Understand Percent

Use the Definition of Percent

How many cents are in one dollar? There are $100$ cents in a dollar. How many years are in a century? There are $100$ years in a century. Does this give you a clue about what the word “percent” means? It is really two words, “per cent,” and means per one hundred. A percent is a ratio whose denominator is $100 .$ We use the percent symbol $%,$ to show percent.

According to data from the American Association of Community Colleges $(2015),$ about $57%$ of community college students are female. This means $57$ out of every $100$ community college students are female, as Figure 1 shows. Out of the $100$ squares on the grid, $57$ are shaded, which we write as the ratio $\frac{57}{100} .$

The figure shows a hundred flat with 57 units shaded. — Among every $100$ community college students, $57$ are female.

Similarly, $25%$ means a ratio of $\frac{25}{100}, 3%$ means a ratio of $\frac{3}{100}$ and $100%$ means a ratio of $\frac{100}{100} .$ In words, "one hundred percent" means the total $100%$ is $\frac{100}{100},$ and since $\frac{100}{100} = 1,$ we see that $100%$ means $1$ whole.

According to the Public Policy Institute of California $(2010), 44%$ of parents of public school children would like their youngest child to earn a graduate degree. Write this percent as a ratio.

Solution

This table illustrates the step-by-step conversion of a percentage (44%) into a ratio.
The amount we want to convert is 44%.	$44%$
Write the percent as a ratio. Remember that percent means per 100.	$\frac{44}{100}$

In $2007,$ according to a U.S. Department of Education report, $21$ out of every $100$ first-time freshmen college students at $4-year$ public institutions took at least one remedial course. Write this as a ratio and then as a percent.

Solution

Steps demonstrating the conversion of a quantity out of 100 into its ratio and percentage forms.
The amount we want to convert is $21$ out of $100$ .	$21$ out of $100$
Write as a ratio.	$\frac{21}{100}$
Convert the 21 per 100 to percent.	$21%$

Convert Percents to Fractions and Decimals

Since percents are ratios, they can easily be expressed as fractions. Remember that percent means per $100,$ so the denominator of the fraction is $100 .$

Convert each percent to a fraction:

ⓐ $36%$
ⓑ $125%$

Solution

This table demonstrates the step-by-step conversion of a percentage (36%) into its simplified fraction form.
ⓐ
	$36%$
Write as a ratio with denominator 100.	$\frac{36}{100}$
Simplify.	$\frac{9}{25}$

Steps demonstrating the conversion of a percentage into a simplified fraction.
ⓑ
	$125%$
Write as a ratio with denominator 100.	$\frac{125}{100}$
Simplify.	$\frac{5}{4}$

The previous example shows that a percent can be greater than $1 .$ We saw that $125%$ means $\frac{125}{100},$ or $\frac{5}{4} .$ These are improper fractions, and their values are greater than one.

Convert each percent to a fraction:

ⓐ $24.5%$
ⓑ $33 \frac{1}{3} %$

Solution

This table illustrates the step-by-step process of converting 24.5% into its simplest fractional form, demonstrating each mathematical operation.
ⓐ
	$24.5%$
Write as a ratio with denominator 100.	$\frac{24.5}{100}$
Clear the decimal by multiplying numerator and denominator by 10.	$\frac{24.5 (10)}{100 (10)}$
Multiply.	$\frac{245}{1000}$
Rewrite showing common factors.	$\frac{5 \cdot 49}{5 \cdot 200}$
Simplify.	$\frac{49}{200}$

Steps to convert the percentage 33 1/3% to its equivalent fraction 1/3, detailing each mathematical transformation.
ⓑ
	$33 \frac{1}{3} %$
Write as a ratio with denominator 100.	$\frac{33 \frac{1}{3}}{100}$
Write the numerator as an improper fraction.	$\frac{\frac{100}{3}}{100}$
Rewrite as fraction division, replacing 100 with $\frac{100}{1}$ .	$\frac{100}{3} \div \frac{100}{1}$
Multiply by the reciprocal.	$\frac{100}{3} \cdot \frac{1}{100}$
Simplify.	$\frac{1}{3}$

In Decimals, we learned how to convert fractions to decimals. To convert a percent to a decimal, we first convert it to a fraction and then change the fraction to a decimal.

