# Percent

It's common to run into percent problems in everyday life. For example, you might notice a 15-percent discount on your favorite shoes. Or a friend has given you 50 percent of their candy bar. Since you will encounter percent a lot in math, it's important to understand what it is and how to work with it.

## What is a percent?

Percent is short for "per centum," which is Latin for "per hundred." So, for example, 15 percent (also written as 15%), means "fifteen out of a hundred."

Of course, things don't typically come in groups of exactly 100, which is why we use percents to indicate proportions. Let's say that we have discovered that 75% of North Americans eat apple pie. This means that for each 100 North Americans, 75 eat apple pie. Let's set up a proportion to show this relationship.

$75100$ = # of Apple Pie Eaters/N

where N is the population of North Americans.

Proportions are what we'll use to solve percent problems. Let's take a closer look.

## Three kinds of percent problems

Based on the statement "x percent of y is z," if any two of the three variables are given, you are able to use algebra to find the missing variable. Since any of the three: x, y, or z could be the unknown, we're going to go over three different kinds of percent problems:

1. Problems where x is the unknown

An example of this type of problem is, "What percent of 44 is 11?"

2. Problems where y is the unknown

An example is "58 is 25% of what number?"

3. Problems where z is the unknown

An example is "What is 88% of 5000?"

Each kind of problem is different, so we'll take a look at how to solve them individually.

## Solving percent problems: where x is the unknown

In the case that x is unknown, you're essentially asking a question like "What percent of 44 is 11?" or "11 is what percent of 44?". In other words, if there are 44 seats in a room (y) and 11 are occupied (z), what percent of the seats are occupied (x)?

Let's write this out as an example.

### Example 1

What percent of 44 is 11?

First, we'll write a proportion.

$x100$ = $1144$

(Note: You can reduce this fraction now if you want; however, we'll divide it at the end anyway.)

Now, let's cross-multiply.

$44x$ = $1100$

Lastly, we'll divide.

$x$ = $25$

So, 25% of 44 is 11.

Let's try one more.

### Example 2

What percent of 200 is 55?

Write the proportion.

$x100$ = $55200$

In this case, you can cross-multiply, or you could notice that the denominator on the right is 2 times the denominator on the left.

To find x, you can just divide the numerator on the right by 2.

$x100$ = $55200$

$x$ = $55$/$2$ = $27.5$

Therefore, 27.5% of 200 is 55.

## Solving percent problems: where y is the unknown

When y is unknown, you're asking a question like "58 is 25% of what number?" Another way to phrase it is "25% of what number is 58?" In this case, you know x (25%) and z (58), but you want to solve for y.

Suppose you know that there are 58 pieces of candy in a box, and these pieces represent 25% of the total candy that was once in the box. Now, you want to know how many pieces of candy were originally in the box.

### Example 1

58 is 25% of what number?

Start by writing a proportion.

$25100$ = $58y$

Now, let's cross-multiply.

$25$$y$ = $5800$

Divide.

$y$ = $232$

Therefore, 58 is 25% of 232.

Now, let's try another example.

### Example 2

348 is 40% of what number?

Write a proportion.

$40100$ = $348y$

Cross-multiply.

$40$$y$ = $34800$

Divide.

$y$ = $870$

So, 348 is 40% of 870.

## Solving percent problems: where z is the unknown

When the z is unknown, an example of the question you're asking is "What is 88% of 5000?" or "88% of 5000 is what number?" In this case, you know x (88%) and y (5000), but you want to solve for z.

Let's say that you have a goal of taking 5000 steps today as a part of your exercise routine, and so far, you've taken 88% of the total steps you want to take. How many steps have you taken?

### Example 1

What is 88% of 5000?

Write a proportion.

$88100$ = $z5000$

Cross-multiply.

$88$$z$ = $440000$

Divide.

$z$ = $4400$

In other words, 4400 is 88% of 5000.

Here is another example.

### Example 2

14% of 6375 is what number?

Write a proportion.

$14100$ = $z6375$

Cross-multiply.

$14$$z$ = $89250$

Divide.

