6.1 Understand Percent

Learning objectives

Learning objectives

• Use the definition of percent

• Convert percents to fractions and decimals

• Convert decimals and fractions to percents

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 $\mathrm{\%,}$ to show percent.

According to data from the American Association of Community Colleges $(2015),$ about $\mathrm{57\%}$ of community college students are female. This means $57$ out of every $100$ community college students are female, as Figure 6.2 shows. Out of the $100$ squares on the grid, $57$ are shaded, which we write as the ratio $\frac{57}{100}.$

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

Example 6.1

Solution

1. Identify the stated percent. The report says that $44\%$ of the parents chose the graduate-degree option, so the amount to convert is $44\%$.
2. Apply the definition of percent. Since percent means per $100$, remove the percent sign and place the percent number over a denominator of $100$, giving $\frac{44}{100}$.
3. Write the requested ratio. The ratio that represents the same value as $44\%$ is $\frac{44}{100}$, which is 44 parts out of every 100.

Takeaway: The $44\%$ statistic in Example 6.1 becomes the ratio $\frac{44}{100}$ because a percent always represents a number out of 100.

Example 6.2

Solution

1. Read the reported ratio. The statement gives $21$ freshmen out of every $100$ freshmen, so the two compared quantities are $21$ and $100$.
2. Form the ratio. Putting the number of freshmen who took a remedial course in the numerator and the total number in the denominator gives $\frac{21}{100}$.
3. Convert the ratio to a percent. Because the denominator is $100$, the numerator tells the number of students per hundred, so the ratio corresponds to $21\%$.
4. Check both requested forms. The ratio $\frac{21}{100}$ and the percent $21\%$ describe the same group, confirming the conversion.

Takeaway: In Example 6.2, the reported rate of $21$ freshmen per $100$ is written as both $\frac{21}{100}$ and $21\%$.

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 $100.$

Example 6.3

Solution

1. Plan both percent conversions. Each percent must be written over $100$ and then simplified by dividing the numerator and denominator by their greatest common factor.
2. Convert the first percent. Write $36\%$ as $\frac{36}{100}$ and divide both parts by $4$, which gives the simplified fraction $\frac{9}{25}$.
3. Convert the second percent. Write $125\%$ as $\frac{125}{100}$ and divide both parts by $25$, which gives the simplified fraction $\frac{5}{4}$.
4. Compare the simplified values. Since $\frac{9}{25}<1$ while $\frac{5}{4}>1$, the results agree with the fact that $36\%$ is below one whole and $125\%$ is above one whole.

Takeaway: Writing the percent numbers over $100$ and simplifying gives $\frac{9}{25}$ for $36\%$ and the improper fraction $\frac{5}{4}$ for $125\%$.

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

Example 6.4

Solution

1. Choose a denominator of 100. To convert each percent to a fraction, use the percent number as the numerator and $100$ as the denominator.
2. Clear the decimal in the first fraction. The ratio $\frac{24.5}{100}$ becomes $\frac{245}{1000}$ after multiplying numerator and denominator by $10$, and dividing both by $5$ gives $\frac{49}{200}$.
3. Rewrite the mixed number in the second fraction. The ratio $\frac{33\frac{1}{3}}{100}$ equals $\frac{\frac{100}{3}}{100}$, which is $\frac{100}{3}\div\frac{100}{1}$.
4. Simplify and state both results. Dividing by $\frac{100}{1}$ means multiplying by $\frac{1}{100}$, so the second fraction becomes $\frac{1}{3}$. Thus the answers are $\frac{49}{200}$ and $\frac{1}{3}$.

Takeaway: For Example 6.4, clearing the decimal in $24.5\%$ and rewriting $33\frac{1}{3}$ as an improper fraction lead to $\frac{49}{200}$ and $\frac{1}{3}$.

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.

Example 6.5

Solution

1. Use a denominator of 100. A percent is a ratio with denominator $100$, so each conversion begins by writing the percent number over $100$.
2. Convert the first percent. For $6\%$, write $\frac{6}{100}$ and divide $6$ by $100$ to obtain $0.06$.
3. Convert the second percent. For $78\%$, write $\frac{78}{100}$ and divide $78$ by $100$ to obtain $0.78$.
4. Check the decimal places. Each division moves the percent number two places to the left, so $6\%$ becomes $0.06$ and $78\%$ becomes $0.78$.

