0.6 Percentages

Jared Eusea; Karen Perilloux; Kristina VanDusen; Lauren Johnson; Prakash Ghimire

0.6 Percentages

Learning Objectives

By the end of this section, you will be able to:

Use the definition of percent
Convert percents to fractions and decimals
Convert decimals and fractions to percents
Translate and solve basic percent equations

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.

Percent

A percent is a ratio whose denominator is 100.

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 [latex]\frac{57}{100}[/latex].

The figure shows a hundred flat with 57 units shaded. — **Figure 1** Among every 100 community college students, 57 are female.

Similarly, [latex]25%[/latex] means a ratio of [latex]\frac{25}{100}[/latex], [latex]3%[/latex] means a ratio of [latex]\frac{3}{100}[/latex], and [latex]100%[/latex] means a ratio of [latex]\frac{100}{100}[/latex].

In words, "one hundred percent" means the total [latex]100%[/latex] is [latex]\frac{100}{100}[/latex], and since [latex]\frac{100}{100}=1[/latex], we see that [latex]100%[/latex] means 1 whole.

Writing Percents as Ratios

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

The amount we want to convert is 44%. Remember that percent means per 100.

The percent as a ratio is [latex]\frac{44}{100}[/latex]

Try It

Write the percent as a ratio.

According to a survey, 89% of college students have a smartphone.

Show Solution

[latex]\frac{89}{100}[/latex]

Try It

Write the percent as a ratio.

A study found that 72% of U.S. teens send text messages regularly.

Show Solution

[latex]\frac{72}{100}[/latex]

Writing Fractions as Ratios and Percents

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

The amount we want to convert is [latex]21 \text{ out of }100[/latex]. As a ratio, [latex]\frac{21}{100}[/latex] and as a percent [latex]21%[/latex].

Try It

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.

Show Solution

[latex]\frac{62}{100}\text{ or }62\%[/latex]

Try It

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.

Show Solution

[latex]\frac{41}{100}\text{ or }41\%[/latex]

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.

How To

Convert a percent to a fraction.

Write the percent as a ratio with the denominator 100.
Simplify the fraction if possible.

Converting Percents to Fractions

Convert each percent to a fraction:

a) [latex]36\%[/latex]

b) [latex]125\%[/latex]

Solution to part (a)

36% [latex]= \frac{36}{100} = \frac{9}{25}[/latex]

Solution to part (b)

125% [latex]= \frac{125}{100} = \frac{5}{4}[/latex]

Try It

Convert each percent to a fraction:

a) [latex]48\%[/latex]

b) [latex]110\%[/latex]

Show Solution

a) [latex]\frac{12}{25}[/latex]

b) [latex]\frac{11}{10}[/latex]

Try It

Convert each percent to a fraction:

a) [latex]64\%[/latex]

b) [latex]150\%[/latex]

Show Solution

a) [latex]\frac{16}{25}[/latex]

b) [latex]\frac{3}{2}[/latex]

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

Converting Percents to Fractions

Convert each percent to a fraction:

a) [latex]24.5\%[/latex]

b) [latex]33\frac{1}{3}\%[/latex]

Solution to part (a)

Explanation	Steps
	$24.5%$
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.	$\frac{49}{200}$

Solution to part (b)

Explanation	Steps
	$33 \frac{1}{3} %$
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 [latex]\frac{100}{1}[/latex].	$\frac{100}{3} \div \frac{100}{1}$
Multiply by the reciprocal.	$\frac{100}{3} \cdot \frac{1}{100}$
Simplify.	$\frac{1}{3}$

Try It

Convert each percent to a fraction:

a) [latex]64.4\%[/latex]

b) [latex]66\frac{2}{3}\%[/latex]

Show Solution

a) [latex]\frac{161}{250}[/latex]

b) [latex]\frac{2}{3}[/latex]

Try It

Convert each percent to a fraction:

a) [latex]42.5\%[/latex]

b) [latex]8\frac{3}{4}\%[/latex]

Show Solution

a) [latex]\frac{17}{40}[/latex]

b) [latex]\frac{35}{400}[/latex]

To convert a percent to a decimal, we first convert it to a fraction and then change the fraction to a decimal.

How To

Convert a percent to a decimal.

