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Number Patterns and Fractions Jen Kershaw Catherine Kwok Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook® textbooks). Copyright © 2016 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/about/ terms-of-use. Printed: July 18, 2016 AUTHORS Jen Kershaw Catherine Kwok www.ck12.org Chapter 1. Number Patterns and Fractions C HAPTER 1 Number Patterns and Fractions C HAPTER O UTLINE 1.1 Factor Pairs 1.2 Divisibility Rules to Find Factors 1.3 Prime and Composite Numbers 1.4 Prime Factorization 1.5 Greatest Common Factor Using Lists 1.6 Greatest Common Factor Using Factor Trees 1.7 Equivalent Fractions 1.8 Fractions in Simplest Form 1.9 Common Multiples 1.10 Least Common Multiple 1.11 Fraction Comparison with Lowest Common Denominators 1.12 Fraction Ordering with Lowest Common Denominators 1.13 Length Measurements to a Fraction of an Inch 1.14 Mixed Numbers as Improper Fractions 1.15 Improper Fractions as Mixed Numbers 1.16 Fraction and Mixed Number Comparison 1.17 Decimals as Fractions 1.18 Decimals as Mixed Numbers 1.19 Fractions as Decimals 1.20 Mixed Numbers as Decimals 1.21 Repeating Decimals Introduction In this chapter, students will engage with the following concepts: number patterns and fractions, prime factorization, greatest common factors, equivalent fractions, least common multiples, ordering fractions, mixed numbers, improper fractions, converting decimals to fractions, and converting fractions to decimals. 1 1.1. Factor Pairs www.ck12.org 1.1 Factor Pairs In this concept, you will learn how to identify factor pairs. The 6th grade class is going on a field trip to the state fair. There are a total of 156 students on the trip. The teachers are trying to decide on how to evenly group the students so the groups are not too small or too large. How many different combinations of equal groups can they make with 156 students? What combination would produce a group size of around 15 students per group? In this concept, you will learn how to identify factor pairs. Finding Factor Pairs A factor is a number or a group of numbers that are multiplied together to make a product. Two factors multiplied together for a product is called a factor pair. Find the factor pairs for 12. First, find all the factors for 12 starting with 1. Any number multiplied by 1 is that number. Therefore, one is a factor of every number. 1 × 12 = 12 After starting with 1, move on to 2, then 3 and so on until you have listed out all of the factors for 12. Knowing the multiplication facts is useful for finding factor pairs. 1 × 12 = 12 2 × 6 = 12 3 × 4 = 12 Then, list the factor pairs for 12. 12 − 1 and 12, 2 and 6, 3 and 4 2 www.ck12.org Chapter 1. Number Patterns and Fractions A factor is also a number that will divide evenly into another number, so you can find factor pairs by dividing as well. This is useful when finding the factors of larger numbers. Find the factor pairs for 72 First, find all the factors for 72. Divide 72 starting with the number 1, and continue on until you have found all of the factors that divide into 72 evenly. 72 ÷ 1 = 72 72 ÷ 2 = 36 72 ÷ 3 = 24 72 ÷ 4 = 18 72 ÷ 6 = 12 72 ÷ 8 = 9 Remember that you only need to find one factor pair once. 72 ÷ 8 = 9 and 72 ÷ 9 = 8 are in the same fact family. Then, list the factor pairs for 72. 72 − 1 and 71, 2 and 36, 3 and 24, 4 and 18, 6 and 12, 8 and 9 Examples Example 1 Earlier, you were given a problem about the 6th grade field trip to the State Fair. The teachers are trying to figure out how to evenly group 156 students. Find all the possible group combinations using factor pairs and the combination that closely produces around 15 students per group. First, find the factor pairs for 156 students. 1 × 156 = 156 2 × 76 = 156 3 × 52 = 156 4 × 39 = 156 6 × 26 = 156 12 × 13 = 156 Then, find total number of group combinations. There are 6 factor pairs, but there are 2 possible combinations per factor pair. For example, the factor pair 1 and 156 can make 1 group of 156 students or 156 groups of 1 student. 6 × 2 = 12 There are 12 possible combinations of groups. The factor pair that produces around 15 students is 12 and 13, 12 groups of 13 students or 13 groups of 12 students. Example 2 List the factor pairs of 18. First, find the factor pairs for 18. 1 × 18 = 18 2 × 9 = 18 3 × 6 = 18 3 1.1. Factor Pairs www.ck12.org Then, list the factor pairs for 18. 18 − 1 and 18, 2 and 9, 3 and 6 Example 3 List the factor pairs for the following number. 36 First, find the factor pairs for 36. 1 × 36 = 39 2 × 18 = 39 3 × 12 = 39 4 × 9 = 39 6 × 6 = 39 Then, list the factor pairs for 36. 36 − 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6 Example 4 List the factor pairs for the following number. 24 First, find the factor pairs for 24. 1 × 24 2 × 12 3×8 4×6 Then, list the factor pairs for 24. 24 − 1 and 24, 2 and 12, 3 and 8, and 4 and 6 Example 5 List the factor pairs for the following number. 4 www.ck12.org Chapter 1. Number Patterns and Fractions 90 First, find the factor pairs for 90. 90 ÷ 1 = 72 90 ÷ 2 = 45 90 ÷ 3 = 30 90 ÷ 5 = 18 90 ÷ 6 = 15 90 ÷ 9 = 10 Then, list the factor pairs for 90. 90 − 1 and 90, 2 and 45, 3 and 30, 5 and 18, 6 and 15, 9 and 10 Review List the factor pairs for each of the following numbers. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 12 10 15 16 56 18 20 22 23 25 27 31 81 48 24 30 Review (Answers) To see the Review answers, open this PDF file and look for section 5.1. 5 1.2. Divisibility Rules to Find Factors www.ck12.org 1.2 Divisibility Rules to Find Factors In this concept, you will learn and apply the divisibility rules to find factors of given numbers. Lisa, Mark, and Stacy are meeting up for lunch. Their total comes out to $28.87. They decide they want to split the bill between the three of them. Will they be able to split the check equally? And how much will each of them pay? In this concept, you will learn and apply the divisibility rules to find factors of given numbers. Finding Factors by Using Divisibility Rules There are some quick tests you can use to see if a large number is divisible by another number. Divisibility rules help determine if a number is divisible by let’s say 2 or 3 or 4. This can help us to identify the factors of a number. Here is a chart that shows all of the basic divisibility rules. 6 www.ck12.org Chapter 1. Number Patterns and Fractions Some of these rules will be more useful than others, but this chart will help you. Find a factor of 1,346 using the divisibility rules. Go through each rule and see if it applies. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. The last digit is even-this number is divisible by 2. The sum of all the digits is 14-this number is not divisible by 3. The last two digits are not divisible by 4-this number is not divisible by 4. The last digit is not zero or five-this number is not divisible by 5. 1, 346 − 12 = 1, 334 - this number is not divisible by 7. The last three numbers are not divisible by 8. The sum of the digits is 14-this number is not divisible by 9 The number does not end in zero-this number is not divisible by 10 The number is not divisible by 3 and 4 The number 1,346 is divisible by 2. Examples Example 1 Earlier, you were given a problem about Lisa and her friends having lunch. They want to split a bill of $28.87 equally between the three of them. Check the divisibility rule and divide to see how much each of them will pay. First, check to see if the sum of all the digits is divisible by 3. 2 + 8 + 8 + 7 = 25 → no Then, divide the total by 3. 7 1.2. Divisibility Rules to Find Factors www.ck12.org $28.87 ÷ 3 = 9.623333 . . . 3 $28.87 is not divisible by 3. Two people will pay $9.62 and one person will pay $9.63. Example 2 Test if 918 divisible by 9. Why or why not? To figure this out, use the divisibility rules. Check to see if the sum of the digits is divisible by 9. 9 + 1 + 8 = 18 18 is divisible by 9, therefore 918 is also divisible by 9. Example 3 Use the divisibility rules to answer the following question. Is 3,450 divisible by 10? First, check to see if the number ends in 0. 3, 450 → yes 3,450 is divisible by 0. Example 4 Use the divisibility rules to answer the following question. Is 1,298 divisible by 3? First, check if the sum of all digits is divisible by 3. 1 + 2 + 9 + 8 = 20 → no 1,298 is not divisible by 3. Example 5 Use the divisibility rules to answer the following question. Is 3,678 divisible by 2? First, check if the last digit is even. 3, 678 → yes 3,678 is divisible by 2. 8 www.ck12.org Chapter 1. Number Patterns and Fractions Review Use the divisibility rules to answer the following questions. Explain your reasoning. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. Is 18 divisible by 3? Is 22 divisible by 2? Is 44 divisible by 6? Is 112 divisible by 2 and 3? Is 27 divisible by 9 and 3? Is 219 divisible by 9? Is 612 divisible by 2 and 3? Is 884 divisible by 4? Is 240 divisible by 5? Is 782 divisible by 7? Is 212 divisible by 4 and 6? Is 456 divisible by 6 and 3? Is 1848 divisible by 8 and 4? Is 246 divisible by 2? Is 393 divisible by 3? Is 7450 divisible by 10? Review (Answers) To see the Review answers, open this PDF file and look for section 5.2. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/166547 9 1.3. Prime and Composite Numbers www.ck12.org 1.3 Prime and Composite Numbers In this concept, you will learn how to classify a number as a prime or composite number. Simon is working on a riddle. “I am a number between 1 and 50. The sum of my digits is not prime, but I myself am prime. In a year, you will only see me 7 times. What number am I?” How can Simon solve the riddle? In this concept, you will learn how to classify a number as a prime or composite number. Classifying Prime and Composite Numbers Numbers can be classified into two categories. The number of factors that a number has determines whether the number is considered a prime number or a composite number. Prime numbers are special numbers that are greater than 1 and only have two factors: 1 and the prime number itself. Composite numbers are numbers that have more than two factors. Most numbers are composite numbers. Here is a prime number. 13 The factors of 13 are only 1 and 13. Therefore, 13 is a prime number. Here is a chart of prime numbers. There are 25 prime numbers between 1 and 100. Take a few minutes to take some notes on prime and composite numbers. Be particularly careful when considering the number “1”. One is neither prime nor composite. 10 www.ck12.org Chapter 1. Number Patterns and Fractions Here is a composite number. 