CHAPTER 4 Ratio, Proportion, and Percent

# 4.2 Understand Percent

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

# Use the Definition of Percent

How many cents are in one dollar? There are cents in a dollar. How many years are in a century? There are 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 . We use the percent symbol to show percent.

Percent

A percent is a ratio whose denominator is .

According to data from the Statistics Canada,  of Canadian Internet users reported a cyber security incident, including being redirected to fraudulent websites that asked for personal information or getting a virus or other computer infection. This means out of every Canadian internet users reported cyber security incidents as (Figure 1) shows. Out of the squares on the grid, are shaded, which we write as the ratio .

Similarly, means a ratio of means a ratio of and means a ratio of . In words, “one hundred percent” means the total is , and since , we see that means whole.

EXAMPLE 1

According to a survey done by Universities Canada of Canada’s Universities are working to include Indigenous representation within their governance or leadership structures.Write this percent as a ratio.

Solution
 The amount we want to convert is 71%. Write the percent as a ratio. Remember that percent means per 100.

TRY IT 1.1

Write the percent as a ratio.

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

TRY  IT 1.2

Write the percent as a ratio.

A study found that of Canadian teens send text messages regularly.

EXAMPLE 2

In , according to a Universities Canada survey, out of every of today’s undergraduates benefit from experiential learning such as co-ops, internships and service learning. Write this as a ratio and then as a percent.

Solution
 The amount we want to convert is out of . out of Write as a ratio. Convert the 56 per 100 to percent.

TRY IT 2.1

Write as a ratio and then as a percent: According to Statistics Canada, only out of young Canadians cross a provincial border to complete their university degree.

TRY IT 2.2

Write as a ratio and then as a percent: According to an international comparison done by the British Council, out of current professional leaders across 30 countries and in all sectors, are liberal arts grads with bachelor’s degrees in the social sciences or humanities.

# Convert Percents to Fractions and Decimals

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

Convert a Percent to a Fraction.

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

EXAMPLE 3

Convert each percent to a fraction:

Solution
 a) Write as a ratio with denominator 100. Simplify.
 b) Write as a ratio with denominator 100. Simplify.

TRY IT 3.1

Convert each percent to a fraction:

TRY IT 3.2

Convert each percent to a fraction:

The previous example shows that a percent can be greater than . We saw that means , or . These are improper fractions, and their values are greater than one.

EXAMPLE 4

Convert each percent to a fraction:

Solution
 a) Write as a ratio with denominator 100. Clear the decimal by multiplying numerator and denominator by 10. Multiply. Rewrite showing common factors. Simplify.
 b) Write as a ratio with denominator 100. Write the numerator as an improper fraction. Rewrite as fraction division, replacing 100 with . Multiply by the reciprocal. Simplify.

TRY IT 4.1

Convert each percent to a fraction:

TRY IT 4.2

Convert each percent to a fraction:

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

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

EXAMPLE 5

Convert each percent to a decimal:

Solution

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

 a) Write as a ratio with denominator 100. Change the fraction to a decimal by dividing the numerator by the denominator.
 b) Write as a ratio with denominator 100. Change the fraction to a decimal by dividing the numerator by the denominator.

TRY IT 5.1

Convert each percent to a decimal:

1. 0.09
2. 0.87

TRY IT 5.2

Convert each percent to a decimal:

1. 0.03
2. 0.91

EXAMPLE 6

Convert each percent to a decimal:

Solution
 a) Write as a ratio with denominator 100. Change the fraction to a decimal by dividing the numerator by the denominator.
 b) Write as a ratio with denominator 100. Change the fraction to a decimal by dividing the numerator by the denominator.

TRY IT 6.1

Convert each percent to a decimal:

1. 1.15
2. 0.235

TRY IT 6.2

Convert each percent to a decimal:

1. 1.23
2. 0.168

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

Percent Decimal

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 as , we can think of as .) 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 the table above and shows visually how to convert them to decimals by moving the decimal point two places to the left.

EXAMPLE 7

Convert the percent to: a) a fraction b) a decimal

Solution
 a) Write as a ratio with denominator 100.
 b) Change the fraction to a decimal by dividing the numerator by the denominator.

TRY IT 7.1

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

Twitter’s share of web traffic jumped when one celebrity tweeted live on air.

TRY IT 7.2

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

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 . Convert the percent to:

1. a fraction
2. a decimal
Solution
 a) Write as a ratio with denominator 100. Simplify.
 b) Change the fraction to a decimal by dividing the numerator by the denominator.

TRY IT 8.1

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

The probability that it will rain Monday is .

TRY IT 8.2

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 .

# 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 , it is easy to change that fraction to a percent.

HOW TO: Convert a Decimal to a Percent

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

EXAMPLE 9

Convert each decimal to a percent: a) b)

Solution
 a) Write as a fraction. The denominator is 100. Write this ratio as a percent.
 b) The denominator is 100. Write this ratio as a percent.

TRY IT 9.1

Convert each decimal to a percent: a) b).

1. 1%
2. 17%

TRY IT 9.2

Convert each decimal to a percent: a) b)

1. 4%
2. 41%

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

EXAMPLE 10

Convert each decimal to a percent: a) b)

Solution
 a) Write as a fraction. Write as an improper fraction. The denominator is 100. Write this ratio as a percent.

Notice that since > , the result is more than

 b) Write as a fraction. The denominator is 1,000. Divide the numerator and denominator by 10, so that the denominator is 100. Write this ratio as a percent.