Takeaway: Dividing a percent number by $100$ gives its decimal equivalent, so Example 6.5 produces $0.06$ from $6\%$ and $0.78$ from $78\%$.

Example 6.6

Solution

1. Write both percents over 100. The two ratios are $\frac{135}{100}$ for $135\%$ and $\frac{12.5}{100}$ for $12.5\%$.
2. Divide for the first decimal. Dividing $135$ by $100$ moves the decimal point two places left and gives $1.35$, which is greater than $1$ as expected.
3. Divide for the second decimal. Dividing $12.5$ by $100$ moves the decimal point two places left and gives $0.125$.
4. Check the relative sizes. The percent above $100\%$ produces a decimal above $1$, while the percent below $100\%$ produces a decimal below $1$, confirming both results.

Takeaway: In Example 6.6, dividing over a denominator of $100$ converts $135\%$ to $1.35$ and $12.5\%$ to $0.125$.

Let's summarize the results from the previous examples in Table 6.1, and look for a pattern we could use to quickly convert a percent number to a decimal number.

Table

6.1

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 $\mathrm{6\%}$ as $\mathrm{6.0\%}.$) Notice that we may need to add zeros in front of the number when moving the decimal to the left.

Figure 6.3 uses the percents in Table 6.1 and shows visually how to convert them to decimals by moving the decimal point two places to the left.

Example 6.7

Solution

1. Identify the survey percent. The reported portion of business leaders is $77\%$, so that percent is the value that must be converted.
2. Write the percent as a fraction. Using the definition of percent, write $77\%$ as $\frac{77}{100}$.
3. Convert the fraction to a decimal. Divide $77$ by $100$, which moves the decimal point two places left and gives $0.77$.
4. Check the two forms. The fraction $\frac{77}{100}$ and the decimal $0.77$ have the same value, so both requested conversions are correct.

Takeaway: The $77\%$ survey result in Example 6.7 is $\frac{77}{100}$ as a fraction and $0.77$ as a decimal.

Example 6.8

Solution

1. Write the probability as a ratio. The probability of drawing a heart is $25\%$, so it is represented by $\frac{25}{100}$.
2. Simplify the fraction. Both $25$ and $100$ can be divided by $25$, giving the equivalent probability $\frac{1}{4}$.
3. Convert to a decimal. Dividing $1$ by $4$ gives $0.25$, which agrees with one quarter of the deck.
4. Check against the suits. A standard deck has four equally represented suits, and one of those four suits gives the probability $\frac{1}{4}=0.25$.

Takeaway: Example 6.8 shows that the $25\%$ probability of selecting a heart equals $\frac{1}{4}$, or $0.25$, of the cards.

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 6.9

Solution

1. Recall the decimal-to-percent rule. Multiplying a decimal by $100$ moves its decimal point two places to the right, and the result is then read as a percent.
2. Convert the first decimal. Multiplying $0.05$ by $100$ gives $5$, so $0.05$ corresponds to $5\%$.
3. Convert the second decimal. Multiplying $0.83$ by $100$ gives $83$, so $0.83$ corresponds to $83\%$.
4. Check the ratios. The decimals $\frac{5}{100}$ and $\frac{83}{100}$ have denominators of $100$, confirming that their percent forms are $5\%$ and $83\%$.

Takeaway: Because $0.05=\frac{5}{100}$ and $0.83=\frac{83}{100}$, Example 6.9 converts them to $5\%$ and $83\%$.

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

Example 6.10

Solution

1. Scale the first decimal to 100. The decimal $1.05$ equals $\frac{105}{100}$, so its percent number is $105$ and the percent form is $105\%$.
2. Scale the second decimal to 100. The decimal $0.075$ equals $\frac{75}{1000}$; dividing both parts by $10$ gives the equivalent ratio $\frac{7.5}{100}$.
3. Read each percent. The ratios show that $1.05$ corresponds to $105\%$ and $0.075$ corresponds to $7.5\%$.
4. Check the sizes. Since $1.05>1$, its percent is greater than $100\%$, while $0.075<1$, so its percent is below $100\%$.

Takeaway: Writing $1.05$ over $100$ and rescaling $0.075$ shows in Example 6.10 that they equal $105\%$ and $7.5\%$.

Let's summarize the results from the previous examples in Table 6.2 so we can look for a pattern.