Write the percent as a ratio with the denominator 100.
Convert the fraction to a decimal by dividing the numerator by the denominator.

Converting Percents to Decimals

Convert each percent to a decimal:

a) [latex]6\%[/latex]

b) [latex]78\%[/latex]

Solution to part (a)

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

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

Solution to part (b)

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

Try It

Convert each percent to a decimal:

a) [latex]9\%[/latex]

b) [latex]87\%[/latex]

Show Solution

a) [latex]0.09[/latex]

b) [latex]0.87[/latex]

Try It

Convert each percent to a decimal:

a) [latex]3\%[/latex]

b) [latex]91\%[/latex]

Show Solution

a) [latex]0.03[/latex]

b) [latex]0.91[/latex]

Converting Percents to Decimals

Convert each percent to a decimal:

a) [latex]135\%[/latex]

b) [latex]12.5\%[/latex]

Solution to part (a)

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

Solution to part (b)

Explanation	Steps
	$12.5%$
Write as a ratio with denominator 100.	$\frac{12.5}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.125$

Try It

Convert each percent to a decimal:

a) [latex]115\%[/latex]

b) [latex]23.5\%[/latex]

Show Solution

a) [latex]1.15[/latex]

b) [latex]0.235[/latex]

Try It

Convert each percent to a decimal:

a) [latex]123\%[/latex]

b) [latex]16.8\%[/latex]

Show Solution

a) [latex]1.23[/latex]

b) [latex]0.168[/latex]

Let's summarize the results from the previous examples in Table 1, and look for a pattern we could use to quickly convert a percent number to a decimal number.

Table 1
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.

The first column is “Percent” with values: 6%, 78%, 135%, 12.5%. The second column is "Decimal" with the values: 0.06, 0.78, 1.35, 0.125. — **Figure 2** uses the percents in Table 1 and shows visually how to convert them to decimals by moving the decimal point two places to the left.

Converting Percents to Fractions and Decimals

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 and a decimal.

Solution

77% as a ratio is [latex]\frac{77}{100}[/latex] and (b) a decimal is 0.77

Try It

Convert the percent to: (a) a fraction and (b) a decimal.

Twitter's share of web traffic jumped 24% when one celebrity tweeted live on air.

Show Solution

a) [latex]\frac{6}{25}[/latex]

b) [latex]0.24[/latex]

Try It

Convert the percent to: (a) a fraction and (b) a decimal.

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

Show Solution

a) [latex]\frac{11}{25}[/latex]

b) [latex]0.44[/latex]

Converting Percents to Fractions and Decimals

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 and a decimal.

The figure shows someone holding a deck of cards. — **Figure 3** (credit: Riles32807, Wikimedia Commons)

Solution

Write as a ratio with denominator 100. [latex]\frac{25}{100} = \frac{1}{4}[/latex].

Change the fraction to a decimal by dividing the numerator by the denominator. 0.25

Try It

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

The probability that it will rain Monday is 30%.

Show Solution

a) [latex]\frac{3}{10}[/latex]

b) [latex]0.3[/latex]

Try It

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

The probability of getting heads three times when tossing a coin three times is 12.5%.

Show Solution

a) [latex]\frac{1}{8}[/latex]

b) [latex]0.125[/latex]

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.

How To

Convert a decimal to a percent.

Write the decimal as a fraction.
If the denominator of the fraction is not 100, rewrite it as an equivalent fraction with denominator 100.
Write this ratio as a percent.

Converting Decimals to Percents

Convert each decimal to a percent:

a) [latex]0.05[/latex]

b) [latex]0.83[/latex]

Solution to part (a)

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

Solution to part (b)

Explanation	Steps
	$0.83$
The denominator is 100.	$\frac{83}{100}$
Write this ratio as a percent.	$83%$

Try It

Convert each decimal to a percent:

a) [latex]0.01[/latex]

b) [latex]0.17[/latex]

Show Solution

a) [latex]1\%[/latex]

b) [latex]17\%[/latex]

Try It

Convert each decimal to a percent:

a) [latex]0.04[/latex]

b) [latex]0.41[/latex]

Show Solution

a) [latex]4\%[/latex]

b) [latex]41\%[/latex]

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

Converting Decimals to Percents

Convert each decimal to a percent:

a) [latex]1.05[/latex]

b) [latex]0.075[/latex]

Solution to part (a)

Explanation	Steps
	$1.05$
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 [latex]1.05>1[/latex], the result is more than [latex]100\%[/latex].