44 The factors for 44 is 1, 2, 4, 11, 22, and 44. Therefore, 44 is a composite number; it is made up of more than two factors. Examples Example 1 Earlier, you were given a problem about Simon’s riddle. Use the prime number cart to help solve the riddle. “I am a number between 1 and 50. The sum of my digits is not prime, but I myself am prime. In a year, you will only see me 7 times. What number am I? First, look at the numbers between 1 an 50 that are prime. There are 15 prime numbers. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 Then, eliminate the numbers where the sum of the digits is prime. 31, 37, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 41, 43, 47 Next, find the number you only see 7 times a year. Think about a calendar. 37 is not on a calendar at all. You see 11, 13, 17, 19 more than 7 times in a year. 31, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47 The answer to the riddle is 31. 11 1.3. Prime and Composite Numbers www.ck12.org Example 2 Prove that 91 is a prime number. First, list all the factors of 91. 91 at the very least has the factors 1 and 91. Keep looking for other factors. Not 2, 3, 4, 5, 6. Let’s divide 91 by 7. 91 ÷ 7 = 13 91 also has the factors 7 and 13. 91 is not a prime number. It is a composite number. Example 3 If a number has more than two factors, the number is prime. True or false? False, numbers with more than two factors are composite numbers. Example 4 The operation associated with factors is addition. True or false? False, the operation associated with factors is multiplication. Example 5 Why is 29 a prime number? 29 is a prime number because the only two factors for 29 are 1 and 29. Review Identify the following values as prime or composite. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 12 12 10 15 16 56 18 20 22 23 25 27 31 81 48 24 30 www.ck12.org Chapter 1. Number Patterns and Fractions Review (Answers) To see the Review answers, open this PDF file and look for section 5.3. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/161991 13 1.4. Prime Factorization www.ck12.org 1.4 Prime Factorization In this concept, you will learn to write the prime factorization of given numbers using a factor tree. Connor is working on prime factorization for his math homework. He needs to find the prime numbers that, when multiplied together, produce the number 82. How can Connor complete this problem? In this concept, you will learn to write the prime factorization of given numbers using a factor tree. Prime Factorization Using Factor Trees When a number is factored, it is broken down into two factors that are either prime numbers or composite numbers. Prime numbers are numbers that have only two factors, one and itself, and composite numbers are numbers that have more than two factors. Some examples of prime numbers are 2, 3, 11, etc. Prime factorization is the process of breaking down a number into a product of all prime numbers. Here is a composite number. 36 The number 36 can be factored several different ways, but let’s factor it with 6 × 6. 36 = 6 × 6 These two factors are not prime factors. Therefore, both factors can be factored again. 36 = 6 × 6 = 2 × 3 × 2 × 3 2 and 3 are both prime numbers. One way to organize the factors is using a factor tree. 14 www.ck12.org Chapter 1. Number Patterns and Fractions 36 \ 6 × 6 \ \ 2×3 2×3 36 = 2 × 2 × 3 × 3 The number is written at the top of the factor tree. Then it is broken down into a factor pair, 6 × 6. 6 can further be factored so the factor pairs are written underneath the 6. Each number is continued to be factored until the factors are all prime numbers. Note that 36 is written as a product of its primes at the bottom of the factor tree. Write the 2s together and the 3s together. Grouping like factors will help keep track of them. The prime factorization is written using exponential notation, a method of writing repeated multiplication. 36 = 2 × 2 × 3 × 3 = 22 × 32 The base is the number being repeated and the exponent is the number of times the number is being multiplied. 2 times 2 is written as base 2 with the exponent 2, the number of times 2 is multiplied by itself. 3 times 3 is written as base 3 with an exponent of 2. Examples Example 1 Earlier, you were given a problem about Connor’s prime factorization math problem. Connor needs to find the prime factorization of 82. Use a factor tree to solve this problem. First, start with 82 at the top of the factor tree. Then, begin by factoring 82 using factor pairs. 82 \ 2 × 41 82 = 2 × 41 Connor finds that 82 is the product of only 2 prime numbers, 2 and 41. Example 2 Write the prime factorization of 25. First, start with 25 at the top of your factor tree. Then, factor 25 into the product of all prime numbers. 25 can be factored into 5 times 5. 5 is a prime number so this tells you that you have reached the bottom of the factor tree. 15 1.4. Prime Factorization www.ck12.org 25 \ 5×5 25 = 5 × 5 Next, write the factors in exponential notation. The base is 5 and the exponent is 2. 25 = 52 The prime factorization of 25 is 52 . Example 3 Write the prime factorization for the following number. 48 First, start with 48 at the top of your factor tree. Then, factor 48 into the product of all prime numbers. 48 \ 4 ×12 \ \ 2 × 2 2 ×6 \ 2×3 48 = 2 × 2 × 2 × 2 × 3 Next, write the factors in exponential notation. 48 = 24 × 3 The prime factorization of 48 is 24 × 3. Example 4 Write the prime factorization for the following number. 100 16 www.ck12.org Chapter 1. Number Patterns and Fractions First, start with 100 at the top of your factor tree. Then, factor 100 into the product of all prime numbers. 100 \ 2 × 50 \ 2 × 25 \ 5×5 100 = 2 × 2 × 5 × 5 Next, write the factors in exponential notation. 100 = 22 × 52 The prime factorization of 100 is 22 × 52 . Example 5 Write the prime factorization for the following number. 144 First, start with 144 at the top of your factor tree. Then, factor 144 into the product of all prime numbers. 144 \ 12 × 12 \ \ 2 ×6 2 × 6 \ \ 2×3 2×3 144 = 2 × 2 × 2 × 2 × 3 × 3 Next, write the factors in exponential notation. 144 = 24 × 32 The prime factorization of 144 is 24 × 32 . 17 1.4. Prime Factorization www.ck12.org Review Write the prime factorization of each number using exponential notation. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 56 14 121 84 50 64 72 16 24 300 128 312 525 169 213 Review (Answers) To see the Review answers, open this PDF file and look for section 5.4. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/162141 18 www.ck12.org Chapter 1. Number Patterns and Fractions 1.5 Greatest Common Factor Using Lists In this concept, you will learn to find the greatest common factors of numbers using lists. Mara is in making flower arrangements for a party. She has 48 carnations and 42 daisies. She wants there to be an equal number of carnations and daisies in each bouquet. What is the most number of bouquets she can make? How many of each flower will they contain? In this concept, you will learn to find the greatest common factors of numbers using lists. Finding The Greatest Common Factor Using Lists The greatest common factor (GCF) is the greatest factor that two or more numbers have in common. One way to find the GCF is to make lists of the factors for two numbers and then choose the greatest factor that the two factors have in common. Find the GCF for 12 and 16. It is helpful to order them from smallest to largest in order to make sure that you cover every factor. First, find all the factors of 12 and 16 and write them in a list in the order of least to greatest. 12 − 1, 2, 3, 4, 6, 12 16 − 1, 2, 4, 8, 16 One way to check if all the factors are listed is to use the rainbow method. Draw a line from one part of a factor pair to the other. The resulting image should resemble a rainbow. 19 1.5. Greatest Common Factor Using Lists www.ck12.org Next, identify the GCF, the largest number that appears in both lists. The GCF for 12 and 16 is 4. Examples Example 1 Earlier, you were given a problem about Mara and her flowers. Mara has 48 carnations and 42 daisies and wants each bouquet to have the same number of flowers. Compare the factors 48 and 42 and find the greatest common factor. First, find all the factors of 48 and 42 and write them from least to greatest. 48 − 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 42 − 1, 2, 3, 6, 7, 14, 21, 42 Then, identify the GCF. The GCF for 48 and 42 is 6. Next, find the number of carnations and daisies in 6 bouquets. carnations : 48 ÷ 6 = 8 daisies : 42 ÷ 6 = 7 The most number of bouquets Mara can make will be 6. Each will have 8 carnations and 7 daisies. Example 2 What is the GCF of 140 and 124? First, find all the factors of 140 and 124 and write them in a list in the order of least to greatest. 20 www.ck12.org Chapter 1. Number Patterns and Fractions 140 − 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140 124 − 1, 2, 4, 31, 62, 124 Next, identify the GCF, the largest number that appears in both lists. The GCF for 140 and 124 is 4. Example 3 Find the GCF for the pair of numbers. 24 and 36 First, find all the factors of 24 and 36 and write them in a list in the order of least to greatest. 24 − 1, 2, 3, 4, 6, 8, 12, 24 36 − 1, 2, 3, 4, 6, 9, 12, 18, 36 Next, identify the GCF, the largest number that appears in both lists. The GCF for 24 and 36 is 12. Example 4 Find the GCF for the pair of numbers. 10 and 18 First, find all the factors of 10 and 18 and write them in a list in the order of least to greatest. 10 − 1, 2, 5, 10 18 − 1, 2, 3, 6, 9, 18 Next, identify the GCF, the largest number that appears in both lists. The GCF for 10 and 18 is 2. Example 5 Find the GCF for the pair of numbers. 18 and 45 First, find all the factors of 18 and 45 and write them in a list in the order of least to greatest. 21 1.5. Greatest Common Factor Using Lists www.ck12.org 18 − 1, 2, 3, 6, 9, 18 45 − 1, 3, 5, 9, 25, 45 Next, identify the GCF, the largest number that appears in both lists. The GCF for 18 and 45 is 9. Review Find the GCF for each pair of numbers. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 9 and 21 4 and 16 6 and 8 12 and 22 24 and 30 35 and 47 35 and 50 44 and 121 48 and 144 60 and 75 21 and 13 14 and 35 81 and 36 90 and 80 22 and 33 11 and 13 15 and 30 28 and 63 67 and 14 18 and 36 Review (Answers) To see the Review answers, open this PDF file and look for section 5.5. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/162149 22 www.ck12.org Chapter 1. Number Patterns and Fractions MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/162151 23 1.6. Greatest Common Factor Using Factor Trees www.ck12.org 1.6 Greatest Common Factor Using Factor Trees In this concept, you will learn to find the greatest common factor using factor trees. Richard is making gift bags. He has 36 pencils and 28 pens. How many gift bags can Richard make if there are the same number of pencils and pens in each bag? Use factor trees to solve this problem. How many pencils and pens will be in each bag? In this concept, you will learn to find the greatest common factor using factor trees. Finding the Greatest Common Factor Using Factor Trees The greatest common factor (GCF) is the greatest factor that two or more numbers have in common. The GCF can be found by making a list and comparing all the factors. A factor tree can also be used to find the GCF. The GCF is the product of the common prime factors. Let’s find the GCF of 20 and 30 using a factor tree. First, make a factor tree for each number. Then, identify the common factors. The numbers 20 and 30 have the factors 2 and 5 in common. 24 www.ck12.org Chapter 1. Number Patterns and Fractions 20 = 2 × 2 × 5 30 = 2 × 3 × 5 Next, multiply the common factors to find the GCF. If there is only one common factor, there is no need to multiply. 2 × 5 = 10 The GCF of 20 and 30 is 10. Note that if the numbers being compared have no factors in common using a factor tree, they still have the factor 1 in common. Examples Example 1 Earlier, you were given a problem about Richard who needs to make gift bags with 36 pencils and 28 pens. Use factor trees to find the most number of bags he can make that have the same number of pencils and pens in each. First, make a factor tree for each number. Then, identify the common factors. The common factors are two 2s. 36 = 2 × 2 × 3 × 3 28 = 2 × 2 × 7 Next, multiply to common factors to find the GCF. 2×2 = 4 Finally, divide the number of pencils and pens by the GCF, 4. pencils = 36 ÷ 4 = 9 pens = 28 ÷ 4 = 7 Richard can make 4 gift bags that have 9 pencils and 7 pens in each bag. 25 1.6. Greatest Common Factor Using Factor Trees www.ck12.org Example 2 Find the GCF of 36 and 54 using factor trees. First, make a factor tree for each number. Then, identify the common factors. The numbers 36 and 54 have the factors 2 and two 3s in common. 36 = 2 × 2 × 3 × 3 54 = 2 × 3 × 3 × 3 Next, multiply the common factors to find the GCF. 2 × 3 × 3 = 18 The GCF of 36 and 54 is 18. Example 3 Find the greatest common factor using factor trees. 14 and 28 First, make a factor tree for each number. 26 www.ck12.org Chapter 1. Number Patterns and Fractions Then, identify the common factors. The numbers 14 and 28 have the factors 2 and 7 in common. 14 = 2 × 7 28 = 2 × 2 × 7 Next, multiply the common factors to find the GCF. 2 × 7 = 14 The GCF of 14 and 28 is 14. Example 4 Find the greatest common factor using factor trees. 24 and 34 First, make a factor tree for each number. 27 1.6. Greatest Common Factor Using Factor Trees www.ck12.org Then, identify the common factors. The numbers 24 and 34 have the factor 2 in common. 24 = 2 × 2 × 2 × 3 The GCF of 12 and 24 is 12. Example 5 Find the greatest common factor using factor trees. 19 and 63 First, make a factor tree for each number. 28 34 = 2 × 17 www.ck12.org Chapter 1. Number Patterns and Fractions Then, identify the common factors. The numbers 19 and 63 have the factor 1 in common. 19 = 1 × 19 63 = 3 × 3 × 7 The GCF of 19 and 63 is 1. Review Find greatest common factor for each pair of numbers. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 14 and 28 14 and 30 16 and 36 24 and 60 72 and 108 18 and 81 80 and 200 99 and 33 27 and 117 63 and 126 89 and 178 90 and 300 56 and 104 63 and 105 72 and 128 Review (Answers) To see the Review answers, open this PDF file and look for section 5.6. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/162149 MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/162151 29 1.7. Equivalent Fractions www.ck12.org 1.7 Equivalent Fractions In this concept, you will learn to find an equivalent fraction. After the party, there was still 34 of the cake left over. Mike wants people to take some cake home with them. If there are 9 people, how much of the cake will each person take with them? In this concept, you will learn to find an equivalent fraction. Finding Equivalent Fractions A fraction is a part of a whole. It describes the relationship between a part of something and the whole thing. A fraction has two numbers separated by a fraction bar. The top number is called the numerator and tells you how many parts there are out of the whole. The bottom number is the denominator. It tells you how many parts the whole has been divided into. Here is a fraction. 4 5 There are 4 parts and the whole is divided into 5 parts. Fractions can also be represented as a picture. Here is a picture of a fraction. The whole has been divided into ten parts. This is our denominator. Five out of ten are shaded. This is our numerator. 30 www.ck12.org Chapter 1. Number Patterns and Fractions 5 10 5 . Notice that half of the image is shaded. Remember The fraction for the parts that are not shaded would also be 10 5 1 that a half is also written as the fraction 2 . Therefore, 10 is the same as 12 . When two fractions have the same value, the fractions are equivalent fractions. The bars below visually represent equivalent fractions. The shaded part of each bar represents a fraction that is equivalent to one half. 1 2 3 4 = = = 2 4 6 8 To find equivalent fractions, multiply the numerator and denominator by the same number. If the numerator and denominator have a common factor other than 1, you can also divide the numerator and denominator by a common factor. Here is a fraction. 6 12 Find two equivalent fractions by multiplying or dividing. First, multiply the numerator and denominator by the same number. Let’s choose 2. 6×2 12 = 12 × 2 24 The numbers 6 and 12 have common factors. 31 1.7. Equivalent Fractions www.ck12.org 6 − 1, 2, 3, 6 12 − 1, 2, 3, 4, 6, 12 Let’s try dividing to find a second equivalent fraction. First, divide the numerator and denominator by a common factor. Let’s choose 3. 2 6÷3 = 12 ÷ 3 4 Two equivalent fraction for 6 12 is 12 24 and 24 . 6 12 2 = = 12 24 4 Examples Example 1 Earlier, you were given a problem about Mike and the cake. Mike wants to evenly divide equals 9. 3 4 of the cake between 9 people. Find an equivalent fraction of 3 4 where the numerator Multiply the numerator and denominator by the same number. 3×3 9 = 4 × 3 12 If Mike divides the leftover cake evenly between 9 people, each person will take home Example 2 Find an equivalent fraction. 3 4 Multiply the numerator and denominator by the same number. Let’s choose 2. 3×2 4×2 6 8 An equivalent fraction of 32 3 4 is 86 . = = 6 8 3 4 1 12 of the original cake. www.ck12.org Chapter 1. Number Patterns and Fractions Example 3 Find an equivalent fraction for the following fraction. 1 4 Multiply the numerator and denominator by the same number. 1×2 2 = 4×2 8 An equivalent fraction for 1 4 is 28 . Example 4 Find an equivalent fraction for the following fraction. 2 3 Multiply the numerator and denominator by the same number. 2×2 6 = 3×3 9 An equivalent fraction for 2 3 is 69 . Example 5 Find an equivalent fraction for the following fraction. 8 10 Divide the numerator and denominator by a common factor, 2. 4 8÷2 = 10 ÷ 2 5 An equivalent fraction for 8 10 is 54 . 33 1.7. Equivalent Fractions www.ck12.org Review Find an equivalent fraction for the following fractions. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 2 1 3 1 4 1 5 2 3 2 5 3 4 3 10 2 9 2 7 Determine if each pair of fractions is equivalent. Use true or false as your answer. 11. 12. 13. 14. 15. 1 2 2 3 2 5 3 7 5 9 and and and and and 3 6 4 9 4 20 9 21 25 45 Review (Answers) To see the Review answers, open this PDF file and look for section 5.7. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/166580 34 www.ck12.org Chapter 1. Number Patterns and Fractions 1.8 Fractions in Simplest Form In this concept, you will earn to write a fraction in simplest form. Tessa made a pan of brownies for the sixth grade social. She cut the brownie pan into sixteen brownies. She sold 12 out of 16 brownies at the bake sale. What fraction of the brownies did she sell? What fraction did she not sell? In this concept, you will learn to write a fraction in simplest form. Finding Fractions in Simplest Form Some fractions can describe large quantities. Simplifying a fraction can make it easier to understand its value. To simplify a fraction, divide the numerator and the denominator by a common factor. Sometimes you will also hear simplifying called reducing a fraction. A fraction that has been simplified by the greatest common factor is in simplest form. Remember that the greatest common factor (GCF) is the greatest factor that two or more numbers have in common. Here is a fraction. 48 60 Simplify the fraction to better understand its value. First, find a common factor for the numerator and denominator. 48 - 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 60 - 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 Then, divide by the GCF common factor. The GCF for 48 and 60 is 12 48 ÷ 12 4 = 60 ÷ 12 5 35 1.8. Fractions in Simplest Form The simplest form of 48 60 www.ck12.org is 45 . Comparing fractions with different denominators can be difficult. Some fraction may look similar, but not be equivalent fractions. You can compare fractions with different denominators by comparing them in their simplest form. Here are two fractions. 3 6 and 4 8 Let’s simplify 36 . To do this, divide the numerator and denominator by the GCF. The GCF of 3 and 6 is 3. 3÷3 1 = 6÷3 2 Let’s simplify 48 . To do this, divide the numerator and the denominator by the GCF. The GCF of 4 and 8 is 4. 4÷4 1 = 8÷4 2 Both 3 6 4 8 and are equivalent to 12 . Therefore, 3 6 and 4 8 are also equivalent fractions. Examples Example 1 Earlier, you were given a problem about Tessa and her brownies. Tessa sold 12 out of 16 brownies at the bake sale. Simply the fraction for the number brownies she sold and did not sell. First, write the fraction for the number of brownies that was sold. 12 16 Then, simplify the fraction by dividing both by the GCF. The GCF is 4. 12 3 = 16 4 She sold 3 4 of the brownies Now, write the fraction for the number of brownies that was not sold. 4 16 Then, simplify the fraction by dividing both by the GCF. The GCF is 4. 4 1 = 16 4 She did not sell 36 1 4 of the brownies. www.ck12.org Chapter 1. Number Patterns and Fractions Example 2 Simplify the fraction. 