Convert each percent to a decimal:

ⓐ $6%$
ⓑ $78%$

Solution

Because we want to change to a decimal, we will leave the fractions with denominator $100$ instead of removing common factors.

This table illustrates the step-by-step conversion of 6% into its equivalent fractional (6/100) and decimal (0.06) representations.
ⓐ
	$6%$
Write as a ratio with denominator 100.	$\frac{6}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.06$

Steps to convert a percentage (78%) into its equivalent ratio and decimal forms.
ⓑ
	$78%$
Write as a ratio with denominator 100.	$\frac{78}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.78$

Convert each percent to a decimal:

ⓐ $135%$
ⓑ $12.5%$

Solution

Steps for converting a percentage to its fraction and decimal representations.
ⓐ
	$135%$
Write as a ratio with denominator 100.	$\frac{135}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$1.35$

Step-by-step conversion of 12.5% from percentage to its decimal equivalent.
ⓑ
	$12.5%$
Write as a ratio with denominator 100.	$\frac{12.5}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.125$

Let's summarize the results from the previous examples in Table 11, and look for a pattern we could use to quickly convert a percent number to a decimal number.

Percent	Decimal
$6%$	$0.06$
$78%$	$0.78$
$135%$	$1.35$
$12.5%$	$0.125$

Do you see the pattern?

To convert a percent number to a decimal number, we move the decimal point two places to the left and remove the $%$ sign. (Sometimes the decimal point does not appear in the percent number, but just like we can think of the integer $6$ as $6.0,$ we can think of $6%$ as $6.0% .$ ) Notice that we may need to add zeros in front of the number when moving the decimal to the left.

Figure 2 uses the percents in Table 11 and shows visually how to convert them to decimals by moving the decimal point two places to the left.

The figures shows two columns and five rows . The first row is a header row and it labels each column “Percent” and “Decimal”. Under the “Percent” column are the values: 6%, 78%, 135%, 12.5%. Under the “Decimal” column are the values: 0.06, 0.78, 1.35, 0.125. There are two jumps for each percent to show how to convert it to a decimal.

Among a group of business leaders, $77%$ believe that poor math and science education in the U.S. will lead to higher unemployment rates.

Convert the percent to: ⓐ a fraction ⓑ a decimal

Solution

This table illustrates the conversion of 77% into its equivalent ratio form, 77/100, by expressing it with a denominator of 100.
ⓐ
	$77%$
Write as a ratio with denominator 100.	$\frac{77}{100}$

Demonstrates converting the fraction 77/100 to the decimal 0.77 by dividing the numerator by the denominator.
ⓑ
	$\frac{77}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.77$

There are four suits of cards in a deck of cards—hearts, diamonds, clubs, and spades. The probability of randomly choosing a heart from a shuffled deck of cards is

25% .

Convert the percent to:

ⓐ a fraction
ⓑ a decimal

The figure shows someone holding a deck of cards. — (credit: Riles32807, Wikimedia Commons)

Solution

Steps to convert a percentage (25%) into a simplified fraction (1/4).
ⓐ
	$25%$
Write as a ratio with denominator 100.	$\frac{25}{100}$
Simplify.	$\frac{1}{4}$

This table illustrates the process of converting the fraction 1/4 to its decimal equivalent, 0.25, by dividing the numerator by the denominator.
ⓑ	$\frac{1}{4}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.25$

Convert Decimals and Fractions to Percents

To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is $100,$ it is easy to change that fraction to a percent.

Convert each decimal to a percent: ⓐ $0.05$ ⓑ $0.83$

Solution

Demonstrates converting the decimal 0.05 into its equivalent fraction (5/100) and percentage (5%) forms with step-by-step instructions.
ⓐ
	$0.05$
Write as a fraction. The denominator is 100.	$\frac{5}{100}$
Write this ratio as a percent.	$5%$

Steps showing the conversion of the decimal 0.83 to its equivalent fraction (83/100) and percentage (83%).
ⓑ
	$0.83$
The denominator is 100.	$\frac{83}{100}$
Write this ratio as a percent.	$83%$

To convert a mixed number to a percent, we first write it as an improper fraction.

