Understand Percent

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 shows. Out of the 100 squares on the grid, 57 are shaded, which we write as the ratio 57100.

Similarly, 25% means a ratio of 25100,3% means a ratio of 3100 and 100% means a ratio of 100100. In words, "one hundred percent" means the total 100% is 100100, and since 100100=1, we see that 100% means 1 whole. 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%. 44% Write the percent as a ratio. Remember that percent means per 100. 44100

Write the percent as a ratio. According to a survey, 89% of college students have a smartphone. 89100 Write the percent as a ratio. A study found that 72% of U.S. teens send text messages regularly. 72100 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 21 out of 100. 21 out of 100 Write as a ratio. 21100 Convert the 21 per 100 to percent. 21%

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. 62100,62% 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. 41100,41%

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. Convert a percent to a fraction. Write the percent as a ratio with the denominator 100. Simplify the fraction if possible. Convert each percent to a fraction: ⓐ 36% ⓑ 125% Solution ⓐ 36% Write as a ratio with denominator 100. 36100 Simplify. 925

ⓑ 125% Write as a ratio with denominator 100. 125100 Simplify. 54

Convert each percent to a fraction: ⓐ 48% ⓑ 110% ⓐ 1225 ⓑ 1110 Convert each percent to a fraction: ⓐ 64% ⓑ 150% ⓐ 1625 ⓑ 32 The previous example shows that a percent can be greater than 1. We saw that 125% means 125100, or 54. These are improper fractions, and their values are greater than one. Convert each percent to a fraction: ⓐ 24.5% ⓑ 3313% Solution ⓐ 24.5% Write as a ratio with denominator 100. 24.5100 Clear the decimal by multiplying numerator and denominator by 10. 24.5(10)100(10) Multiply. 2451000 Rewrite showing common factors. 5·495·200 Simplify. 49200

ⓑ 3313% Write as a ratio with denominator 100. 3313100 Write the numerator as an improper fraction. 1003100 Rewrite as fraction division, replacing 100 with 1001. 1003÷1001 Multiply by the reciprocal. 1003⋅1100 Simplify. 13

Convert each percent to a fraction: ⓐ 64.4% ⓑ 6623% ⓐ 161250 ⓑ 23 Convert each percent to a fraction: ⓐ 42.5% ⓑ 834% ⓐ 1740 ⓑ 780 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.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. Convert each percent to a decimal: ⓐ 6% ⓑ 78% 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. 6100 Change the fraction to a decimal by dividing the numerator by the denominator. 0.06

ⓑ 78% Write as a ratio with denominator 100. 78100 Change the fraction to a decimal by dividing the numerator by the denominator. 0.78

Convert each percent to a decimal: ⓐ 9% ⓑ 87% ⓐ 0.09 ⓑ 0.87 Convert each percent to a decimal: ⓐ 3% ⓑ 91% ⓐ 0.03 ⓑ 0.91 Convert each percent to a decimal: ⓐ 135% ⓑ 12.5% Solution ⓐ 135% Write as a ratio with denominator 100. 135100 Change the fraction to a decimal by dividing the numerator by the denominator. 1.35

ⓑ 12.5% Write as a ratio with denominator 100. 12.5100 Change the fraction to a decimal by dividing the numerator by the denominator. 0.125

Convert each percent to a decimal: ⓐ 115% ⓑ 23.5% ⓐ 1.15 ⓑ 0.235 Convert each percent to a decimal: ⓐ 123% ⓑ 16.8% ⓐ 1.23 ⓑ 0.168 Let's summarize the results from the previous examples in , and look for a pattern we could use to quickly convert a percent number to a decimal number. 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. uses the percents in and shows visually how to convert them to decimals by moving the decimal point two places to the left.

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 Solution ⓐ 77% Write as a ratio with denominator 100. 77100

ⓑ 77100 Change the fraction to a decimal by dividing the numerator by the denominator. 0.77

Convert the percent to: ⓐ a fraction and ⓑ a decimal Twitter's share of web traffic jumped 24% when one celebrity tweeted live on air. ⓐ 625 ⓑ 0.24 Convert the percent to: ⓐ a fraction and ⓑ a decimalThe U.S. Census estimated that in 2013,44% of the population of Boston age 25 or older have a bachelor's or higher degrees. ⓐ 1125 ⓑ 0.44 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

Solution ⓐ 25% Write as a ratio with denominator 100. 25100 Simplify. 14

ⓑ 14 Change the fraction to a decimal by dividing the numerator by the denominator. 0.25

Convert the percent to: ⓐ a fraction, and ⓑ a decimal The probability that it will rain Monday is 30%. ⓐ 310 ⓑ 0.3 Convert the percent to: ⓐ a fraction, and ⓑ a decimal The probability of getting heads three times when tossing a coin three times is 12.5%. ⓐ 18 ⓑ 0.125

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. 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. Convert each decimal to a percent: ⓐ 0.05 ⓑ 0.83 Solution ⓐ 0.05 Write as a fraction. The denominator is 100. 5100 Write this ratio as a percent. 5%