TRY IT 10.1

Convert each decimal to a percent: a) b)

1. 175%
2. 8.25%

TRY IT 10.2

Convert each decimal to a percent: a) b)

1. 225%
2. 9.25%

Let’s summarize the results from the previous examples in the table below so we can look for a pattern.

Decimal Percent

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.3) uses the decimal numbers in the table above and shows visually to convert them to percents by moving the decimal point two places to the right and then writing the sign.

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

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

EXAMPLE 11

Convert each fraction or mixed number to a percent: a) b) c)

Solution

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

 a) Change to a decimal. Write as a percent by moving the decimal two places.
 b) Change to a decimal. Write as a percent by moving the decimal two places.
 c) Write as an improper fraction. Change to a decimal. Write as a percent.

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

TRY IT 11.1

Convert each fraction or mixed number to a percent: a) b) c)

1. 62.5%
2. 275%
3. 340%

TRY IT 11.2

Convert each fraction or mixed number to a percent: a) b) c)

1. 87.5%
2. 225%
3. 160%

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 to a percent.

Solution

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

 Change to a decimal—rounding to the nearest thousandth. Write as a percent.

TRY IT 12.1

Convert the fraction to a percent:

42.9%

TRY IT 12.2

Convert the fraction to a percent:

57.1%

When we first looked at fractions and decimals, we saw that fractions converted to a repeating decimal. When we converted the fraction to a decimal, we wrote the answer as . We will use this same notation, as well as fraction notation, when we convert fractions to percents in the next example.

EXAMPLE 13

Statistics Canada reported in 2018 that approximately of Canadian adults are obese. Convert the fraction to a percent.

Solution
 Change to a decimal. Write as a repeating decimal. Write as a percent.

We could also write the percent as .

TRY IT 13.1

Convert the fraction to a percent:

According to the Canadian Census 2016, about people within the population of Canada are between the ages of and .

TRY IT 13.2

Convert the fraction to a percent:

# 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.

# Glossary

percent
A percent is a ratio whose denominator is .

# Practice Makes Perfect

## Use the Definition of Percents

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

 1. In , the unemployment rate for those with only a high school degree was . 2. In , among the unemployed, were long-term unemployed. 3. The unemployment rate for those with Bachelor’s degrees was in . 4. The unemployment rate in Canada in was . In the following exercises, write as a) a ratio and b) a percent 5. out of nursing candidates received their degree at a community college. 6. out of firefighters and law enforcement officers were educated at a community college. 7. out of first-time freshmen students attend a community college. 8. out of 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 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

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

 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

In the following exercises, convert each percent to

a) a simplified fraction and

b) a decimal

 41. In of home sales had owner financing. (Source: Bloomberg Businessweek, 5/23–29/2011) 42. In of the Canadian population was a visible minority. (Source: www12.statcan.gc.ca) 43. According to government data, in the number of cell phones in India was of the population. 44. According to the Survey of Earned Doctorates, among Canadians age or older who had doctorate degrees in are women. 45. A couple plans to have two children. The probability they will have two girls is . 46. Javier will choose one digit at random from through . The probability he will choose is . 47. According to the local weather report, the probability of thunderstorms in New York City on July is . 48. A club sells tickets to a raffle. Osbaldo bought one ticket. The probability he will win the raffle is .

## Convert Decimals and Fractions to Percents

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

 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

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

 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

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

 81. of washing machines needed repair. 82. of dishwashers needed repair.

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

 83. According to the Government of Canada, in of Canadian adults were overweight or obese. 84. Statistics Canada showed that in of Canadian workers are using more than one language at work.

In the following exercises, complete the table.

85.

Fraction Decimal Percent
86.

Fraction Decimal Percent

## Everyday Math

87. 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 . She knows this is a little less than .

a) Convert to a fraction

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

88. Savings Ryan has of each paycheck automatically deposited in his savings account.

a) Write as a fraction.

b) Use your answer from a) to find the amount that goes to savings from Ryan’s 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.

 Seller Rating
89. Write seller rating as a fraction and a decimal. 90. Write seller rating as a percent and a decimal.
91. Write seller rating as a percent and a decimal. 92. Which seller should Amelio buy from and why?

## Writing Exercises

 93. Convert to fractions. Do you notice a pattern? Explain what the pattern is. 94. Convert , and to percents. Do you notice a pattern? Explain what the pattern is. 95. When the Szetos sold their home, the selling price was of what they had paid for the house ago. Explain what means in this context. 96. According to cnn.com, cell phone use in was of what it had been in . Explain what means in this context.

1. 3. 5.

a)

b)

7.

a)

b)

9. 11.
13. 15. 17.
19. 21. 23.
25. 0.05 27. 0.01 29. 0.63
31. 0.4 33. 1.15 35. 1.5
37. 0.214 39. 0.078 41.

a)

b)

43.

a)

b)

45.

a)

b)

47.

a)

b)

49. 1% 51. 18% 53. 135%
55. 300% 57. 0.9% 59. 8.75%
61. 150% 63. 225.4% 65. 25%
67. 37.5% 69. 175% 71. 680%
73. 41.7% 75. 77. 42.9%
79. 55.6% 81. 25% 83. 64%
85.

Fraction Decimal Percent
87.

a)

b)

89.
91. 80%; 0.8 93. . 95. The Szetos sold their home for five times what they paid 30 years ago.