9.1: Percent, Decimals, Fractions

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Jul 15, 2024
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- 9: Percent
- 9.2: Solving Basic Percent Problems

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In the square shown in Figure $\PageIndex{1}$ , a large square has been partitioned into ten rows of ten little squares in each row. In Figure 7.1, we’ve shaded 20 of 100 possible little squares, or 20% of the total number of little squares.

Screen Shot 2019-09-20 at 3.32.24 PM.png — Figure $\PageIndex{1}$ : Shading 20 of 100 little squares, or 20% of the total number of little squares.

The Meaning of Percent

Percent means “parts per hundred.”

In Figure 7.1, 80 out of a possible 100 squares are left unshaded. Thus, 80% of the little squares are unshaded. If instead we shaded 35 out of the 100 squares, then 35% of the little squares would be shaded. If we shaded all of the little squares, then 100% of the little squares would be shaded (100 out of 100).

So, when you hear the word “percent,” think “parts per hundred.”

Changing a Percent to a Fraction

Based on the discussion above, it is fairly straightforward to change a percent to a fraction.

Percent to Fraction

To change a percent to a fraction, drop the percent sign and put the number over 100.

Example 1

Change 24% to a fraction.

Solution

Drop the percent symbol and put 24 over 100.

$\begin{aligned} 24 \% = \frac{24}{100} ~ & \textcolor{red}{ \text{ Percent: Parts per hundred.}} \\ = \frac{6}{25} ~ & \textcolor{red}{ \text{ Reduce.}} \end{aligned}\nonumber$

Hence, 24% = 6/25.

Exercise

Change 36% to a fraction reduced to lowest terms.

Answer: 9/25

Example 2

Change $14 \frac{2}{7} \%$ to a fraction.

Solution

Drop the percent symbol and put $14 \frac{2}{7}$ over 100.

$\begin{aligned} 14 \frac{2}{7} \% = \frac{14 \frac{2}{7}}{100} ~ & \textcolor{red}{ \text{ Percent: Parts per hundred.}} \\ = \frac{\frac{100}{7}}{100} ~ & \textcolor{red}{ \text{ Mixed to improper fraction.}} \\ = \frac{100}{7} \cdot \frac{1}{100} ~ & \textcolor{red}{ \text{ Invert and multiply.}} \\ = \frac{\cancel{100}}{7} \cdot \frac{1}{\cancel{100}} ~ & \textcolor{red}{ \text{ Cancel.}} \\ = \frac{1}{7} \end{aligned}\nonumber$

Hence, $14 \frac{2}{7} \%=1/7.$

Exercise

Change $11 \frac{1}{9} \%$ to a fraction reduced to lowest terms.

Answer: 1/9

Example 3

Change 28.4% to a fraction.

Solution

Drop the percent symbol and put 28.4 over 100.

$\begin{aligned} 28.4 \% = \frac{28.4}{100} ~ & \textcolor{red}{ \text{ Percent: Parts per hundred.}} \\ = \frac{28.4 \cdot \textcolor{red}{10}}{100 \cdot \textcolor{red}{10}} ~ & \textcolor{red}{ \text{ Multiply numerator and denominator by 10.}} \\ = \frac{284}{1000} ~ & \textcolor{red}{ \text{ Multiplying by 10 moves decimal point one place right.}} \\ = \frac{71 \cdot 4}{250 \cdot 4} ~ & \textcolor{red}{ \text{ Factor.}} \\ = \frac{71}{250} ~ & \textcolor{red}{ \text{ Cancel common factor.}} \end{aligned}\nonumber$

Exercise

Change 87.5% to a fraction reduced to lowest terms.

Answer: 7/8

Changing a Percent to a Decimal

To change a percent to a decimal, we need only remember that percent means “parts per hundred.”

Example 4

Change 23.25% to a decimal.

Solution

Drop the percent symbol and put 23.25 over 100.