Solution to part (b)

Explanation	Steps
	$0.075$
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.	$7.5%$

Try It

Convert each decimal to a percent:

a) [latex]1.75[/latex]

b) [latex]0.0825[/latex]

Show Solution

a) [latex]175\%[/latex]

b) [latex]8.25\%[/latex]

Try It

Convert each decimal to a percent:

a) [latex]2.25[/latex]

b) [latex]0.0925[/latex]

Show Solution

a) [latex]225\%[/latex]

b) [latex]9.25\%[/latex]

Let's summarize the results from the previous examples in Table 2 so we can look for a pattern.

Table 2
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.

We have 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.

How To

Convert a fraction to a percent.

Convert the fraction to a decimal.
Convert the decimal to a percent.

Converting Fractions and Mixed Numbers to Percents

Convert each fraction or mixed number to a percent:

a) [latex]\frac{3}{4}[/latex]

b) [latex]\frac{11}{8}[/latex]

c) [latex]2\frac{1}{5}[/latex]

Solution to part (a)

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

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

Solution to part (b)

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

Solution to part (c)

Explanation	Steps
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.

Try It

Convert each fraction or mixed number to a percent:

a) [latex]\frac{5}{8}[/latex]

b) [latex]\frac{11}{4}[/latex]

c) [latex]3\frac{2}{5}[/latex]

Show Solution

a) [latex]62.5\%[/latex]

b) [latex]275\%[/latex]

c) [latex]340\%[/latex]

Try It

Convert each fraction or mixed number to a percent:

a) [latex]\frac{7}{8}[/latex]

b) [latex]\frac{9}{4}[/latex]

c) [latex]1\frac{3}{5}[/latex]

Show Solution

a) [latex]87.5\%[/latex]

b) [latex]225\%[/latex]

c) [latex]160\%[/latex]

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.

Converting Fractions to Percents

Convert [latex]\frac{5}{7}[/latex] to a percent.

Solution

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

Explanation	Steps
	$\frac{5}{7}$
Change to a decimal—rounding to the nearest thousandth.	$0.714$
Write as a percent.	$71.4%$

Try It

Convert the fraction to a percent: [latex]\frac{3}{7}[/latex].

Show Solution

[latex]42.9\%[/latex]

Try It

Convert the fraction to a percent: [latex]\frac{4}{7}[/latex].

Show Solution

[latex]57.1\%[/latex]

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 [latex]\frac{4}{3}[/latex] to a decimal, we wrote the answer as [latex]1.\bar{3}[/latex].

We will use this same notation, as well as fraction notation, when we convert fractions to percents in the next example.

Converting Fractions to Percents

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

Solution

Explanation	Steps
	$\frac{1}{3}$
Change to a decimal.
Write as a repeating decimal.	$0.333 \dots$
Write as a percent.	$33 \frac{1}{3} %$

We could also write the percent as [latex]33.\bar{3}\%[/latex].

Try It

Convert the fraction to a percent:

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

Show Solution

[latex]11.\bar{1}\%[/latex]

Try It

Convert the fraction to a percent:

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

Show Solution

[latex]16.\bar{6}\%[/latex]

Translate and Solve Basic Percent Equations

We will solve percent equations by using the methods we used to solve equations with fractions or decimals. In the past, you may have solved percent problems by setting them up as proportions. That was the best method available when you did not have the tools of algebra. Now as a prealgebra student, you can translate word sentences into algebraic equations, and then solve the equations.

We'll look at a common application of percent—tips to a server at a restaurant—to see how to set up a basic percent application.

When Aolani and her friends ate dinner at a restaurant, the bill came to $80. They wanted to leave a 20% tip. What amount would the tip be?

To solve this, we want to find what amount is 20% of $80. The $80 is called the base. The amount of the tip would be 0.20(80), or $16. See Figure 4. To find the amount of the tip, we multiplied the percent by the base.

A restaurant receipt with the amount of the bill, $80, and the amount of the tip, $16. There is a group of bills totaling $16. — **Figure 4**: 20% of an $80 restaurant bill comes out to $16.