27 36 First, find the GCF of 27 and 36. The GCF for 27 and 36 is 9. 27 - 1, 3, 9, 27 36 - 1, 2, 3, 4, 6, 9, 12, 18, 36 Then, divide both the numerator and the denominator by 9. 27 ÷ 9 3 = 36 ÷ 9 4 The simplest form of 27 36 is 34 . Example 3 Simplify the fraction. 4 20 First, find the GCF of 4 and 20. The GCF for 4 and 20 is 4 Then, divide both the numerator and the denominator by 4. The simplest form of Solution: 4 20 is 15 . 1 5 Example 4 Simplify the fraction. 8 16 First, find the GCF of 8 and 16. The GCF for 8 and 16 is 8 Then, divide both the numerator and the denominator by 8. The simplest form of 8 16 is 12 . Example 5 Simplify the fraction. 37 1.8. Fractions in Simplest Form www.ck12.org 5 15 First, find the GCF of 5 and 15. The GCF for 5 and 15 is 5 Then, divide both the numerator and the denominator by 5. The simplest form of 5 15 is 13 . Review Simplify each fraction. If the fraction is already in simplest form write simplest form for your answer. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 8 10 4 10 2 10 2 12 3 12 4 9 5 20 12 24 12 36 11 44 20 45 18 20 12 30 22 40 35 63 Review (Answers) To see the Review answers, open this PDF file and look for section 5.8. 38 www.ck12.org Chapter 1. Number Patterns and Fractions 1.9 Common Multiples In this concept, you will learn to identify multiples and find common multiples for pairs of numbers. Charlie is playing a video game. He has to grab the magic token when the guard turns his back. He notices that the magic token appears every 4 seconds and the guard turns his back every 3 seconds. How can Charlie use this information to get the timing right so he can grab the magic token? In this concept, you will learn to identify multiples and find common multiples for pairs of numbers. Finding Common Multiples A multiple is the product of a quantity and a whole number. Here are some multiples for the quantity of 3, multiplied by different whole numbers. 3 × 1 = 3, 3 × 2 = 6, 3 × 3 = 9, 3 × 4 = 12, 3 × 5 = 15, 3 × 6 = 18 Listing out these products is the same as listing out multiples. 3, 6, 9, 12, 15, 18... You can see that this is also the same as counting by threes. The dots at the end mean that these multiples can go on and on and on. Every number has an infinite number of multiples. List six multiples for 4. To do this, think of taking the quantity 4 and multiplying it by 1, 2, 3, 4, 5... 4 × 1 = 4, 4 × 2 = 8, 4 × 3 = 12, 4 × 4 = 16, 4 × 5 = 20 Our answer is 4, 8, 12, 16, 20, 24... A common multiple is a multiple that two or more numbers have in common. List the multiples of the numbers to find the common multiples. Find common multiples for 3 and 4. First, write out the first few multiples for the numbers and then identify the multiples the two numbers have in common. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48 39 1.9. Common Multiples www.ck12.org Some of the common multiples of 3 and 4 are 12, 24, and 36. Examples Example 1 Earlier, you were given a problem about Charlie and his video game. Charlie has to grab the magic token that appears every 4 seconds when the guard turns his back, which is every 3 seconds. Find the common multiples of 3 and 4 to find out when both will occur at the same time. First, list the first few multiples of 3 and 4. 3, 6, 9, 12, 15, 18, 21, 24, 27 4, 8, 12, 16, 20, 24, 28, 32, 36 Then, identify the common multiples. 3, 6, 9, 12, 15, 18, 21, 24, 27 4, 8, 12, 16, 20, 24, 28, 32, 36 Some common multiples of 3 and 4 are 12 and 24. Charlie can grab the magic token 12 or 24 seconds after he enter that stage. Note that 12 and 24 are multiples of 12. Charlie will get a chance to grab the magic token every 12 seconds. Example 2 What are common multiples of 3 and 7? First, write out the first few multiples for the numbers. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 Then, identify the multiples the two numbers have in common. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 A common multiple of 3 and 7 is 21. In this list, the only common multiple between 3 and 7 is 21. You could find more multiples by increasing the length of the list of multiples for 3 and 7 or finding a multiple of the common multiple. 21 × 2 = 42 42 is a multiple of both 3 and 7. Example 3 List eight multiples of 6. First, multiply 6 by 1 through 8. 6 × 1 = 6, 6 × 2 = 12, 6 × 3 = 18, 6 × 4 = 24, 6 × 5 = 30, 6 × 6 = 36, 6 × 7 = 42, 6 × 8 = 48 The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, and 48. 40 www.ck12.org Chapter 1. Number Patterns and Fractions Example 4 List six multiples of 8. First, multiply 8 by 1 through 6. 8 × 1 = 8, 8 × 2 = 16, 8 × 3 = 24, 8 × 4 = 32, 8 × 5 = 40, 8 × 6 = 48 Six multiples of 8 are 8, 16, 24, 32, 40, and 48. Example 5 What are the first two common multiples of 6 and 8? First, list the first few multiples of 6 and 8. 6, 12, 18, 24, 30, 36, 42, 48 ... 8, 16, 24, 32, 40, 48, ... Then, identify the common multiples. 6, 12, 18, 24, 30, 36, 42, 48 ... 8, 16, 24, 32, 40, 48 ... Two common multiples of 6 and 8 are 24 and 48. Review List the first five multiples for each of the following numbers. 1. 2. 3. 4. 5. 3 5 6 7 8 Find two common multiples of each pair of numbers. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 3 and 5 2 and 3 3 and 4 2 and 6 3 and 9 5 and 7 4 and 12 5 and 6 10 and 12 5 and 8 Review (Answers) To see the Review answers, open this PDF file and look for section 5.9. 41 1.9. Common Multiples www.ck12.org Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/166626 MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/166628 42 www.ck12.org Chapter 1. Number Patterns and Fractions 1.10 Least Common Multiple In this concept, you will learn to find the least common multiples of numbers by using lists. Arjay is planning a barbecue. He is at the store to buy hot dogs and hot dog buns. The hot dogs come in packs of 8 while the buns come in packs of 6. At least how many packages of each should he buy to have the same number of hot dogs and buns? In this concept, you will learn to find the least common multiples of numbers by using lists. Finding the Least Common Multiple Using Lists Common multiples are multiples that two or more numbers have in common. The least common multiple (LCM) is the smallest multiple that two numbers have in common. Let’s look back at the common multiples for 3 and 4. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48 Some of the common multiples of 3 and 4 are 12, 24 and 36. The LCM of these two numbers is 12. It is the smallest number that they both have in common. To find the LCM, list the multiples of the two numbers. Stop when you have found the first common multiple. Examples Example 1 Earlier, you were given a problem about Arjay on his shopping trip. 43 1.10. Least Common Multiple www.ck12.org Arjay wants to know at least how many packages of 8 hot dogs and 6 hot dog buns he needs to buy to get the same amount of each. Find the LCM of 8 and 6 to find the number of packages for each. First, list the multiples of both 8 and 6. Stop at the first common multiple. 8 = 8, 16, 24, 32 6 = 6, 12, 18, 24 The LCM of 8 and 6 is 24. Then, find number of packages of hot dogs and buns needed to make 24 hotdogs. 24 ÷ 8 = 3 24 ÷ 6 = 4 Arjay must buy 3 packages of hotdogs and 4 packages of hotdog buns to get the same number of each. Example 2 Find the LCM of 20 and 15. First, list out the multiples of both 20 and 15. Stop at the first common multiple. 20 = 20, 40, 60, 80, 100 15 = 30, 45, 60 The LCM Of 20 and 15 is 60. Example 3 Find the LCM of the pair of numbers. 5 and 3 First, list the multiples of both 5 and 3. Stop at the first common multiple. 5 = 5, 10, 15, 20, 25 3 = 3 6, 9, 12, 15 The LCM of 5 and 3 is 15. Example 4 Find the LCM of the pair of numbers. 2 and 6 First, list the multiples of both 2 and 6. Stop at the first common multiple. 2 = 2, 4, 6, 8 6=6 The LCM of 2 and 6 is 6. Example 5 Find the LCM of the pair of numbers. 44 www.ck12.org Chapter 1. Number Patterns and Fractions 4 and 6 First, list the multiples of both 4 and 6. Stop at the first common multiple. 4 = 4, 8, 12, 16, 20 6 = 6, 12 The LCM of 4 and 6 is 12. Review Find the LCM of each pair of numbers. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 3 and 5 2 and 3 3 and 4 2 and 6 3 and 9 5 and 7 4 and 12 5 and 6 10 and 12 5 and 8 Review (Answers) To see the Review answers, open this PDF file and look for section 5.10. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/166632 45 1.11. Fraction Comparison with Lowest Common Denominators www.ck12.org 1.11 Fraction Comparison with Lowest Common Denominators In this concept, you will learn how to compare fractions using the lowest common denominator. The 6th grade class is having an ice cream social. Terrence and Emilia estimated that one - third of the class will want to eat vanilla ice cream and four - sevenths of the class will want to eat chocolate ice cream. If these two estimates are accurate, which flavor of ice cream will be the most popular? In this concept, you will learn how to compare fractions using the lowest common denominator. Comparing Fractions Using the Lowest Common Denominator Some fractions have different denominators, the bottom number of a fraction. The numerator refers to the top number of a fraction. Here are two fractions with different denominators. 1 2 and 4 3 Remember that the denominator is the number of parts the whole has been divided into. In the first fraction, onefourth, the whole has been divided into four parts. The second fraction, two-thirds, has been divided into three parts. In this example, you cannot compare the numerators because the parts of each fraction have different values. You use greater than (>), less than (<), or equal to (=) to compare two fractions. It is easy to compare fractions with the same denominator. Compare these two fractions. 1 5 3 5 Both fractions represent a whole that is divided into 5 parts. If the fractions were pizzas that were divided into 5 parts, one-fifth of a pizza would be less than with three-fifths of the same pizza. Therefore, you can compare those fractions like this. 46 www.ck12.org Chapter 1. Number Patterns and Fractions 1 3 < 5 5 To compare fractions different denominators, rewrite the fractions so they have a common denominator. Let’s compare the two fractions from earlier. 1 4 2 3 Rewrite the denominators by finding the least common multiple of each denominator. Remember that the least common multiple (LCM) is the smallest multiple that two numbers have in common. This LCM becomes the lowest common denominator (LCD). First, list the multiples for 3 and 4 and find the LCM. 4, 8, 12, 16 3, 6, 9, 12 The LCM for 3 and 4 is 12. Then, rewrite each fraction in terms of twelfths. Make a fraction equivalent to one-fourth in terms of twelfths, and make a fraction equivalent to two-thirds in terms of twelfths. 