Fractions, Decimals and Percents
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C HAPTER
Chapter 1. Fractions, Decimals and Percents
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Fractions, Decimals and Percents
Introduction Fabulous Football
The local high school’s football season has just ended. At lunch, Carla, Mario and Cinda spent some time talking about the season and how the Springstead Raiders had finally made the play-offs. “I love football season. Those Friday night games are so much fun!” Mario said biting into his sandwich. “Definitely, and we did so much better this year than last year. This year we won 10 out of 12 games. Last year, we only won eight out of 12 games,” Carla said. “Yes, so this year the percentage of games that we won definitely increased,” Cinda commented. “Well that is obvious,” Mario said. “What was last year’s percentage compared to this year’s?” “To figure that out, we have to change the fraction of games that we won to a percentage and we have to do that with both last year’s statistics and this year’s statistics,” Carla explained. Carla is on the right track. To understand the percentage of wins, you need to know how to convert fractions to percentages. This lesson is all about fractions, decimals and percentages and how to work with them. You will also see how proportions can be very helpful when doing this work. Pay attention to this lesson and you will be able to use what you have learned at the end of the lesson. What You Will Learn In this lesson, you will learn how to do the following tasks: • Recognize percent as a ratio whose denominator is 100. • Write percents as decimals and decimals as percents. • Write percents as fractions and fractions as percents. 1
www.ck12.org • Find the percent of a number using fraction or decimal multiplication.
Teaching Time I. Recognize Percent as a Ratio Whose Denominator is 100 We see percents all around us every day. If you walk into a store of any kind, you will often see a percent sign. Whether it is a sale sign for 10% off or a sign on a bank for a new mortgage rate, by looking all around you, you will see percents. What is a percent? Simply, a percent is a part of a whole. It is a part of a whole that is being compared to 100. In this way, we can also think of a percent as a ratio. Remember that a ratio is a comparison between two quantities. Fractions and decimals are also parts of a whole. We can say that fractions, decimals and percents are all related. Let’s think about percents in a little more detail.
Look at the word “percent.” Its root “cent” means one hundred and “per” implies division or means “for each.” If you have three candies per person, it means three candies for each person. So “percent” means one for every one hundred. If 25% of the people want chocolate cake, then 25 out of every 100 want chocolate cake. For this reason, percent can be written as a fraction with the denominator being 100. 25% =
25 100
Now think about this again. Anytime you see a percent, you know that the amount is being compared to 100, or is “out of” 100. Example 18% means 18 out of 100 We are comparing the quantity of 18 to the whole of 100. Example 125% means 125 out of 100 Here we have a percent that is greater than 100. This means that we have greater than the total whole included in our percent. 2
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Chapter 1. Fractions, Decimals and Percents
That’s a great question. It is possible because we are thinking about what’s possible not the actual number. It is possible to have 100 percent on a test. However, if there are bonus questions, then you could also have greater than 100 percent. Sales is like this too. If a car salesman needs to sell 5 cars in a week that is 100 percent for him. However, if he sells 8 cars, then he sold greater than 100%.
Remember that a percent is a ratio compared to 100! Write this down in your notebook. II. Write Percents as Decimals and Decimals as Percents In the first section, we talked about how a percent is a part of a whole and that fractions and decimals are parts of a whole too. If this is true, then we can interchange the form that we write these quantities in. We can write a fraction as a decimal and as a percent. We can also write a percent as a decimal or a decimal as a percent and we can do the same with the fractions too. Let’s start by writing percents as decimals and decimals as percents. We can start by thinking about decimals. Decimal places represent powers of ten. The second decimal place is the hundredths place. The decimal .18 means eighteen-hundredths. 3
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Exactly, and now you might be able to see how we can write percent as a decimal. They are both comparing to hundreds. Example Write 56% as a decimal. To do this, we know that the % sign means “out of 100” we can say that decimal places represent tens and hundreds too. Two decimal places represents hundreds, just like the percent sign, %, represents hundreds. To change a percent to a decimal, we drop the percent sign and move the decimal point two places to the left. 56% = .56
Write this down in your notebook. 4
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Chapter 1. Fractions, Decimals and Percents
Example Write 88% as a decimal. To do this, we drop the percent sign and move the decimal point two places to the left. 88% = .88 Example Write 125% as a decimal. Even though this percent is greater than 100, we still follow the same steps. 125% = 1.25 We can work the other way around too and write decimals as percents. Here we will move the decimal point two places to the right and add a percent sign. Example Write .45 as a percent. First, we move the decimal point two places to the right and add a percent sign. .45 = 45% Example Write .345 as a percent. Follow the same rule here. Just move the decimal point two places to the right and add a percent sign. .345 = 34.5%
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www.ck12.org Write this rule down in your notebook. III. Write Percents as Fractions and Fractions as Percents You have seen that percents can be written as ratios with a denominator of 100 or they can be written as decimals. Well, if they can be written as a ratio with a denominator of 100, then those ratios can be simplified as we would simplify any fraction. Likewise, any fraction can be written as a percent using reverse operations. To write a percent as a fraction rewrite it as a fraction with a denominator of 100. Then reduce the fraction to its simplest form. Example 22% First, write this as a fraction with a dominator of 100. 22% =
22 100
Next, simplify the fraction. 22 100
=
11 50
This is our answer. How can we convert a fraction to a percent? To convert a fraction to a percent, we need to be sure that the fraction is being compared to a quantity of 100. Let’s look at an example. Example 28 100
This means that we have 28 out of 100. This fraction is being compared to 100, so we can simply change it to a percent. 28 100
= 28%
Example 3 5
This fraction is not being compared to 100. It is being compared to 5. We have three out of 5. To convert this fraction to a percent, we need to rewrite it as an equal ratio out of 100. We can use proportions to do this. First, write this ratio compared to a second ratio out of 100. 3 = 5 100 We don’t know what the part out of 100 is, so we need to solve the proportion. We can use multiplication to create equal ratios or a proportion.