$\begin{aligned} 23.25 \% = \frac{23.25}{100} ~ & \textcolor{red}{ \text{ Percent: Parts per hundred.}} \\ = 0.2325 ~ & \textcolor{red}{ \text{ Dividing by 100 moves decimal point 2 places left.}} \end{aligned}\nonumber$

Therefore, 23.25% = 0.2325.

Exercise

Change 2.4% to a decimal.

Answer: 0.024

This last example motivates the following simple rule.

Changing a Percent to a Decimal

To change a percent to a decimal, drop the percent symbol and move the decimal point two places to the left.

Example 5

Change $5 \frac{1}{2} \%$ to a decimal.

Solution

Note that 1/2=0.5, then move the decimal 2 places to the left.

$\begin{aligned} 5 \frac{1}{2} \% = 5.5 \% ~ & \textcolor{red}{1/2 = 0.5.} \\ = 0.05 5 ~ & \textcolor{red}{ \begin{array}{l} \text{ Drop % symbol.} \\ \text{ Move decimal point 2 places left.} \end{array}} \\ = 0.055 \end{aligned}\nonumber$

Thus, $5 \frac{1}{2} \%=0.055$ .

Exercise $\PageIndex{1}$

Change $6 \frac{3}{4} \%$ to a decimal.

Answer: 0.0675

Changing a Decimal to a Percent

Changing a decimal to a percent is the exact opposite of changing a percent to a decimal. In the latter case, we drop the percent symbol and move the decimal point 2 places to the left. The following rule does just the opposite.

Changing a Decimal to a Percent

To change a decimal to a percent, move the decimal point two places to the right and add a percent symbol.

Example 6

Change 0.0725 to a percent.

Solution

Move the decimal point two places to the right and add a percent symbol.

$\begin{aligned} 0.0725 = 007.25 \% \\ = 7.25 \% \end{aligned}\nonumber$

Exercise

Change to 0.0375 to a percent.

Answer: 3.75%

Example 7

Change 1.025 to a percent.

Solution

Move the decimal point two places to the right and add a percent symbol.

$\begin{aligned} 1.025 = 102.5 \% \\ = 102.5 \% \end{aligned}\nonumber$

Exercise

Change 0.525 to a percent.

Answer: 52.5%

Changing a Fraction to a Percent

One way to proceed is to first change the fraction to a decimal, then change the resulting decimal to a percent.

Fractions to Percents: Technique #1

To change a fraction to a percent, follow these steps:

Divide numerator by the denominator to change the fraction to a decimal.
Move the decimal point in the result two places to the right and append a percent symbol.

Example 8

Use Technique #1 to change 5/8 to a percent.

Solution

Change 5/8 to a decimal, then change the decimal to a percent.

To change 5/8 to a decimal, divide 5 by 8. Since the denominator is a product of twos, the decimal should terminate.

$Screen Shot 2019-09-20 at 4.13.24 PM.png$

To change 0.625 to a percent, move the decimal point 2 places to the right and append a percent symbol.

0.625 = 0 62.5% = 62.5%

Exercise

Change 5/16 to a percent.

Answer: 31.35%

A second technique is to create an equivalent fraction with a denominator of 100.

Fractions to Percents: Technique #2

To change a fraction to a percent, create an equivalent fraction with a denominator of 100.

Example 9

Use Technique #2 to change 5/8 to a percent.

Solution

Create an equivalent fraction for 5/8 with a denominator of 100.

$\frac{5}{8} = \frac{x}{100}\nonumber$

Solve this proportion for x.