Convert each decimal to a percent: ⓐ $1.05$ ⓑ $0.075$

Solution

Illustrates the step-by-step conversion of the decimal 1.05 into a mixed fraction, improper fraction, and a percentage.
ⓐ
	$1.05$
Write as a fraction.	$1 \frac{5}{100}$
Write as an improper fraction. The denominator is 100.	$\frac{105}{100}$
Write this ratio as a percent.	$105%$

Notice that since $1.05 > 1,$ the result is more than $100%.$

Steps to convert a decimal (0.075) into a fraction and then a percentage.
ⓑ
	$0.075$
Write as a fraction. The denominator is 1,000.	$\frac{75}{1,000}$
Divide the numerator and denominator by 10, so that the denominator is 100.	$\frac{7.5}{100}$
Write this ratio as a percent.	$7.5%$

Let's summarize the results from the previous examples in Table 20 so we can look for a pattern.

Decimal	Percent
$0.05$	$5%$
$0.83$	$83%$
$1.05$	$105%$
$0.075$	$7.5%$

Do you see the pattern? To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.

Figure 4 uses the decimal numbers in Table 20 and shows visually to convert them to percents by moving the decimal point two places to the right and then writing the $%$ sign.

The figure shows two columns and five rows. The first row is a header row and it labels each column “Decimal” and “Percent”. Under the “Decimal” column are the values: 0.05, 0.83, 1.05, 0.075, 0.3. Under the “Percent” column are the values: 5%, 83%, 105%, 7.5%, 30%. There are two jumps for each decimal to show how to convert it to a percent.

In Decimals, we learned how to convert fractions to decimals. Now we also know how to change decimals to percents. So to convert a fraction to a percent, we first change it to a decimal and then convert that decimal to a percent.

Convert each fraction or mixed number to a percent: ⓐ $\frac{3}{4}$ ⓑ $\frac{11}{8}$ ⓒ $2 \frac{1}{5}$

Solution

To convert a fraction to a decimal, divide the numerator by the denominator.

ⓐ
Change to a decimal.	$\frac{3}{4}$
Write as a percent by moving the decimal two places.
	$75%$

ⓑ
Change to a decimal.	$\frac{11}{8}$
Write as a percent by moving the decimal two places.
	$137.5%$

ⓒ
Write as an improper fraction.	$2 \frac{1}{5}$
Change to a decimal.	$\frac{11}{5}$
Write as a percent.
	$220%$

Notice that we needed to add zeros at the end of the number when moving the decimal two places to the right.

Sometimes when changing a fraction to a decimal, the division continues for many decimal places and we will round off the quotient. The number of decimal places we round to will depend on the situation. If the decimal involves money, we round to the hundredths place. For most other cases in this book we will round the number to the nearest thousandth, so the percent will be rounded to the nearest tenth.

Convert $\frac{5}{7}$ to a percent.

Solution

To change a fraction to a decimal, we divide the numerator by the denominator.

Conversions of a fraction (5/7) to its decimal (rounded to thousandths) and percentage forms.
	$\frac{5}{7}$
Change to a decimal—rounding to the nearest thousandth.	$0.714$
Write as a percent.	$71.4%$

When we first looked at fractions and decimals, we saw that some fractions converted to a repeating decimal. For example, when we converted the fraction $\frac{4}{3}$ to a decimal, we wrote the answer as $1 . \bar{3} .$ We will use this same notation, as well as fraction notation, when we convert fractions to percents in the next example.

An article in a medical journal claimed that approximately $\frac{1}{3}$ of American adults are obese. Convert the fraction $\frac{1}{3}$ to a percent.

Solution

	$\frac{1}{3}$
Change to a decimal.	$Long division showing 1 divided by 3 equals the repeating decimal 0.333... This illustrates how a fraction translates to an infinite decimal.$
Write as a repeating decimal.	$0.333 \dots$
Write as a percent.	$33 \frac{1}{3} %$

We could also write the percent as $33 . \overline{3} %$ .

Key Concepts

Convert a percent to a fraction.
1. Write the percent as a ratio with the denominator 100.
2. Simplify the fraction if possible.
Convert a percent to a decimal.
1. Write the percent as a ratio with the denominator 100.
2. Convert the fraction to a decimal by dividing the numerator by the denominator.
Convert a decimal to a percent.
1. Write the decimal as a fraction.
2. If the denominator of the fraction is not 100, rewrite it as an equivalent fraction with denominator 100.
3. Write this ratio as a percent.
Convert a fraction to a percent.
1. Convert the fraction to a decimal.
2. Convert the decimal to a percent.

