Adding and Subtracting Fractions Chapter 3 Adding and Subtracting Fractions © 2010 Pearson Education, Inc. All rights reserved.

3.1 Adding and Subtracting Like Fractions Objectives 1. Define like and unlike fractions. 2. Add like fractions 3. Subtract like fractions. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Fractions with the same denominators are like fractions. Fractions with different denominators are unlike fractions. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Identifying Like and Unlike Fractions a. b. Like fractions All denominators are the same. Unlike fractions Denominators are different. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.1- 5

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Adding Like Fractions Add and write the sum in lowest terms. a. Add numerators Same denominator Copyright © 2010 Pearson Education, Inc. All rights reserved.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Adding Like Fractions Add and write the sum in lowest terms. b. Add numerators Step 1 12 16 Step 2 Sum of numerators Same denominator Step 3 In lowest terms Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.1- 7

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Subtracting Like Fractions Find the difference and simplify the answer. a. Subtract numerators Step 1 Step 2 Step 3 In lowest terms Copyright © 2010 Pearson Education, Inc. All rights reserved.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Subtracting Like Fractions Find the difference and simplify the answer. b. Subtract numerators To simplify the answer, write as a mixed number. Copyright © 2010 Pearson Education, Inc. All rights reserved.

3.2 Least Common Multiples Objectives 1. Find the least common multiple. 2. Find the least common multiple using multiples of the largest number. 3. Find the least common multiple using prime factorization. 4. Find the least common multiple using an alternative method. 5. Write a fraction with an indicated denominator. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.2- 11

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.2- 12

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Finding the Least Common Multiple Find the least common multiple of 4 and 6. List the multiples of 4. List the multiples of 6 4 • 1 4 • 2 4 • 3 4 • 4 4 • 5 4 • 6 4 • 7 4 • 8 4, 8, 12, 16, 20, 24, 28, 32,… 6 • 1 6 • 2 6 • 3 6 • 4 6 • 5 6 • 6 6 • 7 6 • 8 6, 12, 18, 24, 30, 36, 42, 48,… The smallest number found in both lists is 12, so 12 is the least common multiple of 4 and 6. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.2- 13

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Finding the Least Common Multiple Use multiples of the larger number to find the least common multiple of 6 and 10. List the first few multiples of 10. 10, 20, 30, 40, 50, 60, 70, … Now, we check each multiple of 10 to see if it is divisible by 6. The first multiple of 10 that is divisible by 6 is 30. The least common multiple of 6 and 10 is 30. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.2- 14

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Applying Prime Factorization Knowledge Use prime factorization to find the least common multiple of 12 and 45. Find the prime factorization of each number. 12 = 2  2  3 45 = 3  3  5 LCM = 2  2  3  3  5 = 180 Check to see that 180 is divisible by 12 (yes) and (45) yes. The smallest whole number divisible by both 12 and 45 is 180. Factors of 12 Factors of 45 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.2- 15

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Using Prime Factorization Find the least common multiple of 20, 25, and 30. Find the prime factorization of each number. 20 = 2  2  5 25 = 5  5 30 = 2  3  5 LCM = 2  2  3  5  5 = 300 Check to see that 300 is divisible by 20 (yes), 25 (yes) and (30) yes. The smallest whole number divisible by 20, 25, and 30 is 300. Factors of 20 Factors of 25 Factors of 30 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.2- 16

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 5 Finding the Least Common Multiple Find the least common multiple for 7, 15, and 21. Find the prime factorization of each number. 7 = 7 15 = 3  5 21 = 3  7 LCM = 3  5  7 = 105 The least common multiple of 7, 15, and 21 is 105. Factors of 15 Factors of 7 Factors of 21 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.2- 17

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 5 Finding the Least Common Multiple Find the least common multiple for 18, 20, and 30. Find the prime factorization of each number. 18 = 2  3  3 20 = 2  2  5 30 = 2  3  5 LCM = 2  2  3  3  5 = 180 The least common multiple of 18, 20, and 30 is 180. Factors of 18 Factors of 20 Factors of 30 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.2- 18

Some people like the following alternative method for finding the least common multiple for larger numbers. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3.2- 19

Alternative Method for Finding the Least Common Multiple Parallel Example 6 Alternative Method for Finding the Least Common Multiple Find the least common multiple of 18 and 21. Start by trying to divide 18 and 21 by the first prime number in the list of primes: 2, 3, 5, 7, 11. 2 Multiply these numbers to get the LCM. 3 3 7 All quotients are 1. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.2- 20

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 6 Alternative Method for Finding the Least Common Multiple Find the least common multiple of 12, 30, and 50. 2 2 3 5 5 All quotients are 1. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.2- 21

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 7 Writing a Fraction with a New Denominator Write the fraction with a denominator of 18. To find the new numerator, first divide 18 by 6. Multiply both numerator and denominator of the fraction by 3. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.2- 22

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 8 Writing Fractions with a New Denominator Rewrite each fraction with the indicated denominator. a. b. Divide 32 by 8, getting 4. Now multiply both the numerator and denominator by 4. Multiply numerator and denominator by 4. Multiply numerator and denominator by 5. 45 ÷ 9 = 5 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.2- 23

3.3 Adding and Subtracting Unlike Fractions Objectives 1. Add unlike fractions. 2. Add unlike fractions vertically. 3. Subtract unlike fractions. 4. Subtract unlike fractions vertically. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.3- 24