ⓑ 0.83 The denominator is 100. 83100 Write this ratio as a percent. 83%

Convert each decimal to a percent: ⓐ 0.01 ⓑ 0.17. ⓐ 1% ⓑ 17% Convert each decimal to a percent: ⓐ 0.04 ⓑ 0.41 ⓐ 4% ⓑ 41% To convert a mixed number to a percent, we first write it as an improper fraction. Convert each decimal to a percent: ⓐ 1.05 ⓑ 0.075 Solution ⓐ 1.05 Write as a fraction. 15100 Write as an improper fraction. The denominator is 100. 105100 Write this ratio as a percent. 105%

Notice that since 1.05>1, the result is more than 100%. ⓑ 0.075 Write as a fraction. The denominator is 1,000. 751,000 Divide the numerator and denominator by 10, so that the denominator is 100. 7.5100 Write this ratio as a percent. 7.5%

Convert each decimal to a percent: ⓐ 1.75 ⓑ 0.0825 ⓐ 175% ⓑ 8.25% Convert each decimal to a percent: ⓐ 2.25 ⓑ 0.0925 ⓐ 225% ⓑ 9.25% Let's summarize the results from the previous examples in so we can look for a pattern. 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. uses the decimal numbers in 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.Convert a fraction to a percent. Convert the fraction to a decimal. Convert the decimal to a percent. Convert each fraction or mixed number to a percent: ⓐ 34 ⓑ 118 ⓒ 215 Solution To convert a fraction to a decimal, divide the numerator by the denominator. ⓐ Change to a decimal. 34 Write as a percent by moving the decimal two places.

75%

ⓑ Change to a decimal. 118 Write as a percent by moving the decimal two places.

137.5%

220%

Notice that we needed to add zeros at the end of the number when moving the decimal two places to the right. Convert each fraction or mixed number to a percent: ⓐ 58 ⓑ 114 ⓒ 325 ⓐ 62.5% ⓑ 275% ⓒ 340% Convert each fraction or mixed number to a percent: ⓐ 78 ⓑ 94 ⓒ 135 ⓐ 87.5% ⓑ 225% ⓒ 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. Convert 57 to a percent. Solution To change a fraction to a decimal, we divide the numerator by the denominator. 57 Change to a decimal—rounding to the nearest thousandth. 0.714 Write as a percent. 71.4%

Convert the fraction to a percent: 37 42.9% Convert the fraction to a percent: 47 57.1% 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 43 to a decimal, we wrote the answer as 1.3¯. We will use this same notation, as well as fraction notation, when we convert fractions to percents in the next example. An article in a medical journal claimed that approximately 13 of American adults are obese. Convert the fraction 13 to a percent. Solution 13 Change to a decimal.

Write as a repeating decimal. 0.333… Write as a percent. 3313%

We could also write the percent as 33.3_%. Convert the fraction to a percent: According to the U.S. Census Bureau, about 19 of United States housing units have just 1 bedroom. 11.1–%,or1119% Convert the fraction to a percent: According to the U.S. Census Bureau, about 16 of Colorado residents speak a language other than English at home. 16.6–%,or1623%

Practice Makes PerfectUse 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%. 6100 In 2015, among the unemployed, 29% were long-term unemployed. The unemployment rate for those with Bachelor's degrees was 3.2% in 2014. 321000 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. ⓐ 57100 ⓑ 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. ⓐ 42100 ⓑ 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% 125 8% 17% 17100 19% 52% 1325 78% 125% 54 135% 37.5% 38 42.5% 18.4% 23125 46.4% 912% 19200 812% 513% 475 623% In the following exercises, convert each percent to a decimal. 5% 0.05 9% 1% 0.01 2% 63% 0.63 71% 40% 0.4 50% 115% 1.15 125% 150% 1.5 250% 21.4% 0.214 39.3% 7.8% 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) ⓐ 3200 ⓑ 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. ⓐ 702310,000 ⓑ 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%. ⓐ 14 ⓑ 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%. ⓐ 35 ⓑ 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 1% 0.03 0.18 18% 0.15 1.35 135% 1.56 3 300% 4 0.009 0.9% 0.008 0.0875 8.75% 0.0625 1.5 150% 2.2 2.254 225.4% 2.317 In the following exercises, convert each fraction to a percent. 14 25% 15 38 37.5% 58 74 175% 98 645 680% 514 512 4123%or41.6–% 1112 223 266.6–% 123 37 42.9% 67 59 55.5–% 49 In the following exercises, convert each fraction to a percent. 14 of washing machines needed repair. 25% 15 of dishwashers needed repair. In the following exercises, convert each fraction to a percent. According to the National Center for Health Statistics, in 2012,720 of American adults were obese. 35% 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. Fraction Decimal Percent 12 0.45 18% 13 0.008 2

Fraction Decimal Percent 12 0.5 50% 920 0.45 45% 950 0.18 18% 13 0.3 33.3% 1125 0.008 0.8% 2 2.0 200%

Fraction Decimal Percent 14 0.65 22% 23 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. ⓐ 110 ⓑ 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. 1720;0.85 Write seller B’s rating as a percent and a decimal. Write seller A’s rating as a percent and a decimal. 80%; 0.8 Which seller should Amelio buy from and why?

Writing Exercises Convert 25%,50%,75%,and100% to fractions. Do you notice a pattern? Explain what the pattern is. 14,12,34,1. Convert 110,210,310,410,510,610,710,810, and 910 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. 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.

ⓑ 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.