1.3 Converting Between Fractions, Decimals, and Percents

Narrated lesson: Converting Between Fractions, Decimals, and Percents

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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 $open parenthesis 2015 close parenthesis,$ about $57%$ of community college students are female. This means $57$ out of every $100$ community college students are female, as Figure 1.4 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. — **Figure 1.4** 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.

Example 1

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

Show solution

The amount we want to convert is 44%.	$44%$
Write the percent as a ratio. Remember that percent means per 100.	$\frac{44}{100}$

Example 2

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.

Show solution

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 $100 .$

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.

Example 4

Convert each percent to a fraction:

ⓐ $24.5%$
ⓑ $33 \frac{1}{3} %$

Show solution

ⓐ
	$24.5%$
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.	$\frac{49}{200}$

ⓑ
	$33 \frac{1}{3} %$
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.	$\frac{1}{3}$

Earlier, 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.

Example 5

Convert each percent to a decimal:

ⓐ $6%$
ⓑ $78%$

Show solution

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

ⓐ
	$6%$
Write as a ratio with denominator 100.	$\frac{6}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.06$

ⓑ
	$78%$
Write as a ratio with denominator 100.	$\frac{78}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.78$

Example 6

Convert each percent to a decimal:

ⓐ $135%$
ⓑ $12.5%$

Show solution

ⓐ
	$135%$
Write as a ratio with denominator 100.	$\frac{135}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$1.35$

ⓑ
	$12.5%$
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 1.4, and look for a pattern we could use to quickly convert a percent number to a decimal number.

**Table 1.4**
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 1.5 uses the percents in Table 1.4 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. — **Figure 1.5**

Example 7

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

Show solution

ⓐ
	$77%$
Write as a ratio with denominator 100.	$\frac{77}{100}$

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

Example 8

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

Figure 1.6 (credit: Riles32807, Wikimedia Commons)

Show solution

ⓐ
	$25%$
Write as a ratio with denominator 100.	$\frac{25}{100}$
Simplify.	$\frac{1}{4}$

ⓑ	$\frac{1}{4}$
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.

Example 9

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

Show solution

ⓐ
	$0.05$
Write as a fraction. The denominator is 100.	$\frac{5}{100}$
Write this ratio as a percent.	$5%$

ⓑ
	$0.83$
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.

Example 10

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

Show solution

ⓐ
	$1.05$
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 $1.05 greater than 1,$ the result is more than $100%.$

ⓑ
	$0.075$
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 1.5 so we can look for a pattern.

**Table 1.5**
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 1.7 repeats percent–decimal equivalents. Read each row from the Decimal column to the Percent column when converting a decimal to a percent; move the decimal point two places to the right and write the $%$ sign.

A table with columns Percent and Decimal. Four rows pair 6% with 0.06, 78% with 0.78, 135% with 1.35, and 12.5% with 0.125. The percent entries show leading zeros and decimal points, with two curved marks beneath each entry. — **Figure 1.7**

Earlier, 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.

Example 11

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

Show 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.

Example 12

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

Show solution

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

	$\frac{5}{7}$
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 $1.3 repeating.$ We will use this same notation, as well as fraction notation, when we convert fractions to percents in the next example.

Example 13

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.

Show 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.	$0.333 and so on$
Write as a percent.	$33 \frac{1}{3} %$

We could also write the percent as $33.3 _ %$ .

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 $6.0%.$

Show 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 $2014 .$

Show 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.

Show 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.

Show 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.

$4%$

Show solution

$\frac{1}{25}$

$8%$

$17%$

Show solution

$\frac{17}{100}$

$19%$

$52%$

Show solution

$\frac{13}{25}$

$78%$

$125%$

Show solution

$\frac{5}{4}$

$135%$

$37.5%$

Show solution

$\frac{3}{8}$

$42.5%$

$18.4%$

Show solution

$\frac{23}{125}$

$46.4%$

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

Show solution

$\frac{19}{200}$

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

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

Show solution

$\frac{4}{75}$

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

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

$5%$

Show solution

0.05

$9%$

$1%$

Show solution

0.01

$2%$

$63%$

Show solution

0.63

$71%$

$40%$

Show solution

0.4

$50%$

$115%$

Show solution

1.15

$125%$

$150%$

Show solution

1.5

$250%$

$21.4%$

Show solution

0.214

$39.3%$

$7.8%$

Show 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)

Show 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.

Show 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 $25%.$

Show 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 $60%.$

Show 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.

$0.01$

Show solution

$0.03$

$0.18$

Show solution

18%

$0.15$

$1.35$

Show solution

135%

$1.56$

$3$

Show solution

300%

$4$

$0.009$

Show solution

0.9%

$0.008$

$0.0875$

Show solution

8.75%

$0.0625$

$1.5$

Show solution

150%

$2.2$

$2.254$

Show solution

225.4%

$2.317$

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

$\frac{1}{4}$

Show solution

25%

$\frac{1}{5}$

$\frac{3}{8}$

Show solution

37.5%

$\frac{5}{8}$

$\frac{7}{4}$

Show solution

175%

$\frac{9}{8}$

$6 \frac{4}{5}$

Show solution

680%

$5 \frac{1}{4}$

$\frac{5}{12}$

Show solution

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

$\frac{11}{12}$

$2 \frac{2}{3}$

Show solution

$266.6 repeating %$

$1 \frac{2}{3}$

$\frac{3}{7}$

Show solution

42.9%

$\frac{6}{7}$

$\frac{5}{9}$

Show solution

$55.5 repeating %$

$\frac{4}{9}$

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

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

Show 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.

Show 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. For this exercise, convert the stated percent to a simplified fraction.

In the following exercises, complete the table.

**Table 1.6**
Fraction	Decimal	Percent
$\frac{1}{2}$
	$0.45$
		$18 %$
$\frac{1}{3}$
	$0.008$
$2$

Show solution

**Table 1.7**
Fraction	Decimal	Percent
$\frac{1}{2}$	$0.5$	$50 %$
$\frac{9}{20}$	$0.45$	$45 %$
$\frac{9}{50}$	$0.18$	$18 %$
$\frac{1}{3}$	$0.333$	$33.3 %$
$\frac{1}{125}$	$0.008$	$0.8 %$
$2$	$2.0$	$200 %$

**Table 1.8**
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.

Show 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.

Show 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.

Show 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.

Show 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.

Show 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. — **Figure 1.8**

ⓑ 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.

Corequisite Skills for Introductory Statistics is a custom course edition compiled for Introductory Statistics corequisite courses from OpenStax Contemporary Mathematics, Elementary Algebra 2e, Intermediate Algebra 2e, Prealgebra 2e. Content by OpenStax is reproduced as published, apart from any corrections recorded for this edition. Sections are renumbered to follow the course outline. The course outline is adapted from Larry Green's Support Course for Elementary Statistics (LibreTexts, https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Support_Course_for_Elementary_Statistics), used under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Lessons 2.4 and 6.7 are written for this edition by XYZ Homework. Every OpenStax lesson is abridged to the subsections and practice exercises this course uses. Lessons are headed with their course titles, not the source section titles. Access the complete originals free at openstax.org. License: CC-BY-NC-SA-4.0.

ⓐ
	$36%$
Write as a ratio with denominator 100.	$\frac{36}{100}$
Simplify.	$\frac{9}{25}$

ⓑ
	$125%$
Write as a ratio with denominator 100.	$\frac{125}{100}$
Simplify.	$\frac{5}{4}$