Practice Makes Perfect

Use the Definition of Percents

In the following exercises, write each percent as a ratio.

In $2014,$ the unemployment rate for those with only a high school degree was $6.0% .$

Solution

$\frac{6}{100}$

In $2015,$ among the unemployed, $29%$ were long-term unemployed.

The unemployment rate for those with Bachelor's degrees was $3.2%$ in $2014 .$

Solution

$\frac{32}{1000}$

The unemployment rate in Michigan in $2014$ was $7.3% .$

In the following exercises, write as

ⓐ a ratio and
ⓑ a percent

$57$ out of $100$ nursing candidates received their degree at a community college.

Solution

ⓐ $\frac{57}{100}$
ⓑ $57%$

$80$ out of $100$ firefighters and law enforcement officers were educated at a community college.

$42$ out of $100$ first-time freshmen students attend a community college.

Solution

ⓐ $\frac{42}{100}$
ⓑ $42%$

$71$ out of $100$ full-time community college faculty have a master's degree.

Convert Percents to Fractions and Decimals

In the following exercises, convert each percent to a fraction and simplify all fractions.

$4%$

Solution

$\frac{1}{25}$

$8%$

$17%$

Solution

$\frac{17}{100}$

$19%$

$52%$

Solution

$\frac{13}{25}$

$78%$

$125%$

Solution

$\frac{5}{4}$

$135%$

$37.5%$

Solution

$\frac{3}{8}$

$42.5%$

$18.4%$

Solution

$\frac{23}{125}$

$46.4%$

$9 \frac{1}{2} %$

Solution

$\frac{19}{200}$

$8 \frac{1}{2} %$

$5 \frac{1}{3} %$

Solution

$\frac{4}{75}$

$6 \frac{2}{3} %$

In the following exercises, convert each percent to a decimal.

$5%$

Solution

0.05

$9%$

$1%$

Solution

0.01

$2%$

$63%$

Solution

0.63

$71%$

$40%$

Solution

0.4

$50%$

$115%$

Solution

1.15

$125%$

$150%$

Solution

1.5

$250%$

$21.4%$

Solution

0.214

$39.3%$

$7.8%$

Solution

0.078

$6.4%$

In the following exercises, convert each percent to

ⓐ a simplified fraction and
ⓑ a decimal

In $2010, 1.5%$ of home sales had owner financing. (Source: Bloomberg Businessweek, 5/23–29/2011)

Solution

ⓐ $\frac{3}{200}$
ⓑ $0.015$

In $2000, 4.2%$ of the United States population was of Asian descent. (Source: www.census.gov)

According to government data, in $2013$ the number of cell phones in India was $70.23%$ of the population.

Solution

ⓐ $\frac{7023}{10,000}$
ⓑ $0.7023$

According to the U.S. Census Bureau, among Americans age $25$ or older who had doctorate degrees in $2014, 37.1%$ are women.

A couple plans to have two children. The probability they will have two girls is $25% .$

Solution

ⓐ $\frac{1}{4}$
ⓑ $0.25$

Javier will choose one digit at random from $0$ through $9 .$ The probability he will choose $3$ is $10% .$

According to the local weather report, the probability of thunderstorms in New York City on July $15$ is $60% .$

Solution

ⓐ $\frac{3}{5}$
ⓑ $0.6$

A club sells $50$ tickets to a raffle. Osbaldo bought one ticket. The probability he will win the raffle is $2% .$

Convert Decimals and Fractions to Percents

In the following exercises, convert each decimal to a percent.

$0.01$

Solution

$0.03$

$0.18$

Solution

18%

$0.15$

$1.35$

Solution

135%

$1.56$

$3$

Solution

300%

$4$

$0.009$

Solution

0.9%

$0.008$

$0.0875$

Solution

8.75%

$0.0625$

$1.5$

Solution

150%

$2.2$

$2.254$

Solution

225.4%

$2.317$

In the following exercises, convert each fraction to a percent.