5 × 20 = 100 3 × 20 = 60 3 60 = = 60% 5 100 This is our answer. Sometimes, you will be able to complete this step in your head using mental math. That is terrific! But if not, you can always create proportions and then write percents based on these equal fractions. 6
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Chapter 1. Fractions, Decimals and Percents
IV. Find the Percent of a Number using Fraction or Decimal Multiplication We said that percent are very useful. Part of the usefulness is being able to find percentages of a number. In other words, if you are planning a barbecue and the butcher tells you that 30% of the people want chicken. . . well, how many people is that? For how many people should you buy chicken? The key words here are “of a number” this means that you multiply to solve the problem. We can convert the percent to a decimal and multiply or convert it to a fraction and multiply.
Example Let’s suppose that you invited 58 people to the barbecue. If 30% prefer chicken, then we need to know how many people that is. First, convert 30% to either a decimal or a fraction
30% = .30 or 30% =
30 3 = 100 10
Using the decimal or fraction, you now multiply by the number of people you invited: .30 × 58 = 17.4 or about 17 people Or 3 10
× 58 =
174 10
= 17.4 or about 17 people.
This is the answer. Now let’s go back to the problem from the introduction and look at applying percents to this dilemma.
Real-Life Example Completed Fabulous Football Here is the original problem once again. First, reread it. There will be four parts to your answer. First, write two fractions to show the fraction of games won last year and this year. Then convert each fraction to a percent. The local high school’s football season has just ended. At lunch, Carla, Mario and Cinda spent some time talking about the season and how the Springstead Raiders had finally made the play-offs. 7
www.ck12.org “I love football season. Those Friday night games are so much fun!” Mario said biting into his sandwich. “Definitely, and we did so much better this year than last year. This year we won 10 out of 12 games. Last year, we only won eight out of 12 games,” Carla said. “Yes, so this year the percentage of games that we won definitely increased,” Cinda commented. “Well that is obvious,” Mario said. “What was last year’s percentage compared to this year’s?” “To figure that out, we have to change the fraction of games that we won to a percentage and we have to do that with both last year’s statistics and this year’s statistics,” Carla explained.
Now complete each part of this problem. Remember that there are four parts to your answer. Solution to Real –Life Example First, let’s write two fractions to represent the given data for games won last year and this year by the football team. Last year: 8 out of 12 games were won. 8 12 This year: 10 out of 12 games were won. 10 12 Now we can take these two fractions and create proportions to determine the correct percent of games won. 8 x = 12 100 Next, we cross multiply and solve for the percent.
12x = 800 x = 66.6% or 67% is last year’s percentage of games won. Now let’s figure out this year’s scores. 10 x = 12 100 12x = 1000 8
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Chapter 1. Fractions, Decimals and Percents
x = 83.3% or 83% of this year’s games were won.
Vocabulary Here are the vocabulary words that are found in this lesson. Ratio a comparison between two quantities. Percent a ratio that is being compared to the quantity of 100. Percent means out of 100. Fraction a part of a whole written using a numerator and a denominator. Decimal a part of a whole written in base ten place value. Proportion two equal ratios form a proportion.
Time to Practice Write the following percents as a ratio with a denominator of 100. 1. 2. 3. 4.
64% 3% 119% 4.7%
Write the following percents as decimals. 5. 6. 7. 8.
18% 35.7% 6.09% .008%
Write the following decimals as percents. 9. 10. 11. 12.
.52 .02 1.17 5
Write the following percents as fractions in simplest form. 9
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16% 40% 2% 450%
Write the following fractions as percents. Round to the nearest tenths place. 17. 18. 19. 20.
2 3 23 30 4 75 21 2
Find the percent of the given number by converting the percent to a fraction or decimal. 21. 22. 23. 24.
30% of 90 2% of 800 150% of 21 45% of 75
Read the following situations carefully and answer the questions accordingly. 25. For every paycheck you receive, your employer pays 6% to social security. Write this percent as a ratio with a denominator of 100. 26. Jimmy’s height is 1.78m. Write his height as a percent of a meter. 27. A store did a survey and found that 54 of its customers have shopped at its competitors’ stores within in the previous month. What percent is that? 28. At an amusement park there is an average of 30,000 visits daily. Over 75% of those people ride on the biggest roller coasters. About how many people ride the roller coasters?
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