1 = 4 12 To make equivalent fractions, multiply or divide the numerator and the denominator by the same number to create the equal fraction. 4 is multiplied by 3 to get 12. Complete the equivalent fraction by also multiplying the numerator by 3. 1 3 = 4 12 Now work on rewriting two-thirds in terms of twelfths. 3 is multiplied by 4 to get 12. Multiply the numerator by 4. 2 8 = 3 12 Next, compare the fractions now that both fractions have been written in terms of twelfths. 3 8 < 12 12 so 1 2 < 4 3 1 4 is less than 23 . 47 1.11. Fraction Comparison with Lowest Common Denominators www.ck12.org Examples Example 1 Earlier, you were given a problem about the ice cream social. Terrence and Emilia estimated that one - third of the class will want to eat vanilla ice cream and four - sevenths of the class will want to eat chocolate ice cream. Compare the fractions to see which flavor will be more popular. 1 3 4 7 First, find the LCM of 3 and 7. The lowest common denominator will be 21. Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of 7. Multiply the numerator and denominator of 74 by 3. 1 3 4 7 = = 1 3 by 6 9 by 7 21 12 21 Next, compare the equivalent fractions. 7 12 < 21 21 Chocolate will be more popular. Example 2 Rewrite each fraction with the lowest common denominator and compare using <, >, or =. 6 9 3 4 First, find the LCM of 9 and 4. The lowest common denominator will be 36. 9 = 9, 18, 27, 36 4 = 4, 12, 16, 20, 24, 28, 32, 36 Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of 4. Multiply the numerator and denominator of 43 by 9. 6 9 = 24 36 3 4 = 27 36 Next, compare the equivalent fractions. 24 27 < 36 36 6 9 is less than 34 . 48 www.ck12.org Chapter 1. Number Patterns and Fractions Example 3 Compare the fractions. 2 5 6 10 First, find the LCM of 5 and 10. The lowest common denominator will be 10. Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of 6 2. 10 is already a fraction of tenths. 2 5 by 2 3 by 4 2 = 5 10 Next, compare the equivalent fractions. 4 6 < 10 10 2 5 is less than 6 10 . Example 4 Compare the fractions. 2 3 1 9 First, find the LCM of 3 and 9. The lowest common denominator will be 9. Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of 3. 91 does not change. 2 6 = 3 9 Next, compare the equivalent fractions. 6 1 > 9 9 2 3 is greater than 19 . 49 1.11. Fraction Comparison with Lowest Common Denominators www.ck12.org Example 5 Compare the fractions. 3 4 6 8 First, find the LCM of 4 and 8. The lowest common denominator will be 8. Then, rewrite each fraction with the lowest common denominator. Multiply the numerator and denominator of 2. 86 does not change. 3 6 = 4 8 Next, compare the equivalent fractions. 6 6 = 8 8 3 4 is equal to 86 . Review Rename each in terms of tenths. 1. 2. 3. 4. 1 5 3 5 1 2 4 5 Complete each equal fraction. 5. 1 3 = 9 6. 2 3 = 18 7. 5 6 2 7 4 9 = 18 = 14 = 36 8. 9. 10. 3 4 = 48 Identify the lowest common multiple for each pair of numbers. 11. 3 and 6 12. 4 and 10 13. 5 and 3 14. 7 and 2 15. 8 and 4 50 3 4 by www.ck12.org Chapter 1. Number Patterns and Fractions 16. 6 and 4 17. 8 and 5 18. 12 and 5 19. 9 and 2 20. 6 and 7 Compare the following fractions using <, >, or = 22. 1 2 2 3 1 3 3 9 23. 4 6 2 3 24. 6 10 9 18 21. 25. 4 5 3 6 Review (Answers) To see the Review answers, open this PDF file and look for section 5.11. 51 1.12. Fraction Ordering with Lowest Common Denominators www.ck12.org 1.12 Fraction Ordering with Lowest Common Denominators In this concept, you will learn to order fractions using lowest common denominators. Sam surveyed his classmates for a class assignment. He asked his classmates about some of their interests and hobbies. The results of the survey were: 7 8 1 4 5 8 3 4 3 8 watched TV rode bikes read books played sports played an instrument or sang Sam is using this information to make a presentation. What is the order of activities if Sam were to rank them from the most popular to least? In this concept, you will learn to order fractions using lowest common denominators. Using the Least Common Denominator to Order Fractions Sometimes, you will need to write fractions in order from least to greatest or from greatest to least. This becomes very simple if the fractions have the same denominator. Write in order from least to greatest. 4 2 8 3 6 , , , , 9 9 9 9 9 Since all of these fractions have a common denominator, use the numerators and arrange them in order from the smallest numerator to the largest numerator. The answer is 29 , 39 , 94 , 69 , 89 To order fractions that do not have a common denominator, rewrite all of the fractions using the lowest common denominator (LCD). 52 www.ck12.org Chapter 1. Number Patterns and Fractions Order these fractions from least to greatest. 2 1 1 5 , , , 3 4 2 6 First, find the LCD. The lowest common multiple of 3, 4, 2, and 6 is 12. Remember, if you cannot figure out the LCD in your head, list the multiples to find the least common multiple (LCM). 3 − 3, 6, 9, 12 4 − 4, 8, 12 2 − 2, 4, 6, 8, 10, 12 6 − 6, 12 Then, rewrite each fraction with the denominator 12. 2 3 1 4 1 2 5 6 = = = = 8 12 3 12 6 12 10 12 Next, order the fractions from least to greatest. 1 1 2 5 , , , 4 2 3 6 Examples Example 1 Earlier, you were given a problem about Sam’s survey. Sam wants to find the order of activities from most popular to least popular. Compare the fractions and list them from greatest to least. 7 8 1 4 5 8 3 4 3 8 watched TV rode bikes read books played sports played an instrument or sang First, find the LCD. The LCD of 4 and 8 is 8. Then, rewrite each fraction with the denominator of 8. Three of the fractions already have denominators of 8. Find the equivalent fraction for 14 and 34 . 1 4 = 4 8 3 4 = 6 8 Next, order the fractions from greatest to least. 7 6 5 4 3 , , , , 8 8 8 8 8 53 1.12. Fraction Ordering with Lowest Common Denominators www.ck12.org Sam should list the activities as: Most Popular to Least Popular Activity 1. 2. 3. 4. 5. Watch TV Play sports Read books Ride bike Play instrument or sing Example 2 Write the following fractions in order from least to greatest. 4 2 5 , , 7 3 7 First, find the lowest common denominator. The lowest common multiple of 3 and 7 is 21. Then, rewrite each fraction with the denominator 12. 4 7 2 3 5 7 = = = 12 21 14 21 15 21 Next, order the fractions from least to greatest. 4 2 5 , , 7 3 7 Notice that the original order was in order from least to greatest. Example 3 What would be the LCD for fractions with the denominators of 3, 5, and 6? 3 − 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 5 − 5, 10, 15, 20, 25, 30 The LCD would be 30. Example 4 Rewrite the fractions with a common denominator. 4 1 2 , , 5 5 3 54 6 − 6, 12, 18, 24, 30 www.ck12.org Chapter 1. Number Patterns and Fractions 24 6 20 , , 60 30 30 The LCD is 15. 4 5 1 5 2 3 = = = 12 15 3 15 10 15 Example 5 Write the fractions above in order from greatest to least. 4 2 1 , , 5 3 5 Review Write each series in order from least to greatest. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 5 1 4 6, 3, 9 6 1 2 7, 4, 3 6 4 2 6, 5, 3 1 3 2 2, 5, 3 2 1 3 7, 4, 6 1 2 2 6, 9, 5 4 4 3 16 , 5 , 7 9 4 3 10 , 5 , 4 4 1 2 5, 2, 3 9 2 3 11 , 3 , 4 4 1 3 7, 5, 8 6 1 2 7, 3, 5 7 4 1 8, 5, 3 1 4 2 6, 5, 4 1 4 2 7 9, 7, 9, 8 Review (Answers) To see the Review answers, open this PDF file and look for section 5.12. Resources 55 1.12. Fraction Ordering with Lowest Common Denominators www.ck12.org MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/166984 56 www.ck12.org Chapter 1. Number Patterns and Fractions 1.13 Length Measurements to a Fraction of an Inch In this concept, you will learn how to measure lengths to a fraction of an inch. Meg is looking to replace a door hinge that broke. She’s shopping online for the hinge, but the measurements are all listed in fractions of inches. She measured the size of the hinge on a ruler. Here is what her ruler says: What size hinge should she be looking for? In this concept, you will learn how to measure lengths to a fraction of an inch. Measuring Lengths Using Fractions Think about using a ruler, sometimes you will have something that measures evenly, meaning that the item measures in whole inches. More often, you will have an item that does not measure evenly. When this happens, you will need to measure the item to a fraction of an inch. 16 4 Each whole inch has sixteen lines. This is because one inch is 16 of an inch long. Count four lines, you are at 16 or 8 1 1 4 (a quarter) of an inch. Count to the eighth line, you are at 16 or 2 (one half) of an inch. Count to the twelfth line 12 is 10 or 43 (three fourths) of an inch. Look at this ruler. The arrow is above a line that does not indicate one whole inch. Use fractions to write this fraction of an inch. Count the lines. The arrow is above the eighth line. Write the measurement as a fraction of an inch. 8 16 Measurements are often written as a fraction in simplest form. 8 is 16 or 21 inch. 8 16 in simplest form is equal to 12 . The measurement Here is an example of a whole number measurement with a fraction of another inch. 57 1.13. Length Measurements to a Fraction of an Inch This measurement is 1 inch and a fraction of another inch. The arrow is above 1 12 16 . 3 12 measurement is 1 16 or 1 4 inches. www.ck12.org 12 16 in simplest form is 34 . This Notice that a ruler has lines with fourdifferent lengths. The longest line on the ruler indicates a whole inch. The 1 1 second longest line indicates a half 2 inch. The line shorter than the half inch line marks a quarter 4 inch. Some rulers have marks for an eighth 18 of an inch, like the image below. Examples Example 1 Earlier, you were given a problem about Meg’s door hinge. Find the measurement of her door hinge to a fraction of an inch. Here is what her ruler says: The measurement is between 3 and 4 inches. The arrow is pointing to the quarter inch mark. Meg’s is looking for a door hinge that is 3 14 inch long. Example 2 What is the measurement of the crayon? First, look at the measurement of the crayon. It is between 3 and 4 inches long. Count the number of lines after the 3 inch mark. It is 10 16 of an inch beyond the 3 inch mark. Then, simplify 10 16 = 5 8 The crayon is 58 10 16 . 35 8 inches. www.ck12.org Chapter 1. Number Patterns and Fractions Example 3 What is the second quarter inch between 3 and 4? The second quarter inch is also 24 . 