "100". But over time those words became customarily squished together, and now we say percent.

Since this is a math class, we should be more formal.

Definition

Percentmeans "out of 100". We know four ways to do "out of 100" with arithmetic:

- ÷ 100
- two decimal point scoots to the left
- changing a whole numbr into a fraction with denominator 100
- ×
^{1}/_{100}

We could also use a grid of 100 boxes, or a circle with 100 tic marks, to draw pictures for "out of 100".

Often percents come as a set of values that add up to 100%, and a missing value must be found.

Example 2

How much orange juice is sold?

But this almost never happens in our textbook problems.

Some percents have shortcuts for finding that percent of a value.

- To find 10% of a value, move the decimal point
**one place to the left**. - To find 20% of a value, first find 10% of the value and then
**double**that. - To find 30% of a value, first find 10% of the value and then
**triple**that. - To find 5% of a value, first find 10% of the value and then
**halve**that. - To find 15% of a value, first find 10% of the value, then find 5% of the value, and then
**add those together**.

Before continuing, please make sure you understand a big difference between two concepts! We just said that being in percent format is equivalent to *two* decimal point scoots. Why does the shortcut for finding 10% then involve *only one* decial point scoot?

Let's do some examples using tipping at a restaurant, which for many people is the most common application of percents.

Example 3

A restaurant meal costs $30. You decide to tip 20% because the service was good. How much is the tip?

Example 4

A restaurant meal costs $30. You decide to tip 15% because the service was average. How much is the tip?

Example 5

A restaurant meal costs $30. You decide to tip 10% because the service was abysmal. How much is the tip?

Tipping is one instance when estimation is very useful. If my restaurant bill was $21.87, the tip will not change much if I estimate based on $22.00.