In the next examples, we will find the amount. We must be sure to change the given percent to a decimal when we translate the words into an equation.

Khan Academy. Percentage of a whole number. YouTube. https://youtu.be/-lUEWEEpmIo

Finding Percentages

What number is 35% of 90?

Solution

Explanation	Steps
Translate into algebra. Let n equal the number.
Multiply.	[latex]n=31.5[/latex]
	31.5 is 35% of 90.

Try It

What number is 45% of 80?

Show Solution

36

Try It

What number is 55% of 60?

Show Solution

33

Finding Percentages

125% of 28 is what number?

Solution

Explanation	Steps
Translate into algebra. Let 'a' equal the number.
Multiply.	[latex]35=a[/latex]
	$125% of 28 is 35.$

Remember that a percent over 100 is a number greater than 1. We found that 125% of 28 is 35, which is greater than 28.

Try It

150% of 78 is what number?

Show Solution

117

Try It

175% of 72 is what number?

Show Solution

126

Khan Academy. Percentage word problem example 5. YouTube. https://youtu.be/DAikW24_O0A

Finding Percentages

Translate and solve: 36 is 75% of what number?

Solution

Explanation	Steps
Translate. Let b = the number.
Divide both sides by 0.75.	[latex]\frac{36}{0.75}=\frac{0.75b}{0.75}[/latex]
Simplify.	[latex]48=b[/latex]
	36 is 75% of 48.

Try It

17 is 25% of what number?

Show Solution

68

Try It

40 is 62.5% of what number?

Show Solution

64

Finding Percentages

6.5% of what number is $1.17?

Solution

Explanation	Steps
Translate. Let b = the number.
Divide both sides by 0.065.	[latex]\frac{0.065n}{0.065}=\frac{1.17}{0.065}[/latex]
Simplify.	[latex]n=18[/latex]
	6.5% of $18 is $1.17.

Try It

7.5% of what number is $1.95?

Show Solution

$26

Try It

8.5% of what number is $3.06?

Show Solution

$36

In the next examples, we will solve for the percent.

Khan Academy. Find a percentage. YouTube. https://youtu.be/FaDtge_vkbg

Finding Percentages

What percent of 36 is 9?

Solution

Explanation	Steps
Translate into algebra. Let p = the percent.
Divide by 36.	[latex]\frac{36p}{36}=\frac{9}{36}[/latex]
Simplify.	[latex]p=\frac{1}{4}[/latex]
Convert to decimal form.	[latex]p=0.25[/latex]
Convert to percent.	[latex]p=25\%[/latex]
	25% of 36 is 9.

Try It

What percent of 76 is 57?

Show Solution

75%

Try It

What percent of 120 is 96?

Show Solution

80%

Finding Percentages

144 is what percent of 96?

Solution

Explanation	Steps
Translate into algebra. Let p = the percent.
Divide by 96.	[latex]\frac{144}{96}=\frac{96p}{96}[/latex]
Simplify.	[latex]1.5=p[/latex]
Convert to percent.	[latex]150\%=p[/latex]
	144 is 150% of 96.

Try It

110 is what percent of 88?

Show Solution
125%

Try It

126 is what percent of 72?

Show Solution
175%

Key Concepts

Convert a percent to a fraction.

- Step 1. Write the percent as a ratio with the denominator 100.
- Step 2. Simplify the fraction if possible.

Convert a percent to a decimal.

- Step 1. Write the percent as a ratio with the denominator 100.
- Step 2. Convert the fraction to a decimal by dividing the numerator by the denominator.

Convert a decimal to a percent.

- Step 1. Write the decimal as a fraction.
- Step 2. If the denominator of the fraction is not 100, rewrite it as an equivalent fraction with denominator 100.
- Step 3. Write this ratio as a percent.

Convert a fraction to a percent.

- Step 1. Convert the fraction to a decimal.
- Step 2. Convert the decimal to a percent.

Section Exercises

Use the Definition of Percents

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

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

Solution

[latex]\frac{6}{100}[/latex]

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

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

Solution

[latex]\frac{32}{1000}[/latex]

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

In the following exercises, write as (a) a ratio and (b) a percent.

5. 57 out of 100 nursing candidates received their degree at a community college.