2 4 = 1 2 The second quarter inch between 3 and 4 is 3 12 inch. Example 4 Locate 2 12 inches on a ruler. Check your answer with a friend. Look on the ruler. The line should be exactly half - way between 2 and 3. Example 5 How many parts is an inch divided into on a ruler? An inch is divided into 16 parts. Review Work with a partner and follow the directions. You will need a ruler. 1. 2. 3. 4. 5. Find 15 items of all different sizes. Next, make a list of the names of the items that you have selected. Measure each item and write down the measurements. Do this independently. Compare measurements with your partner. Discuss any discrepancies and make adjustments as needed. Review (Answers) To see the Review answers, open this PDF file and look for section 5.13. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/166986 59 1.14. Mixed Numbers as Improper Fractions www.ck12.org 1.14 Mixed Numbers as Improper Fractions In this concept, you will learn how to rewrite mixed numbers as an improper fraction. Casey ordered eight pizzas for the drama club to enjoy. Each pizza had ten slices. At the end of the pizza party, there were two whole pizzas and two slices left. How many slices weren’t eaten? How can you express this as an improper fraction of pizza slices? In this concept, you will learn how to rewrite mixed numbers as an improper fraction. Writing Mixed Numbers as Improper Fractions A mixed number is a number that has both wholes and parts in it. Here is a mixed number. 5 1 4 There are five whole items and one-fourth of a whole. The opposite of a mixed number is an improper fraction. An improper fraction is a fraction that has a larger numerator than the denominator. Here is an improper fraction. 12 5 The denominator tells you how many parts the whole has been divided into. This whole has been divided into 5 parts. The numerator tells you the number of parts. In this case, there are twelve parts. There are more parts than there are in 1 whole. 60 www.ck12.org Chapter 1. Number Patterns and Fractions To write a mixed number as an improper fraction, write the mixed number as a fraction in terms of parts instead of in terms of wholes and parts. Remember that a whole number can also be written as a fraction. The numerator is equal to the denominator. 1= 5 5 Change 2 13 to an improper fraction. First, multiply the whole number by the denominator to convert the whole to a fraction and add the numerator. This will give you the new numerator. 2×3+1 = 7 Then, put the sum over the denominator. The denominator is 3. 1 7 2 = 3 3 The mixed number 2 13 is also written as 73 . Change the following mixed numbers to improper fractions. Examples Example 1 Earlier, you were given a problem about Casey and the pizzas. Convert 2 pizzas and 2 slices to an improper fraction to find the number of uneaten slices of pizza. 2 Two whole pizzas and two slices = 2 10 First, multiply the whole number by the denominator and add the numerator. 2 × 10 + 2 = 22 Then, put the sum over the denominator. The denominator is 10. 2 2 10 = 22 10 There were a total of 22 slices of pizza left uneaten. Example 2 Express 4 78 as an improper fraction. First, multiply the whole number by the denominator and add the numerator. 4 × 8 + 7 = 39 Then, put the sum over the denominator. The denominator is 8. 61 1.14. Mixed Numbers as Improper Fractions www.ck12.org 7 39 4 = 8 8 The mixed number 4 78 is expressed as 39 8. Example 3 3 13 First, multiply the whole number by the denominator and add the numerator. 3 × 3 + 1 = 10 Then, put the sum over the denominator. The denominator is 3. 3 31 = 10 3 The mixed number 3 13 is expressed as 10 3. Example 4 5 23 First, multiply the whole number by the denominator and add the numerator. 3 × 5 + 2 = 17 Then, put the sum over the denominator. The denominator is 3. 3 31 = 10 3 The mixed number 5 23 is expressed as 17 3. Example 5 6 18 First, multiply the whole number by the denominator and add the numerator. 6 × 8 + 1 = 49 Then, put the sum over the denominator. The denominator is 3. 6 81 = 49 8 The mixed number 6 18 is expressed as 49 8. Review Write each mixed number as an improper fraction. 1. 2. 3. 4. 5. 6. 62 2 12 3 14 5 13 4 23 6 14 6 25 www.ck12.org 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. Chapter 1. Number Patterns and Fractions 7 13 8 25 7 45 8 27 8 34 9 56 6 58 9 23 5 12 16 41 Review (Answers) To see the Review answers, open this PDF file and look for section 5.14. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/166990 63 1.15. Improper Fractions as Mixed Numbers www.ck12.org 1.15 Improper Fractions as Mixed Numbers In this concept, you will learn to write improper fractions as mixed numbers. The school 6th grade class had a bake sale. Missy brought 48 muffins to sell. At the end of the day, there were still 15 muffins left. How many dozen muffins were left? Write the amount as a mixed number. In this concept, you will learn to write improper fractions as mixed numbers. Writing Improper Fractions as Mixed Numbers An improper fraction is a fraction where the numerator is larger than the denominator. An improper fraction can be written as a mixed number. A mixed number is composed of a whole number and a fraction. To change an improper fraction to a mixed number, divide the numerator by the denominator. This will tell you the number of wholes. If there is a remainder, it is the fraction part of a mixed number. Here is an improper fraction. 18 4 There are 18 parts and the whole has only been divided into 4 parts. Remember that when the numerator is larger than the denominator, there is more than one whole. 1 whole = Convert 64 18 4 to a mixed number. 4 4 www.ck12.org Chapter 1. Number Patterns and Fractions First, divide the numerator by the denominator. 18 ÷ 4 = 4R2 Then, write the quotient as a mixed number with the remainder as a fraction. The remainder is the numerator of the fraction. 18 2 =4 4 4 Next, look at the fraction. Simplify the fraction if you can. Divide the numerator and denominator by the greatest common factor, 2. 2 1 = 4 2 The improper fraction 18 4 is expressed as 4 42 or 4 12 . Sometimes, you will have an improper fraction that converts to a whole number and not a mixed number. 18 9 Here 18 divided by 9 is 2. There is no remainder, so there is no fraction. This improper fraction converts to a whole number. The improper fraction 18 9 is expressed as 2. Examples Example 1 Earlier, you were given a problem about Missy and her muffins. Missy had 15 muffins left over from the bake sale and a dozen contains 12 muffins. Convert 15 muffins as a fraction out of 12 to find the number of dozen muffins left. Muffins left over = 15 12 First, divide the numerator by the denominator. 15 ÷ 12 = 1R3 Then, write the quotient as a mixed number with the remainder as a fraction. 65 1.15. Improper Fractions as Mixed Numbers www.ck12.org 3 15 =1 12 12 Next, look at the fraction. Simplify the fraction if you can. Divide the numerator and denominator by the greatest common factor, 3. 3 1 = 12 4 There were 1 14 dozen muffins left over. Example 2 Express this improper fraction as a mixed number. 82 5 First, divide the numerator by the denominator. 82 ÷ 5 = 16R2 Then, write the quotient as a mixed number with the remainder as a fraction. The remainder is the numerator of the fraction. 82 2 = 16 5 5 The improper fraction 82 5 is expressed as 16 52 . Example 3 Express this improper fraction as a mixed number. 24 5 First, divide the numerator by the denominator. 24 ÷ 5 = 4R4 Then, write the quotient as a mixed number with the remainder as a fraction. The remainder is the numerator of the fraction. 24 4 =4 5 5 The improper fraction 66 24 5 is expressed as 4 54 . www.ck12.org Chapter 1. Number Patterns and Fractions Express this improper fraction as a mixed number. 21 3 First, divide the numerator by the denominator. 21 ÷ 3 = 7 This fraction has no remainder and is not a mixed number. The improper fraction 23 3 is equal to 7. Example 5 Express this improper fraction as a mixed number. 32 6 First, divide the numerator by the denominator. 32 ÷ 6 = 5R2 Then, write the quotient as a mixed number with the remainder as a fraction. The remainder is the numerator of the fraction. 2 32 =5 6 6 Next, look at the fraction. Simplify the fraction if you can. Divide the numerator and denominator by the greatest common factor, 2. 2 1 = 6 3 The improper fraction 32 6 is expressed as 5 62 or 5 13 . Review Convert each improper fraction to a mixed number. Simplify when necessary. 1. 22 3 67 1.15. Improper Fractions as Mixed Numbers 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. www.ck12.org 44 5 14 3 7 2 10 3 47 9 50 7 60 8 43 8 19 5 39 7 30 4 11 7 26 5 89 8 70 14 Review (Answers) To see the Review answers, open this PDF file and look for section 5.15. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/167001 68 www.ck12.org Chapter 1. Number Patterns and Fractions 1.16 Fraction and Mixed Number Comparison In this concept, you will learn how to compare and order improper fractions and mixed numbers. Keith and his sister were assigned the task of cleaning up after a party. Keith took all of the leftover tuna sandwiches and his sister took all of the left over ham sandwiches. 69 1.16. Fraction and Mixed Number Comparison Keith has 15 2 www.ck12.org of tuna sandwiches. His sister has 6 34 of ham sandwiches. Who has more sandwiches? In this concept, you will learn how to compare and order improper fractions and mixed numbers. Comparing Improper Fractions and Mixed Numbers An improper fraction is a fraction where the numerator is larger than the denominator. A mixed number is composed of a whole number and a fraction. To compare a mixed number and an improper fraction, first make sure that they are in the same form. Convert the improper fraction to a mixed number or the mixed number to an improper fraction, then compare. 6 Convert 15 4 1 2 15 4 into a mixed number. Divide 15 by 4 and write the quotient as a whole number and a fraction. 3 15 =3 4 4 Compare the numbers. 1 3 6 >3 2 4 6 12 is greater than 15 4. If the whole number is the same, compare the fractions. You may have to convert the fractions using the lowest common denominator. You can order improper fractions and mixed numbers in the same way. Convert them all to the same form and then write them in order. Order these fractions from least to greatest. 1 10 4 1 4 , , ,7 2 6 3 9 First, change the fractions so that they are all in the same form. Let’s change them all to mixed numbers. Simplify if you can. 10 5 6 = 3 = 4 1 3 = 13 1 23 Now you can write them in order from least to greatest. 