Table

6.2

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 6.5 uses the decimal numbers in Table 6.2 and shows visually to convert them to percents by moving the decimal point two places to the right and then writing the $\%$ sign.

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.

Example 6.11

Solution

1. Choose the conversion method. Convert each fraction or mixed number to a decimal by dividing, and then multiply that decimal by $100$ to obtain a percent.
2. Convert the first fraction. Dividing $3$ by $4$ gives $0.75$, and multiplying by $100$ gives $75\%$.
3. Convert the second fraction. Dividing $11$ by $8$ gives $1.375$, and multiplying by $100$ gives $137.5\%$.
4. Convert the mixed number. First write $2\frac{1}{5}$ as $\frac{11}{5}$. Dividing gives $2.2$, and multiplying by $100$ gives $220\%$.

Takeaway: Dividing and then multiplying by $100$ converts the three values in Example 6.11 to $75\%$, $137.5\%$, and $220\%$.

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 course we will round the number to the nearest thousandth, so the percent will be rounded to the nearest tenth.

Example 6.12

Solution

1. Divide the fraction to obtain a decimal. Dividing $5$ by $7$ gives the repeating decimal $0.714285\ldots$, which rounds to $0.714$ to the nearest thousandth.
2. Convert the rounded decimal to a percent. Multiplying $0.714$ by $100$ gives $71.4$, so the fraction is approximately $71.4\%$.
3. Account for rounding. The omitted repeating digits change the percent by less than $0.1\%$ when rounded to the nearest tenth of a percent.
4. State the approximation. The exact fraction does not terminate as a decimal, so the correct percent form is approximately $71.4\%$.

Takeaway: Because $\frac{5}{7}$ has the repeating decimal $0.714285\ldots$, Example 6.12 gives the rounded 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.

Example 6.13

Solution

1. Rewrite the fraction as a division. The fraction $\frac{1}{3}$ means that $1$ is divided by $3$, and the accompanying long-division figure shows the quotient repeating as $0.333\ldots$.
2. Multiply by 100 to convert to a percent. The calculation is $\frac{1}{3}\times 100=\frac{100}{3}=33.333\ldots$.
3. Write the approximate percent. Because the digits repeat indefinitely, the fraction is approximately $33.333\ldots\%$, often written $33.3\%$ to the nearest tenth of a percent. Thus, approximately one third of American adults is represented by about $33.3\%$.

Takeaway: To convert a repeating fraction to a percent, multiply by 100 and use the repeating pattern to state an appropriate approximation.

Section 6.1 Exercises

Section 6.1 Exercises

Use the Definition of Percents

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

In the following exercises, write as

1. ⓐ a ratio and

2. ⓑ a percent

Convert Percents to Fractions and Decimals

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

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

In the following exercises, convert each percent to

1. ⓐ a simplified fraction and

2. ⓑ a decimal

Convert Decimals and Fractions to Percents

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

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

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

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

In the following exercises, complete the table.

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

Writing Exercises

Self Check

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

ⓑ 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.

1. Convert each percent to a fraction:

1. ⓐ $\,\mathrm{64\%}\,$

2. ⓑ $\,\mathrm{150\%}$

2. Convert each percent to a fraction:

1. ⓐ $\,\mathrm{64.4\%}\,$

2. ⓑ $\,66\frac{2}{3}\%$

3. Convert each percent to a fraction:

1. ⓐ $\,\mathrm{42.5\%}\,$

2. ⓑ $\,8\frac{3}{4}\%$

4. Convert each percent to a decimal:

1. ⓐ $\,\mathrm{9\%}\,$

2. ⓑ $\,\mathrm{87\%}$

5. Convert each percent to a decimal:

1. ⓐ $\,\mathrm{3\%}\,$

2. ⓑ $\,\mathrm{91\%}$

1. Convert each percent to a decimal:

1. ⓐ $\,\mathrm{115\%}\,$

2. ⓑ $\,\mathrm{23.5\%}$

2. Convert each percent to a decimal:

1. ⓐ $\,\mathrm{123\%}\,$

2. ⓑ $\,\mathrm{16.8\%}$

3. Convert the percent to: ⓐ a fraction and ⓑ a decimal

Twitter's share of web traffic jumped $\mathrm{24\%}$ when one celebrity tweeted live on air.