Solution

[latex]\frac{57}{100}[/latex]

6. 80 out of 100 firefighters and law enforcement officers were educated at a community college.

7. 42 out of 100 first-time freshmen students attend a community college.

Solution

[latex]\frac{42}{100}[/latex]

8. 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.

9. [latex]4\%[/latex]

Solution

[latex]\frac{1}{25}[/latex]

10. [latex]8\%[/latex]

11. [latex]17\%[/latex]

Solution

[latex]\frac{17}{100}[/latex]

12. [latex]19\%[/latex]

13. [latex]52\%[/latex]

Solution

[latex]\frac{13}{25}[/latex]

14. [latex]78\%[/latex]

15. [latex]125\%[/latex]

Solution

[latex]\frac{5}{4}[/latex]

16. [latex]135\%[/latex]

17. [latex]37.5\%[/latex]

Solution

[latex]\frac{3}{8}[/latex]

18. [latex]42.5\%[/latex]

19. [latex]18.4\%[/latex]

Solution

[latex]\frac{23}{125}[/latex]

20. [latex]46.4\%[/latex]

21. [latex]9\frac{1}{2}\%[/latex]

Solution

[latex]\frac{19}{200}[/latex]

22. [latex]8\frac{1}{2}\%[/latex]

23. [latex]5\frac{1}{3}\%[/latex]

Solution

[latex]\frac{16}{300}[/latex]

24. [latex]6\frac{2}{3}\%[/latex]

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

25. [latex]5\%[/latex]

Solution

[latex]0.05[/latex]

26. [latex]9\%[/latex]

27. [latex]1\%[/latex]

Solution

[latex]0.01[/latex]

28. [latex]2\%[/latex]

29. [latex]63\%[/latex]

Solution

[latex]0.63[/latex]

30. [latex]71\%[/latex]

31. [latex]40\%[/latex]

Solution

[latex]0.4[/latex]

32. [latex]50\%[/latex]

33. [latex]115\%[/latex]

Solution

[latex]1.15[/latex]

34. [latex]125\%[/latex]

35. [latex]150\%[/latex]

Solution

[latex]1.5[/latex]

36. [latex]250\%[/latex]

37. [latex]21.4\%[/latex]

Solution

[latex]0.214[/latex]

38. [latex]39.3\%[/latex]

39. [latex]7.8\%[/latex]

Solution

[latex]0.078[/latex]

40. [latex]6.4\%[/latex]

In the following exercises, convert each percent to (a) a simplified fraction and (b) a decimal.

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

Solution

a) [latex]\frac{3}{200}[/latex]

b) [latex]0.015[/latex]

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

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

Solution

a) [latex]\frac{7023}{10000}[/latex]

b) [latex]0.7023[/latex]

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

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

Solution

a) [latex]\frac{1}{4}[/latex]

b) [latex]0.25[/latex]

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

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

Solution

a) [latex]\frac{3}{5}[/latex]

b) [latex]0.6[/latex]

48. 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.

49. [latex]0.01[/latex]

Solution

[latex]1\%[/latex]

50. [latex]0.03[/latex]

51. [latex]0.18[/latex]

Solution

[latex]18\%[/latex]

52. [latex]0.15[/latex]

53. [latex]1.35[/latex]

Solution

[latex]135\%[/latex]

54. [latex]1.56[/latex]

55. [latex]3[/latex]

Solution

[latex]300\%[/latex]

56. [latex]4[/latex]

57. [latex]0.009[/latex]

Solution

[latex]0.9\%[/latex]

58. [latex]0.008[/latex]

59. [latex]0.0875[/latex]

Solution

[latex]8.75\%[/latex]

60. [latex]0.0625[/latex]

61. [latex]1.5[/latex]

Solution

[latex]150\%[/latex]

62. [latex]2.2[/latex]

63. [latex]2.254[/latex]

Solution

[latex]225.4\%[/latex]

64. [latex]2.317[/latex]

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

65. [latex]\frac{1}{4}[/latex]

Solution

[latex]25\%[/latex]

66. [latex]\frac{1}{5}[/latex]

67. [latex]\frac{3}{8}[/latex]

Solution

[latex]37.5\%[/latex]

68. [latex]\frac{5}{8}[/latex]

