Common Multiples. List the multiples of 3. The multiples of 3 are 3 1, 3 2, 3 3, 3 4,...


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1 .2 Common Multiples.2 OBJECTIVES 1. Find the least common multiple (LCM) of two numbers 2. Find the least common multiple (LCM) of a group of numbers. Compare the size of two fractions In this chapter, we are discussing the process used for adding or subtracting two fractions. One of the most important concepts we use in the addition and subtraction of fractions is that of multiples. Definitions: Multiples The multiples of a number are the product of that number with the natural numbers 1, 2,,,,.... Example 1 Listing Multiples List the multiples of. The multiples of are 1, 2,,,... or NOTE Notice that the multiples, except for itself, are larger than., 6, 9, 12,... The three dots indicate that the list will go on without stopping. An easy way of listing the multiples of is to think of counting by threes. CHECK YOURSELF 1 List the first seven multiples of. Sometimes we need to find common multiples of two or more numbers. Definitions: Common Multiples If a number is a multiple of each of a group of numbers, it is called a common multiple of the numbers; that is, it is a number that is evenly divisible by all the numbers in the group. 2
2 2 CHAPTER ADDING AND SUBTRACTING FRACTIONS Example 2 Finding Common Multiples NOTE 1, 0,, and 60 are multiples of both and. Find four common multiples of and. Some common multiples of and are 1, 0,, 60 CHECK YOURSELF 2 List the first six multiples of 6. Then look at your list from Check Yourself 1 and list some common multiples of and 6. For our later work, we will use the least common multiple of a group of numbers. Definitions: Least Common Multiple The least common multiple (LCM) of a group of numbers is the smallest number that is a multiple of each number in the group. It is possible to simply list the multiples of each number and then find the LCM by inspection. Example Finding the Least Common Multiple (LCM) Find the least common multiple of 6 and 8. Multiples NOTE 8 is also a common multiple of 6 and 8, but we are looking for the smallest such number. 6: 6, 12, 18, 2, 0, 6, 2, 8,... 8: 8, 16, 2, 2, 0, 8,... We see that 2 is the smallest number common to both lists. So 2 is the LCM of 6 and 8. CHECK YOURSELF Find the least common multiple of 20 and 0 by listing the multiples of each number. The technique of the last example will work for any group of numbers. However, it becomes tedious for larger numbers. Let s outline a different approach.
3 COMMON MULTIPLES SECTION.2 2 Step by Step: Finding the Least Common Multiple NOTE For instance, if a number appears three times in the factorization of a number, it must be included at least three times in forming the least common multiple. Step 1 Step 2 Step Write the prime factorization for each of the numbers in the group. Find all the prime factors that appear in any one of the prime factorizations. Form the product of those prime factors, using each factor the greatest number of times it occurs in any one factorization. Some students prefer a slightly different method of lining up the factors to help in remembering the process of finding the LCM of a group of numbers. Example Finding the Least Common Multiple (LCM) To find the LCM of 10 and 18, factor: NOTE Line up the like factors vertically Bring down the factors. 2 and appear, at most, one time in any one factorization. And appears two times in one factorization So 90 is the LCM of 10 and 18. CHECK YOURSELF Use the method of Example to find the LCM of 2 and 6. The procedure is the same for a group of more than two numbers. Example Finding the Least Common Multiple (LCM) To find the LCM of 12, 18, and 20, we factor: NOTE The different factors that appear are 2,, and and appear twice in one factorization, and appears just once So 180 is the LCM of 12, 18, and 20.