$z$ = $892.5$

So, 14% of 6375 is 892.5.

## Converting percents to decimals

Decimals are a shorthand way of writing fractions. They are also different numerical expressions of the same values represented in percentages. For instance, the percent 25% can also be represented as the decimal 0.25.

Since there is a strong relationship between percents and decimals, it's important to know how to convert one to the other.

First, let's look at how to convert percents to decimals.

In order to convert percents to decimals, you'll remove the % sign and divide by 100.

### Example 1

Express 37% as a decimal.

First, remove the % sign.

37

Now, divide by 100.

$37100$ = $0.37$

So, 37% expressed as a decimal is 0.37.

But there is a faster way to convert a percent to a decimal. Simply move the decimal point two places to the left, and remove the % sign.

### Example 2

Express 2.5% as a decimal the quick way.

Move the decimal point two places to the left, and remove the % sign.

$0.025$

Therefore, 2.5% expressed as a decimal is 0.025.

### Example 3

Express 0.7% as a decimal using both methods.

First method:

Remove the % sign.

0.7

Divide by 100.

$0.7100$ = $0.007$

Second method:

Move the decimal point two places to the left, and remove the % sign.

$0.7$% = $0.007$

So, 0.7% expressed as a decimal is 0.007.

## Converting decimals to percents

Now, let's look at how to convert decimals to percents.

In order to convert decimals to percents, you'll multiply by 100 and add the % sign.

### Example 1

Express 0.45 as a percent.

Multiply by 100.

$0.45$ * $100$ = $45$

Add the % sign.

45%

So, 0.45 expressed as a percent is 45%.

But there is a quicker way to convert a decimal to a percent. Simply move the decimal point two places to the right, and add the % sign.

### Example 2

Express 0.03 as a percent the quick way.

Move the decimal point two places to the right, and add the % sign.

$0.03$ = 3%

Therefore, 0.03 expressed as a percent is 3%.

### Example 3

Express 0.012 as a percent using both methods.

First method:

Multiply by 100.

$0.012$ * $100$ = $1.2$

Add the % sign.

1.2%

Second method:

Move the decimal point two places to the right, and add the % sign.

$0.012$ = $1.2$%

So, 0.012 expressed as a percent is 1.2%.

## Converting percents to fractions

Fractions are another way of expressing proportions, and percents are a type of proportion. Since fractions and percents both represent the same idea (a part of a whole), you can convert between them.

In order to convert percents to fractions, you'll write the percent as a fraction with a denominator of 100 and then simplify the fraction if possible.

### Example 1

Express 25% as a fraction.

Write the percent as a fraction with a denominator of 100.

$25100$

Simplify the fraction by dividing both the numerator and denominator by their greatest common factor, which is 25.

$14$

So, 25% expressed as a fraction is 1/4.

Now, let's try another example.

### Example 2

Express 60% as a fraction.

Write the percent as a fraction with a denominator of 100.

$60100$

Simplify the fraction.

Divide both the numerator and denominator by their greatest common factor, which is 20.

$35$

So, 60% expressed as a fraction is 3/5.

Remember, you can also convert fractions to percents by dividing the numerator by the denominator and then multiplying by 100.

## Converting fractions to percents

As mentioned earlier, you can convert between fractions and percents. To convert a fraction to a percent, you'll divide the numerator by the denominator and then multiply by 100.

### Example 1

Express the fraction 3/8 as a percent.

Divide the numerator (3) by the denominator (8).

$38$ = 3/8

Multiply the result by 100 to convert to a percent.

3/8 * 100 = 37.5%

So, 3/8 expressed as a percent is 37.5%.

### Example 2

Express the fraction 7/10 as a percent.

Divide the numerator (7) by the denominator (10).

$710$ = 7/10

Multiply the result by 100 to convert to a percent.

7/10 * 100 = 70%

So, 7/10 expressed as a percent is 70%.

These are the basic concepts and methods for working with percents, fractions, and decimals. Understanding how to convert between them and solve percent problems is fundamental for many mathematical and real-world applications.

If you have specific questions or would like to see more examples, please feel free to ask!