69. [latex]\frac{7}{4}[/latex]

Solution

[latex]175\%[/latex]

70. [latex]\frac{9}{8}[/latex]

71. [latex]6\frac{4}{5}[/latex]

Solution

[latex]680\%[/latex]

72. [latex]5\frac{1}{4}[/latex]

73. [latex]\frac{5}{12}[/latex]

Solution

[latex]41.\bar{6}\%[/latex]

74. [latex]\frac{11}{12}[/latex]

75. [latex]2\frac{2}{3}[/latex]

Solution

[latex]266.\bar{6}\%[/latex]

76. [latex]1\frac{2}{3}[/latex]

77. [latex]\frac{3}{7}[/latex]

Solution

[latex]42.9\%[/latex]

78. [latex]\frac{6}{7}[/latex]

79. [latex]\frac{5}{9}[/latex]

Solution

[latex]55.\bar{5}\%[/latex]

80. [latex]\frac{4}{9}[/latex]

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

81. [latex]\frac{1}{4}[/latex] of washing machines needed repair.

Solution

[latex]25\%[/latex]

82. [latex]\frac{1}{5}[/latex]of dishwashers needed repair.

83. According to the National Center for Health Statistics, in 2012, [latex]\frac{7}{20}[/latex] of American adults were obese.

Solution

[latex]35\%[/latex]

84. 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.

85.

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

Solution

Fraction

Decimal

Percent

[latex]\frac{1}{2}[/latex]

[latex]0.5[/latex]

[latex]50\%[/latex]

[latex]\frac{9}{20}[/latex]

[latex]0.45[/latex]

[latex]45\%[/latex]

[latex]\frac{9}{50}[/latex]

[latex]0.18[/latex]

[latex]18\%[/latex]

[latex]\frac{1}{3}[/latex]

[latex]0.\bar{3}[/latex]

[latex]33.\bar{3}\%[/latex]

[latex]\frac{1}{125}[/latex]

[latex]0.008[/latex]

[latex]0.8\%[/latex]

[latex]2[/latex]

[latex]200\%[/latex]

86.

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

Translate and Solve Basic Percent Equations

In the following exercises, translate and solve.

87. What number is 45% of 120?

Solution
54

88. What number is 65% of 100?

89. What number is 24% of 112?

Solution
26.9

90. What number is 36 of 124?

91. 250% of 65 is what number?

Solution
162.5

92. 150% of 90 is what number?

93. 800% of 2,250 is what number?

Solution
18,000

94. 600% of 1,740 is what number?

95. 28 is 25% of what number?

Solution
112

96. 36 is 25% of what number?

97. 81 is 75% of what number?

Solution
108

98. 93 is 75% of what number?

99. 8.2% of what number is $2.87?

Solution
$35

100. 6.4% of what number is $2.88?

101. 11.5% of what number is $108.10?

Solution
$940

102. 12.3% of what number is $92.25?

103. What percent of 260 is 78?

Solution
30%

104. What percent of 215 is 86?

105. What percent of 1,500 is 540?

Solution
36%

106. What percent of 1,800 is 846?

107. 30 is what percent of 20?

Solution
150%

108. 50 is what percent of 40?

109. 840 is what percent of 480?

Solution
175%

110. 790 is what percent of 395?

Everyday Math

111. 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%.

a) Convert 10% to a fraction.

b) Use your answer from (a) to estimate the sales tax Felipa would pay on a $95 dress.

Solution

a) [latex]\frac{1}{10}[/latex]

b) About [latex]\$9.50[/latex].

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

a) Write 25% as a fraction.

b) Use your answer from (a) 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
$A$	$4/5$
$B$	$3.5/4$
$C$	$85%$

113. Write seller C’s rating as (a) a fraction and (b) a decimal.

Solution

a) [latex]\frac{17}{20}[/latex]

b) [latex]0.85[/latex]

114. Write seller B’s rating as (a) a percent and (b) a decimal.

115. Write seller A’s rating as (a) a percent and (b) a decimal.

Solution

a) [latex]80\%[/latex]

b) [latex]0.8[/latex]

116. Which seller should Amelio buy from and why?

Writing Exercises

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

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

119. 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.

120. 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.

Glossary

percent

A percent is a ratio whose denominator is 100.

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Everyday Math

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