4 26 CHAPTER ADDING AND SUBTRACTING FRACTIONS CHECK YOURSELF Find the LCM of,, and 6. The process of finding the least common multiple is very useful when adding, subtracting, or comparing unlike fractions (fractions with different denominators). Suppose you are asked to compare the sizes of the fractions and. Because each unit in the diagram is divided into seven parts, it is easy to see that is larger than. Four parts of seven are a greater portion than three parts. Now compare the size of the 2 fractions and. 2 2 We cannot compare fifths with sevenths! and are not like fractions. Because they name different ways of dividing the whole, deciding which fraction is larger is not nearly so easy. To compare the sizes of fractions, we change them to equivalent fractions having a common denominator. This common denominator must be a common multiple of the original denominators. Rules and Properties: The Fundamental Principle of Fractions a b a c b c
5 COMMON MULTIPLES SECTION.2 2 Example 6 Finding Common Denominators to Order Fractions 2 Compare the sizes of and. The original denominators are and. Because is a common multiple of and, let s use as our common denominator. 2 1 NOTE and are equivalent fractions. They name the same part of a whole. 2 1 Think, What must we multiply by to get? The answer is. Multiply the numerator and denominator by that number. 1 Multiply the numerator and denominator by. NOTE 1 of parts represents a greater portion of the whole than 1 parts Because and, we see that is larger than. CHECK YOURSELF 6 Which is larger, or? 9 Let s consider an example that uses the inequality notation. Example Using an Inequality Symbol with Two Fractions NOTE The inequality symbol points to the smaller quantity. Use the inequality symbol or to complete the statement below. 8 Once again we must compare the sizes of the two fractions, and this is done by converting the fractions to equivalent fractions with a common denominator. Here we will use 0 as that denominator Because or is larger than or, we write
6 28 CHAPTER ADDING AND SUBTRACTING FRACTIONS CHECK YOURSELF Use the symbol or to complete the statement CHECK YOURSELF ANSWERS 1. The first seven multiples of are, 8, 12, 16, 20, 2, and , 12, 18, 2, 0, 6; some common multiples of and 6 are 12, 2, and 6.. The multiples of 20 are 20, 0, 60, 80, 100, 120,... ; the multiples of 0 are 0, 60, 90, 120, 10,... ; the least common multiple of 20 and 0 is 60, the smallest number common to both lists is larger
7 Name.2 Exercises Section Date Find the least common multiple (LCM) for each of the following groups of numbers. Use whichever method you wish and 2. and ANSWERS and 6. 6 and and and and and and and and and and and and and ,, and , 8, and , 21, and , 1, and , 0, and , 20, and
8 ANSWERS 2. Arrange the given fractions from smallest to largest , , , 9 10, , 1, 1 12, 18, ,, 6 Complete the statements, using the symbol or , 9 16, A company uses two types of boxes, 8 cm and 10 cm long. They are packed in larger cartons to be shipped. What is the shortest length container that will accommodate boxes of either size without any room left over? (Each container can contain only boxes of one size no mixing allowed.) 20
9 ANSWERS 0. There is an alternate approach to finding the least common multiple of two numbers. The LCM of two numbers can be found by dividing the product of the two numbers by the greatest common factor (GCF) of those two numbers. For example, the GCF of 2 and 6 is 12. If we use the above formula, we obtain LCM of 2 and (a) Use the above formula to find the LCM of 10 and 80. (b) Verify the result by finding the LCM using the method of prime factorization. (c) The above approach can be extended so that it can be used to find the LCM of three numbers. Describe this extension. (d) Use the results of part (c) to find the LCM of 8, 1, and Solve the following applications. 1. Drill bits. Three drill bits are marked Which drill bit is largest? 8, 11, and Bolt size. Bolts can be purchased with diameters of inches (in.). Which 8, 1, or 16 is smallest?. Plywood size. Plywood comes in thicknesses of Which size is 8,, 1 2, and 8 in. thickest? 1. Doweling. Doweling is sold with diameters of in. Which size is 2, 9 16, 8, and 8 smallest? 1. Elian is asked to create a fraction equivalent to His answer is What did he do.. wrong? What would be a correct answer? 21
10 ANSWERS A sign on a busy highway says Exit A is mile away and Exit B is mile away. 8 Which exit is first?. Complete the following Crossword puzzle. ACROSS 2. The LCM of 11 and 1. The GCF of 120 and The GCF of 1 and 2 8. The GCF of 60 and 0 DOWN 1. The LCM of 8, 1, and The LCM of 16 and 12. The LCM of 2,, and 1 6. The GCF of and 90 8 Answers ; 1 ; the LCM is , 1, , 6, 11, 1, in. 22
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