$\begin{aligned} 8x = 500 ~ & \textcolor{red}{ \text{ Cross multiply.}} \\ \frac{8x}{8} = \frac{500}{8} ~ & \textcolor{red}{ \text{ Divide both sides by 8.}} \\ x = \frac{125}{2} ~ & \textcolor{red}{ \text{ Reduce: Divide numerator and denominator by 4.}} \\ x = 62.5 ~ & \textcolor{red}{ \text{ Divide.}} \end{aligned}\nonumber$

Thus,

$\frac{5}{8} = \frac{62.5}{100} = 62.5 \%.\nonumber$

Alternate Ending

We could also change 125/2 to a mixed fraction; i.e., 125/2 = 62 1 2 . Then,

$\frac{5}{8} = \frac{62 \frac{1}{2}}{100} = 62 \frac{1}{2} \%.\nonumber$

Same answer.

Exercise

Change 4/9 to a percent.

Answer: $44 \frac{4}{9} \%$

Sometimes we will be content with an approximation.

Example 10

Change 4/13 to a percent. Round your answer to the nearest tenth of a percent.

Solution

We will use Technique #1.

To change 4/13 to a decimal, divide 4 by 13. Since the denominator has factors other than 2’s and 5’s, the decimal will repeat. However, we intend to round to the nearest tenth of a percent, so we will carry the division to four decimal places only. (Four places are necessary because we will be moving the decimal point two places to the right.)

$Screen Shot 2019-09-20 at 4.13.24 PM.png$

To change the decimal to a percent, move the decimal point two places to the right.

0.3076 ≈ 0 30.76% ≈ 30.76%

To round to the nearest tenth of a percent, identify the rounding and test digits.

$Screen Shot 2019-09-20 at 4.23.25 PM.png$

Because the test digit is greater than or equal to 5, add 1 to the rounding digit and truncate. Thus,

0.03076 ≈ 30.8%.

Exercise

Change 4/17 to a percent. Round your answer to the nearest tenth of a percent.

Answer: 23.5%

Exercises

Suggested exercises for PrePALS: 1, 3, 5, 7, 11, 17, 19, 25, 27, 35, 39, 41, 49, 51, 55, 69, 71, 73, 75, 81, 83

In Exercises 1-18, convert the given percent to a fraction, and simplify the result.

1. $4 \frac{7}{10} \%$

2. $7 \frac{1}{4} \%$

3. $7 \frac{2}{9} \%$

4. $4 \frac{9}{10} \%$

5. 11.76%

6. 15.2%

7. 13.99%

8. 18.66%

9. $4 \frac{1}{2} \%$

10. $8 \frac{5}{8} \%$

11. 192%

12. 5%

13. 86%

14. 177%

15. 130%

16. 80%

17. 4.07%

18. 6.5%

In Exercises 19-34, convert the given percent to a decimal.

19. 124%

20. 4%

21. 0.6379%

22. 0.21%

23. 28%

24. 5.4%

25. 0.83%

26. 0.3344%

27. 8%

28. 3%

29. 59.84%

30. 0.17%

31. 155%

32. 7%

33. 36.5%

34. 39.7%

In Exercises 35-50, convert the given decimal to a percent.

35. 8.888

36. 5.1

37. 0.85

38. 0.08

39. 1.681

40. 3.372

41. 0.14

42. 4.89

43. 8.7

44. 8.78

45. 0.38

46. 1.67

47. 0.02

48. 0.07

49. 0.044

50. 0.29

In Exercises 51-68, convert the given fraction to a percent.

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

52. $\frac{29}{8}$

53. $\frac{5}{2}$

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

55. $\frac{8}{5}$

56. $\frac{7}{20}$

57. $\frac{14}{5}$

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

59. $\frac{9}{2}$

60. \(\frac{18}{25}|)