4 10 1 1 , ,4 ,7 3 6 2 9 70 www.ck12.org Chapter 1. Number Patterns and Fractions Examples Example 1 Earlier, you were given a problem about Keith and the sandwiches. 3 Keith has 15 2 tuna sandwiches and his sister has 6 4 ham sandwiches. Compare the fractions to see who has more sandwiches. First, convert the improper fraction to a mixed number. 15 1 =7 2 2 Then, compare the two quantities. 3 1 7 >6 2 4 Keith has more sandwiches. For the following examples, compare the fractions. Example 2 29 3 7 1 3 First, convert the improper fraction to a mixed number. 29 2 =9 3 3 Compare the numbers. 2 1 9 >7 3 3 29 3 is greater than 7 13 . Example 3 4 1 2 12 5 First, change the improper fraction to a mixed number 12 2 =2 5 5 71 1.16. Fraction and Mixed Number Comparison www.ck12.org Then, compare the numbers. 1 2 4 >2 2 5 4 12 is greater than 12 5. Example 4 16 3 22 5 Both fractions are improper. Let’s try comparing the fractions using the lowest common denominator of 3 and 5. The LCD is 15. First, find the equivalent fraction for each with the denominator of 15. 16 3 = 80 15 22 5 = 66 15 Then, compare the fractions. 80 66 > 15 15 16 3 is greater than 22 5. Example 5 17 4 4 1 4 First, convert the mixed fraction to an improper fraction. Multiply the whole number by the denominator and add the numerator. Write it as a fraction over 4. 4 × 4 + 1 = 17 4 41 = 17 4 Then, compare the fractions. 17 17 = 4 4 17 4 is equal to 4 41 . Review Compare each set of values using <, >or =. 72 www.ck12.org 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 12 5 16 5 44 9 45 7 19 4 16 8 49 5 99 10 69 4 70 3 80 8 75 3 18 3 99 3 78 4 Chapter 1. Number Patterns and Fractions 2 41 3 21 6 31 6 21 4 43 2 6 32 10 8 52 10 47 40 4 25 24 6 33 11 10 89 Review (Answers) To see the Review answers, open this PDF file and look for section 5.16. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/167001 73 1.17. Decimals as Fractions www.ck12.org 1.17 Decimals as Fractions In this concept, you will learn to convert decimals to fractions. About 0.71 of the entire Earth is made up of water. That means 0.29 of the Earth is land. What fraction of the Earth is water? What fraction of the Earth is land? In this concept, you will learn to convert decimals to fractions. Converting Decimals to Fractions Decimals and fractions are related. They both represent a part of a whole. With a decimal, the part of a whole is written using a decimal point. With a fraction, the part of a whole is written using a fraction bar and has a numerator and a denominator. Because fractions and decimals are related, decimals can be written as fractions. Use the place value of the decimal to convert it to a fraction. Here is a decimal number. 0.67 The chart represents the place value of the decimal number. TABLE 1.1: Tens Ones Decimal Point . Tenths Hundredths 6 7 Thousandths TenThousandths The fraction is described by reading the decimal, “sixty-seven hundredths.” The numerator is 67 and the denominator is 100. The place value of the decimal number will indicate the denominator of the fraction. 0.67 = 67 100 Here is another decimal number 0.5 74 www.ck12.org Chapter 1. Number Patterns and Fractions TABLE 1.2: Tens Ones Decimal Point . Tenths Hundredths Thousandths TenThousandths 5 Convert this decimal number to a fraction. This decimal number is read as “five tenths.” The numerator is the five and the denominator is the place value of tenths. 5 10 0.5 = You can simplify the fraction. The greatest common factor (GCF) of 5 and 10 is 5. 5 10 = 1 2 0.5 is written as 5 10 or 12 . Examples Example 1 Earlier, you were given a problem about the Earth’s water and land. About 0.71 of the entire Earth is water and 0.29 of the Earth is land. Convert the decimals to fractions to find the fraction of the Earth that is water and land. First, convert the decimal 0.71 to a fraction. 0.71 is 71 hundredths. 0.71 = 71 100 71 is a prime number and cannot be simplified. Then, convert the decimal 0.29 to a fraction. 0.29 is 29 hundredths. 0.29 = 29 100 29 is also a prime number and cannot be simplified. The Earth is 71 100 water and 29 100 land. Example 2 Jessie has completed 0.85 of her homework. If she was going to express this number as a fraction what would the fraction be? Write your answer in simplest form. First, convert the decimal number to a fraction. 0.85 is eighty-five hundredths. The numerator is 85 and the denominator is 100. 85 100 Then, simplify thefraction. The greatest common factor of 85 and 100 is 5. 85 100 = 17 20 0.85 is written as 17 20 . 75 1.17. Decimals as Fractions Example 3 Convert the decimal number to a fraction in simplest form. 0.8 First, convert the decimal to a fraction. 0.8 is 8 tenths. 8 10 0.8 = Then, simplify the fraction. The GCF of 8 and 10 is 2. 8 10 = 4 5 0.8 is written as 4 5 in simplest form. Example 4 Convert the decimal number to a fraction in simplest form. 0.25 First, convert the decimal to a fraction. 0.25 is 25 hundredths. 0.25 = 25 100 Then, simplify the fraction. The GCF of 25 and 100 is 25. 25 100 = 1 4 0.25 is written as 1 4 in simplest form. Example 5 Convert the decimal number to a fraction in simplest form. 0.75 First, convert the decimal to a fraction. 0.75 is 75 hundredths. 0.75 = 75 100 Then, simplify the fraction. The GCF of 75 and 100 is 25. 75 100 = 3 4 0.75 is written as 3 4 in simplest form. Review Write each decimal as a fraction. Do not simplify. 1. 2. 3. 4. 5. 6. 7. 8. 76 0.67 0.33 0.45 0.27 0.56 0.7 0.98 0.32 www.ck12.org www.ck12.org 9. 10. 11. 12. 13. 14. 15. Chapter 1. Number Patterns and Fractions 0.04 0.07 0.056 0.897 0.372 0.652 0.032 Review (Answers) To see the Review answers, open this PDF file and look for section 5.17. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/167024 MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/167026 77 1.18. Decimals as Mixed Numbers www.ck12.org 1.18 Decimals as Mixed Numbers In this concept, you will learn to convert decimals to mixed numbers. Henry is working on building a dog house. He needs to get lumber that is 6 inches wide and 1.5 inches thick. He goes to the hardware store and sees that they have 6 by 1 14 , 6 by 1 12 , and 6 by 34 . Which one does Henry need? In this concept, you will learn to convert decimals to mixed numbers. Converting Decimals to Mixed Numbers Some decimal numbers represent both a part and a whole. These decimal numbers can be written as mixed numbers. The decimal number must have both a whole and a part to be written as a mixed number. The mixed number and the decimal are equal because they both have the same value. Here is a decimal number. 4.5 Let’s write this decimal in a place value chart. TABLE 1.3: Tens Ones 4 Decimal Point . Tenths Hundredths Thousandths TenThousandths 5 This decimal number has 4 ones and 5 tenths. The 4 represents the wholes. The 5 tenths represents the fraction. The five is the numerator and the tenths is the denominator. 78 www.ck12.org Chapter 1. Number Patterns and Fractions 4.5 = 4 5 10 Next, check and see if the fraction can be simplified. In this case, five-tenths can be simplified to one-half. 5 1 = 10 2 4.5 can be written as 4 21 . A decimal value can only be expressed one way. However, many fractions can be written to express the same value. 0.75 can be written as Simplify 75 100 . You can make an equivalent fraction that has the same value. 75 100 . 3 75 = 100 4 You can keep on creating equivalent fractions that have the same value as 0.75. 3 6 9 75 = = = 100 4 8 12 Finding equivalent fractions for mixed numbers is similar. The whole number stays the same, but the fraction can vary. Here is a decimal number. 4.56 Convert the decimal number to a mixed number and find equivalent fractions. First, convert the decimal to a mixed number. 4.56 is read as four and fifty-six hundredths, the four is the whole number, the fifty-six is the numerator, and the denominator is the hundredths. 4 56 100 Then, simplify the fraction part of this mixed number to get another mixed number that is equivalent to the one above. The greatest common factor of 56 and 100 is 4. 4 56 14 =4 100 25 Examples Example 1 Earlier, you were given a problem about Henry building a dog house. 79 1.18. Decimals as Mixed Numbers www.ck12.org He needs lumber that is 6 inches wide and 1.5 inches thick. Convert 1.5 inches into a fraction to find the lumber he needs. First, convert the decimal to a fraction. 1.5 is 1 and 5 tenths. 1.5 = 1 5 10 Then, write the fraction in simplest form. The GCF of 5 and 10 is 5. 1 5 1 =1 10 2 Henry needs to by the 6 by 1 12 inch pieces of wood. Example 2 Convert the following decimal to a mixed number in simplest form. 6.55 First, convert the decimal to a mixed number. 6.55 is read as six and fifty-five hundredths. 6 is the whole number, 55 is the numerator, and 100 is the denominator. 6.55 = 6 55 100 Then, write the fraction in simplest form. The GCF of 55 and 100 is 5. 6 55 11 =6 100 20 11 in simplest form. 6.55 is written as 6 20 Example 3 Convert the decimal to a mixed number in simplest form. 7.8 First, convert the decimal to a mixed number. 7.8 is 7 and 8 tenths. 7.8 = 7 8 10 Then, write the fraction in simplest form. The GCF of 8 and 10 is 2. 7 7.8 is written as 7 45 in simplest form. 80 8 4 =7 10 5 www.ck12.org Chapter 1. Number Patterns and Fractions Example 4 Convert the decimal to a mixed number in simplest form. 4.45 First, convert the decimal to a mixed number. 4.45 is 4 and 45 hundredths. 4.45 = 4 45 100 Then, write the fraction in simplest form. The GCF of 45 and 100 is 5. 4 9 45 =4 100 20 9 4.45 is written as 4 20 in simplest form. Example 5 Convert the decimal to a mixed number in simplest form. 2.25 First, convert the decimal to a mixed number. 2.25 is 2 and 25 hundredths. 2.25 = 2 25 100 Then, write the fraction in simplest form. The GCF of 25 and 100 is 25. 2 25 1 =2 100 4 2.25 is written as 2 41 in simplest form. Review Convert each decimal to a mixed number in simplest form. 1. 2. 3. 4. 5. 6. 7. 8. 9. 3.5 2.4 13.2 25.6 3.45 7.17 18.18 9.20 7.65 81 1.18. Decimals as Mixed Numbers 10. 11. 12. 13. 14. 15. www.ck12.org 13.11 7.25 9.75 10.10 4.33 8.22 Review (Answers) To see the Review answers, open this PDF file and look for section 5.18. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/167083 82 www.ck12.org Chapter 1. Number Patterns and Fractions 1.19 Fractions as Decimals In this concept, you will learn to convert fractions to decimals. Michelle and Terry are shooting hoops. Michelle made 7 out of the last 10 shots. Terry made 6 out the last 8 shots. Compare their results using decimals. Who had better results? In this concept, you will learn to convert fractions to decimals. Converting Fractions to Decimals Decimals and fractions both represent quantities that are part of a whole. Fractions can also be converted to a decimal number. There are two ways to convert a fraction to a decimal. The first way is to think in terms of place value. If a fraction that has ten as a denominator, you can think of that fraction as tenths. Here is a fraction of a tenth and the decimal equivalent. 6 = .6 10 There is one decimal place in tenths, so this decimal is accurate. This is a very useful method when the denominator is a base ten value like: 10, 100, 1, 000 . . . Here is a fraction with a base ten value of 1,000. 125 1000 There are three decimal places in a thousandth decimal. There are three digits in the numerator. This fraction converts easily to a decimal. 83 1.19. Fractions as Decimals www.ck12.org 125 = 0.125 1000 The second way is to use division. The fraction bar is also a symbol for division. The numerator is the dividend and the denominator is the divisor. Here is another fraction. 3 5 To change 35 to a decimal number, divide 3 by 5. Remember that you are looking for a decimal number. Use zero placeholders to help find the decimal value. 0.6 5)3.0 − 3.0 0 The decimal value of 3 5 is 0.6. Examples Example 1 Earlier, you were given a problem about Michelle and Terry playing basketball. Michelle made 7 out of the last 10 shots and Terry made 6 out the last 8 shots. 7 out of 10 is also also 68 . Convert the fractions to decimals and compare their results. First, convert 7 10 to a decimal. The denominator is a base ten number. 7 = 0.7 10 Then, convert 6 8 to a decimal. Divide 6 by 8. 0.75 8)6.00 −56 40 −40 0 Next, compare the decimals. The better player has the larger decimal number. 0.7 < 0.75 Terry made more of his shots than Michelle. 84 7 10 . 6 out of 8 is www.ck12.org Chapter 1. Number Patterns and Fractions Example 2 Write the following fraction as a decimal. 1 4 One way is to use base ten values. First, find an equivalent fraction of 1 4 with a denominator of 100. 1 25 = 4 100 Then, convert the fraction to a decimal. 25 100 is also 25 hundredths. 25 = 0.25 100 The decimal value of 1 4 is 0.25. The other way is to use division. Divide 1 by 4. Use zero place holders if needed. 0.25 4)1.00 −8 20 −20 0 The decimal value of 1 4 is 0.25. Example 3 Convert the fraction to a decimal. 8 10 This fraction has a base ten value in the denominator. Place the 8 in the tenth place. 8 = 0.8 10 The decimal value of 8 10 is 0.8. 85 1.19. Fractions as Decimals www.ck12.org Example 4 Convert the fraction to a decimal. 5 100 This fraction has a base ten value in the denominator. Place 5 in the hundredths place. 5 = 0.05 100 The decimal value of 5 100 is 0.05. Example 5 Convert the fraction to a decimal. 4 5 Divide the numerator by the denominator. Use zero placeholders if needed. 0.8 5)4.0 − 4.0 0 The decimal value of 4 5 is 0.8. Review Convert the following fractions as decimals. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 86 3 10 23 100 9 100 8 10 182 1000 25 100 6 10 125 1000 1 10 2 100 1 2 1 4 3 4 3 6 3 5 www.ck12.org Chapter 1. Number Patterns and Fractions Review (Answers) To see the Review answers, open this PDF file and look for section 5.19. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/167101 87 1.20. Mixed Numbers as Decimals www.ck12.org 1.20 Mixed Numbers as Decimals In this concept, you will learn to convert mixed numbers to decimals. Kara is making curtains for her room. She has a total of 6 45 yards of fabric to work with. Each curtain needs to be 3.45 yards and she is making 2 curtains. Will she have enough fabric for both? In this concept, you will learn to convert mixed numbers to decimals. Converting Mixed Numbers to Decimals You can convert fractions to decimals using two methods. One method involves using fractions with base ten denominators and place values. Another method involves division. These methods can also be use to convert mixed numbers to decimal numbers. Here is a mixed number. 5 3 10 You can also think of a mixed number as the sum of a whole number and a fraction. 5+ 3 10 To find the decimal value of a mixed number, convert the fraction to a decimal number and add the whole number to the decimal value of the fraction. Remember that the whole number in a fraction has the same value as a whole number in a decimal number. Whole numbers are placed on the left side of the decimal point. 88 www.ck12.org The fraction 3 10 Chapter 1. Number Patterns and Fractions is 3 tenths. Place the whole number, 5, to the left of the decimal point and the 3 in the tenths place. 5 3 = 5.3 10 Here is another mixed number. 8 1 5 First, find the decimal value of 15 . Divide 1 by 5 to get the decimal part of the number. 0.2 5)1.0 − 1.0 0 Then, place the whole number to the left of the decimal point. 1 8 = 8.2 5 The decimal value of 8 51 is 8.2. Examples Example 1 Earlier, you were given a problem about Kara and her curtains. Kara wants to make 2 curtains that are 3.45 yards each and has 6 45 yards of fabric. Convert the mixed number to see if she will have enough. First, convert the fraction part into a decimal. 4 = 0.8 5 Then, place the whole number to the left of the decimal point. 4 6 = 6.8 5 Next, find the total amount she needs. 3.45 × 2 = 6.9 Kara will not have enough fabric and will be short 0.1 yards of fabric. 89 1.20. Mixed Numbers as Decimals www.ck12.org Example 2 Write the following mixed number as a decimal. 16 3 4 First, convert the fraction part into a decimal. Divide or find an equivalent fraction with the denominator as a base ten value. 3 75 = = 0.75 4 100 Then, place the whole number to the left of the decimal point. 3 16 = 16.75 4 The decimal value of 16 43 is 16.75. Example 3 Write the mixed number as a decimal. 6 First, convert the fraction part into a decimal. 13 100 13 100 is 13 hundredths. 13 = 0.13 100 Then, place the whole number to the left of the decimal point. 6 13 The decimal value of 6 100 is 6.13. Example 4 Write the mixed number as a decimal. 90 13 = 6.13 100 www.ck12.org Chapter 1. Number Patterns and Fractions 15 First, convert the fraction part into a decimal. 9 10 9 10 is 9 tenths. 9 = 0.9 10 Then, place the whole number to the left of the decimal point. 9 is 15.9. The decimal value of 15 10 Example 5 Write the mixed number as a decimal. 6 1 4 First, convert the fraction part into a decimal. 0.25 4)1.00 −8 20 −20 0 Then, place the whole number to the left of the decimal point. 1 6 = 6.25 4 The decimal value of 6 41 is 6.25. Review Write each mixed number as a decimal. 1. 2. 3. 4. 5. 6. 7. 1 4 10 8 6 10 6 14 100 18 7 100 9 12 10 11 24 100 19 8 100 91 1.20. Mixed Numbers as Decimals 8. 9. 10. 11. 12. 13. 14. 15. www.ck12.org 10 5 20 4 12 7 13 2 5 10 9 18 2 10 100 1 46 4 65 45 Review (Answers) To see the Review answers, open this PDF file and look for section 5.20. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/167109 92 www.ck12.org Chapter 1. Number Patterns and Fractions 1.21 Repeating Decimals In this concept, you will learn to write fractions and mixed numbers as repeating decimals. Jose has 10 bars of chocolate that he needs to give to 3 of his friends. How many bars of chocolate does each friend receive? In this concept, you will learn to write fractions and mixed numbers as repeating decimals. Writing Fractions and Mixed Numbers as Repeating Decimals A terminating decimal is a decimal number that does not go on forever. The word “terminate” means to end. Most of the fractions you have been working with are terminating decimals. Here is a fraction with a terminating decimal. 1 4 Divide 1 by 4 to find the decimal value. 93 1.21. Repeating Decimals www.ck12.org 0.25 4)1.00 −8 20 −20 0 You use zero placeholders, but ultimately, the decimal will divide evenly. A decimal that does not end and repeats the same number or numbers over and over again is called a repeating decimal. When you divide the numerator by the denominator and keep ending up with the same number, you might have a repeating decimal. Convert 2 3 to a decimal. First, this does not have a base ten denominator. Divide the numerator by the denominator. 0.666 4)2.000 − 18 20 −18 20 −18 2 The same remainder keeps showing up and the quotient becomes a series of 6’s. It does not matter if you keep adding zero placeholders. A repeating decimal is indicated by adding a line over the last digit or series of digits in the quotient that repeats itself. The decimal value of 2 3 is 0.6̄ . Examples Example 1 Earlier, you were given a problem about Jose and his chocolate bars. Jose wants to give 10 chocolate bars to 3 of his friends. Divide 10 by 3 to find how many chocolate bar each friend receives. Divide 10 by 3. 94 www.ck12.org Chapter 1. Number Patterns and Fractions 3.333 3)10.000 −9 10 −9 10 −9 10 −9 1 The answer is a repeating decimal 3.3̄. Jose can give each friend 3.3̄ bars of chocolate. Example 2 Is 4 9 a repeating decimal or a terminating decimal? Convert the fraction to a decimal. Divide 4 by 9. 0.4444 9)4.0000 − 36 40 −36 40 −36 40 −36 4 The same remainder keeps showing up and the quotient will go on and on as a series of 4s. The decimal value of 4 9 is a repeating decimal, 0.4̄. Example 3 Determine if the fraction is a repeating or terminating decimal. 1 3 Convert the fraction to a decimal. Divide 1 by 3. 95 1.21. Repeating Decimals www.ck12.org 0.3333 3)1.0000 −9 10 −9 10 −9 10 −9 1 The decimal value of 1 3 is a repeating decimal, 0.3̄. Example 4 Determine if the fraction is a repeating or terminating decimal. 1 8 Convert the fraction to a decimal. Divide 1 by 8. 0.125 8)1.000 −8 20 −16 40 −40 0 The decimal value of 1 8 is a terminating decimal, 0.125. Example 5 Determine if the fraction is a repeating or terminating decimal. 5 1 2 First, convert the fraction part to a decimal. 1 = 0.5 2 Then, place the whole number to the left of the decimal point. 96 www.ck12.org Chapter 1. Number Patterns and Fractions 1 5 = 5.5 2 The decimal value of 5 21 is a terminating decimal, 5.5. Review Determine if the fractions are repeating or terminating decimals. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 14 3 34 9 23 3 17 4 19 6 12 5 3 13 8 12 9 23 11 54 16 14 44 3 66 7 18 4 74 7 Review (Answers) To see the Review answers, open this PDF file and look for section 5.21. Resources MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/162265 MEDIA Click image to the left or use the URL below. URL: https://www.ck12.org/flx/render/embeddedobject/162269 97