Copyright © 2010 Pearson Education, Inc. All rights reserved. To add unlike fractions, we must first change them to like fractions (fractions with the same denominator.) Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.3- 25

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Adding Unlike Fractions Add The least common multiple of 6 and 12 is 12. Write the fractions as like fractions with a denominator of 12. This is the least common denominator (LCD). Step 1 Step 2 Step 3 Step 3 is not needed because the fraction is in lowest terms. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.3- 26

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Adding Fractions Add the fractions using the three steps. Simplify all answers. The least common multiple of 4 and 8 is 8. Rewritten as like fractions Step 1 Step 2 Step 3 Step 3 is not needed because the fraction is in lowest terms. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.3- 27

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Add unlike fractions vertically Add the following fractions vertically. a. Rewritten as like fractions Add the numerators. Denominator is 24, the LCD. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.3- 28

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Add unlike fractions vertically Add the following fractions vertically. b. Rewritten as like fractions Add the numerators. Denominator is 24, the LCD. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.3- 29

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Subtracting Unlike Fractions Subtract. Simplify all answers. Rewritten as like fractions Step 1 Subtract numerators. Step 2 Step 3 Step 3 is not needed because the fraction is in lowest terms. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.3- 30

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Subtracting Unlike Fractions Subtract. Simplify all answers. Rewritten as like fractions Step 1 Subtract numerators. Step 2 Step 3 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.3- 31

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 5 Vertical Subtraction of Fractions Subtract the following fractions vertically. a. Rewritten as like fractions Subtract the numerators. Denominator is 15, the LCD. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.3- 32

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 5 Vertical Subtraction of Fractions Subtract the following fractions vertically. b. Rewritten as like fractions Subtract the numerators. Denominator is 15, the LCD. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.3- 33

3.4 Adding and Subtracting Mixed Numbers Objectives 1. Estimate an answer, then add or subtract mixed numbers. 2. Estimate an answer, then subtract mixed numbers by regrouping. 3. Add or subtract mixed numbers using an alternate method. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.4- 34

Copyright © 2010 Pearson Education, Inc. All rights reserved. Add or subtract mixed numbers by adding or subtracting the fraction parts and then the whole number parts. It is a good idea to estimate the answer first. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.4- 35

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Adding and Subtracting Mixed Numbers First estimate the answer. Then add or subtract to find the exact answer. a. Estimate Exact Rounds to Rounds to Sum of whole numbers. Sum of fractions. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.4- 36

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Adding and Subtracting Mixed Numbers First estimate the answer. Then add or subtract to find the exact answer. b. 30 is the least common denominator. Estimate Exact Rounds to Rounds to Subtract whole numbers. Subtract fractions. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.4- 37

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Simplify and Regroup When Adding Mixed Numbers First estimate, and then add Estimate Exact Rounds to Becomes Rounds to Sum of whole numbers. Sum of fractions. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.4- 38

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Regroup When Subtracting Mixed Numbers a. First estimate, and then subtract. Estimate Exact Rounds to Rounds to Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.4- 40

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Regroup When Subtracting Mixed Numbers b. First estimate, and then subtract. Estimate Exact Rounds to Rounds to Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.4- 41

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Regroup When Subtracting Mixed Numbers First estimate, and then subtract. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.4- 42

Copyright © 2010 Pearson Education, Inc. All rights reserved. An alternate method for adding or subtracting mixed numbers is to first change the mixed numbers to improper fractions. Then, rewrite the unlike fractions as like fractions. Finally, add or subtract the numerators and write the answer in lowest terms. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.4- 43

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Adding or Subtracting Mixed Numbers Using an Alternate Method Add. 6 is the least common denominator Change to improper fractions. Mixed number Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.4- 44

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 continued Adding or Subtracting Mixed Numbers Using an Alternate Method Subtract. 15 is the least common denominator Change to improper fractions. Mixed number Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.4- 45

3.5 Order Relations and the Order of Operations Objectives 1. Identify the greater of two fractions. 2. Use exponents with fractions. 3. Use the order of operations with fractions. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.5- 46

Copyright © 2010 Pearson Education, Inc. All rights reserved. There are times when we want to compare the size of two numbers. For example, we might want to know which is the greater amount, the larger size, or the longer distance. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.5- 47

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Using Less-Than and Greater-Than Symbols Parallel Example 1 Using Less-Than and Greater-Than Symbols Rewrite the following using < and > symbols. a. b. The number farther to the left on the number line is less. The number farther to the right on the number line is greater. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.5- 49

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Identifying the Greater Fraction Parallel Example 2 Identifying the Greater Fraction Determine which fraction in each pair is greater. a. b. Rewrite both fractions with a common denominator of 40 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.5- 51

Copyright © 2010 Pearson Education, Inc. All rights reserved. Exponents were used to write repeated multiplication in an earlier chapter. exponent exponent Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.5- 52

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Using Exponents with Fractions Simplify. a. b. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.5- 53

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Using Exponents with Fractions Simplify. c. 4 4 2 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.5- 54

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Using the Order of Operations with Fractions Simplify by using the order of operations. a. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.5- 56

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Using the Order of Operations with Fractions Simplify by using the order of operations. b. Simplify the exponent. Multiply. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 3.5- 57