$\frac{1}{4}$

Solution

25%

$\frac{1}{5}$

$\frac{3}{8}$

Solution

37.5%

$\frac{5}{8}$

$\frac{7}{4}$

Solution

175%

$\frac{9}{8}$

$6 \frac{4}{5}$

Solution

680%

$5 \frac{1}{4}$

$\frac{5}{12}$

Solution

$41 \frac{2}{3} % o r 41. \bar{6} %$

$\frac{11}{12}$

$2 \frac{2}{3}$

Solution

$266 . \bar{6} %$

$1 \frac{2}{3}$

$\frac{3}{7}$

Solution

42.9%

$\frac{6}{7}$

$\frac{5}{9}$

Solution

$55. \bar{5} %$

$\frac{4}{9}$

In the following exercises, convert each fraction to a percent.

$\frac{1}{4}$ of washing machines needed repair.

Solution

25%

$\frac{1}{5}$ of dishwashers needed repair.

In the following exercises, convert each fraction to a percent.

According to the National Center for Health Statistics, in $2012, \frac{7}{20}$ of American adults were obese.

Solution

35%

The U.S. Census Bureau estimated that in $2013, 85%$ of Americans lived in the same house as they did $1$ year before.

In the following exercises, complete the table.

Fraction	Decimal	Percent
$\frac{1}{2}$
	$0.45$
		$18 %$
$\frac{1}{3}$
	$0.008$
$2$

Solution

Fraction	Decimal	Percent
$\frac{1}{2}$	$0.5$	$50 %$
$\frac{9}{20}$	$0.45$	$45 %$
$\frac{9}{50}$	$0.18$	$18 %$
$\frac{1}{3}$	$0.3$	$33.3 %$
$\frac{1}{125}$	$0.008$	$0.8 %$
$2$	$2.0$	$200 %$

Fraction	Decimal	Percent
$\frac{1}{4}$
	$0.65$
		$22 %$
$\frac{2}{3}$
	$0.004$
$3$

Everyday Math

Sales tax Felipa says she has an easy way to estimate the sales tax when she makes a purchase. The sales tax in her city is $9.05% .$ She knows this is a little less than $10% .$

ⓐ Convert $10%$ to a fraction.
ⓑ Use your answer from ⓐ to estimate the sales tax Felipa would pay on a $$95$ dress.

Solution

ⓐ $\frac{1}{10}$
ⓑ $approximately $9.50$

Savings Ryan has $25%$ of each paycheck automatically deposited in his savings account.

ⓐ Write $25%$ as a fraction.
ⓑ Use your answer from ⓐ to find the amount that goes to savings from Ryan's $$2,400$ paycheck.

Amelio is shopping for textbooks online. He found three sellers that are offering a book he needs for the same price, including shipping. To decide which seller to buy from he is comparing their customer satisfaction ratings. The ratings are given in the chart.

Use the chart to answer the following questions

Seller	Rating
$A$	$4/5$
$B$	$3.5/4$
$C$	$85%$

Write seller $C’s$ rating as a fraction and a decimal.

Solution

$\frac{17}{20}; 0.85$

Write seller $B’s$ rating as a percent and a decimal.

Write seller $A’s$ rating as a percent and a decimal.

Solution

80%; 0.8

Which seller should Amelio buy from and why?

Writing Exercises

Convert $25%, 50%, 75%, and 100%$ to fractions. Do you notice a pattern? Explain what the pattern is.

Solution

$\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 .$

Convert $\frac{1}{10}, \frac{2}{10}, \frac{3}{10}, \frac{4}{10}, \frac{5}{10}, \frac{6}{10}, \frac{7}{10}, \frac{8}{10},$ and $\frac{9}{10}$ to percents. Do you notice a pattern? Explain what the pattern is.

When the Szetos sold their home, the selling price was $500%$ of what they had paid for the house $30 years$ ago. Explain what $500%$ means in this context.

Solution

The Szetos sold their home for five times what they paid 30 years ago.

According to cnn.com, cell phone use in $2008$ was $600%$ of what it had been in $2001 .$ Explain what $600%$ means in this context.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A self-assessment chart for students to evaluate their understanding of percentages, including defining percent and converting between percents, fractions, and decimals.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

percent

A perfecnt is a ratio whose denominator is

100