61. $\frac{9}{4}$

62. $\frac{7}{8}$

63. $\frac{7}{5}$

64. $\frac{4}{25}$

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

66. $\frac{23}{8}$

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

68. $\frac{13}{2}$

69. Convert 24/29 to a percent, and round your answer to the nearest hundredth of a percent.

70. Convert 5/3 to a percent, and round your answer to the nearest hundredth of a percent.

71. Convert 15/7 to a percent, and round your answer to the nearest tenth of a percent.

72. Convert 10/7 to a percent, and round your answer to the nearest tenth of a percent.

73. Convert 7/24 to a percent, and round your answer to the nearest hundredth of a percent.

74. Convert 5/6 to a percent, and round your answer to the nearest hundredth of a percent.

75. Convert 8/3 to a percent, and round your answer to the nearest tenth of a percent.

76. Convert 22/21 to a percent, and round your answer to the nearest tenth of a percent.

77. Convert 9/23 to a percent, and round your answer to the nearest tenth of a percent.

78. Convert 11/9 to a percent, and round your answer to the nearest tenth of a percent.

79. Convert 17/27 to a percent, and round your answer to the nearest hundredth of a percent.

80. Convert 22/27 to a percent, and round your answer to the nearest hundredth of a percent.

81. Crime rates. Preliminary crime rates for the first six months of 2009 compared to the same period in 2008 are shown below. Associated Press-Times-Standard 12/22/09 Despite recession, the national crime rates keep falling.

$\begin{array}{c c} \text{Murder } & ~ −10.0 \% \\ \text{Forcible rape } & ~ −3.3 \% \\ \text{Robbery } & ~ −6.5 \% \\ \text{Aggravated assault } & ~ −3.2 \% \\ \text{Burglary } ~ & −2.5 \% \\ \text{Larceny-theft } & ~ −5.3 \% \\ \text{Motor vehicle theft } & ~ −18.75 \% \\ \text{Arson } & ~ −8.2 \% \end{array}\nonumber$

Source: Federal Bureau of Investigation

i) What do the negative signs indicate?

ii) Which type of crime decreased the most?

iii) Which type of crime decreased the least?

82. Major Hurricanes. 5 of the 8 hurricanes in 2008 were categorized as major. Write the fractional number of major hurricanes in 2008 as a percent. NOAA Associated Press 5/22/09

83. Chance of flood. These excerpts are from the story Corps: Dam work lessens Seattle-area flood chance published in the Times-Standard on Nov. 6, 2009. Write all four of the odds of flooding as a percent chance. Round to the nearest tenth of a percent if necessary.

i) Col. Anthony Wright, from the U.S. Army Corps of Engineers, speaking of the repairs to the Green River Dam, reported there was now a 1-in-25 chance that a storm would force the corps to release enough water from the dam’s reservoir to cause a flood downstream in the Green River Valley.

ii) The odds of widespread flooding in the valley improve to 1-in-32 when all the sandbagging and flood-protection efforts are factored in.

iii) Previously, the Corps of Engineers said the chance of widespread flooding was 1-in-4.

iv) When the dam operates at capacity, there is a 1-in-140 chance of flooding.

Answers

1. $\frac{47}{1000}$

3. $\frac{13}{180}$

5. $\frac{147}{1250}$

7. $\frac{1399}{10000}$

9. $\frac{9}{200}$

11. $\frac{48}{25}$

13. $\frac{43}{50}$

15. $\frac{13}{10}$

17. $\frac{407}{10000}$

19. 1.24

21. 0.006379

23. 0.28

25. 0.0083

27. 0.08

29. 0.5984

31. 1.55

33. 0.365

35. 888.8%

37. 85%

39. 168.1%

41. 14%

43. 870%

45. 38%

47. 2%

49. 4.4%

51. 50%

53. 250%

55. 160%

57. 280%

59. 450%

61. 225%

63. 140%

65. 120%

67. 240%

69. 82.76%

71. 214.3%

73. 29.17%

75. 266.7%

77. 39.1%

79. 62.96%

81.

i) The negative signs indicate the crime rate has decreased from previous measures.

ii) Motor vehicle theft decreased the most with an 18.75% decrease.

iii) Burglary decreased the least with a 2.5% decrease.

83.

i) 4% chance of flood

ii) 3.1% chance of flood

iii) 25% chance of flood

iv) 0.7% chance of flood

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