4. Convert the percent to: ⓐ a fraction and ⓑ a decimal

The U.S. Census estimated that in $2013,\mathrm{44\%}$ of the population of Boston age $25$ or older have a bachelor's or higher degrees.

5. Convert the percent to: ⓐ a fraction, and ⓑ a decimal

The probability that it will rain Monday is $\mathrm{30\%}.$

1. Convert the percent to: ⓐ a fraction, and ⓑ a decimal

The probability of getting heads three times when tossing a coin three times is $\mathrm{12.5\%}.$

2. Convert each decimal to a percent: ⓐ $\,0.01\,$ ⓑ $\,0.17.$

3. Convert each decimal to a percent: ⓐ $\,0.04\,$ ⓑ $\,0.41$

4. Convert each decimal to a percent: ⓐ $\,1.75\,$ ⓑ $\,0.0825$

5. Convert each decimal to a percent: ⓐ $\,2.25\,$ ⓑ $\,0.0925$

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

2. Convert each fraction or mixed number to a percent: ⓐ $\,\frac{7}{8}\,$ ⓑ $\,\frac{9}{4}\,$ ⓒ $\,1\frac{3}{5}$

3. Convert the fraction to a percent: $\frac{3}{7}$

4. Convert the fraction to a percent: $\frac{4}{7}$

5. Convert the fraction to a percent:

According to the U.S. Census Bureau, about $\frac{1}{9}$ of United States housing units have just $1$ bedroom.

1. Convert the fraction to a percent:

According to the U.S. Census Bureau, about $\frac{1}{6}$ of Colorado residents speak a language other than English at home.

2. In $2014,$ the unemployment rate for those with only a high school degree was $\mathrm{6.0\%}.$

3. The unemployment rate for those with Bachelor's degrees was $\mathrm{3.2\%}$ in $2014.$

4. $57$ out of $100$ nursing candidates received their degree at a community college.

5. $42$ out of $100$ first-time freshmen students attend a community college.

1. $\mathrm{4\%}$

2. $\mathrm{17\%}$

3. $\mathrm{52\%}$

4. $\mathrm{125\%}$

5. $\mathrm{37.5\%}$

1. $\mathrm{18.4\%}$

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

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

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

5. According to government data, in $2013$ the number of cell phones in India was $\mathrm{70.23\%}$ of the population.

1. A couple plans to have two children. The probability they will have two girls is $\mathrm{25\%}.$

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

3. $0.01$

4. $0.18$

5. $1.35$

1. $3$

2. $0.009$

3. $0.0875$

4. $1.5$

5. $2.254$

1. $\frac{1}{4}$

2. $\frac{3}{8}$

3. $\frac{7}{4}$

4. $6\frac{4}{5}$

5. $\frac{5}{12}$

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

2. $\frac{3}{7}$

3. $\frac{5}{9}$

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

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

1. Write the percent as a ratio.

According to a survey, $\mathrm{89\%}$ of college students have a smartphone.

2. Write the percent as a ratio.

A study found that $\mathrm{72\%}$ of U.S. teens send text messages regularly.

3. Write as a ratio and then as a percent: The American Association of Community Colleges reported that $62$ out of $100$ full-time community college students balance their studies with full-time or part time employment.

4. Write as a ratio and then as a percent: In response to a student survey, $41$ out of $100$ Santa Ana College students expressed a goal of earning an Associate's degree or transferring to a four-year college.

5. Convert each percent to a fraction:

1. ⓐ $\,\mathrm{48\%}\,$

2. ⓑ $\,\mathrm{110\%}$

1. | Fraction | Decimal | Percent |

| --- | --- | --- |

| $\frac{1}{2}$ | | |

| | $0.45$ | |

| | | $18\%$ |

| $\frac{1}{3}$ | | |

| | $0.008$ | |

| $2$ | | |

Table

6.3

2. 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 $\mathrm{9.05\%}.$ She knows this is a little less than $\mathrm{10\%}.$

1. ⓐ Convert $\mathrm{10\%}$ to a fraction.

2. ⓑ Use your answer from ⓐ to estimate the sales tax Felipa would pay on a $\text{\textdollar 95}$ dress.

3. Write seller $\mathrm{C's}$ rating as a fraction and a decimal.

4. Write seller $\mathrm{A's}$ rating as a percent and a decimal.

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

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