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5.1 Rational Numbers Goal: Write fractions as decimals and vice versa. Vocabulary Rational number: Terminating decimal: Repeating decimal: Example 1 Identifying Rational Numbers Show that the number is rational by writing it as a quotient of two integers. 2 3 b. 12 a. 3 1 4 d. 2 c. 4 Solution a. Write the integer 3 as . b. Write the integer 12 as are or . These fractions . 2 3 c. Write the mixed number 4 as the improper fraction 1 4 d. Think of 2 as the opposite of 1 4 Then you can write 2 as . First write . To write . as . as a quotient of two integers, you can assign the negative sign to either the or the . You can write or . 84 Chapter 5 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 5.1 Rational Numbers Goal: Write fractions as decimals and vice versa. Vocabulary Rational A rational number is a number that can be written as number: the quotient of two integers. Terminating decimal: Repeating decimal: Example 1 A decimal that has a final digit A decimal that has one or more digits that repeat without end Identifying Rational Numbers Show that the number is rational by writing it as a quotient of two integers. 2 3 b. 12 a. 3 1 4 d. 2 c. 4 Solution 3 a. Write the integer 3 as 1 . 12 12 . These fractions b. Write the integer 12 as 1 or 1 are equivalent . 2 14 c. Write the mixed number 4 as the improper fraction 3 . 3 1 1 1 9 d. Think of 2 as the opposite of 2 4 . First write 2 4 as 4 . 4 1 9 9 Then you can write 2 as 4 . To write 4 as a quotient 4 of two integers, you can assign the negative sign to either the 9 4 numerator or the denominator . You can write or 9 . 4 84 Chapter 5 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Writing Fractions as Decimals Example 2 5 16 4 9 a. Write as a decimal. b. Write as a decimal. a. b. 0.3125 16.0 50 0 0 4 8 20 1 6 40 3 2 80 8 0 0 0.44. . . 9.0 40 36 40 3 6 Answer: Use a bar to show the in the decimal: 4 0.4 . 9 Answer: The remainder is , so the decimal is a decimal: 5 0.3125. 16 Checkpoint Write the fraction or mixed number as a decimal. 3 5 7 20 2. 1. 12 25 23 40 3. 2 4. Using Decimals to Compare Fractions Example 3 42 48 27 30 Compare and . 42 48 27 30 and Answer: Compare the decimals. 27 30 so Copyright © Holt McDougal. All rights reserved. Divide. > , 42 . 48 Chapter 5 • Pre-Algebra Notetaking Guide 85 Writing Fractions as Decimals Example 2 4 9 5 16 a. Write as a decimal. b. Write as a decimal. a. b. 0.3125 16.0 50 0 0 4 8 20 1 6 40 3 2 80 8 0 0 0.44. . . 9.0 40 36 40 3 6 Answer: Use a bar to show the repeating digit in the repeating decimal: 4 0.4 . 9 Answer: The remainder is 0 , so the decimal is a terminating decimal: 5 0.3125. 16 Checkpoint Write the fraction or mixed number as a decimal. 3 5 7 20 2. 1. 0.6 0.35 12 25 23 40 3. 2 4. 2.48 0.575 Using Decimals to Compare Fractions Example 3 42 48 27 30 Compare and . 42 0.875 48 27 30 and 0.9 Answer: Compare the decimals. Divide. 0.9 > 0.875 , 27 42 so > . 30 Copyright © Holt McDougal. All rights reserved. 48 Chapter 5 • Pre-Algebra Notetaking Guide 85 Example 4 Writing Terminating Decimals as Fractions place, 3 is in a. 0.03 so denominator is place, 4 is in b. 9.4 so denominator is Example 5 . . Simplify fraction. Writing a Repeating Decimal as a Fraction To write 0.8 1 as a fraction, let x 0.8 1 . 1. Because 0.8 1 has 2 repeating digits, multiply each side of 1 by x 0.8 , or . Then x . x 2. Subtract x from x. ( x 0.8 1 ) x 3. Solve for x and simplify. x x Answer: The decimal 0.8 1 is equivalent to the fraction 86 Chapter 5 • Pre-Algebra Notetaking Guide . Copyright © Holt McDougal. All rights reserved. Example 4 Writing Terminating Decimals as Fractions 3 is in hundredths’ place, 3 a. 0.03 100 so denominator is 100 . 4 b. 9.4 9 10 2 9 5 Example 5 4 is in tenths’ place, so denominator is 10 . Simplify fraction. Writing a Repeating Decimal as a Fraction To write 0.8 1 as a fraction, let x 0.8 1 . 1. Because 0.8 1 has 2 repeating digits, multiply each side of 1 . 1 by 102 , or 100 . Then 100 x 81.8 x 0.8 1 100 x 81.8 2. Subtract x from 100 x. ( x 0.8 1 ) 99 x 81 3. Solve for x and simplify. 99 81 x 99 99 x 9 11 Answer: The decimal 0.8 1 is equivalent to the fraction 9 . 11 86 Chapter 5 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Checkpoint Write the decimal as a fraction or mixed number. 5. 0.8 6. 3.75 7. 0.1 2 8. 5.3 Ordering Rational Numbers Example 6 17 11 13 Order the numbers , 1.35, 5.67, 6, , from least 2 3 4 to greatest. Graph the numbers on the number line. You may want to write improper fractions as mixed numbers. 8 6 4 2 0 2 4 6 8 Answer: Read the numbers graphed on the number line from left to right: Copyright © Holt McDougal. All rights reserved. , , , , , . Chapter 5 • Pre-Algebra Notetaking Guide 87 Checkpoint Write the decimal as a fraction or mixed number. 6. 3.75 5. 0.8 4 5 3 4 3 7. 0.1 2 8. 5.3 1 3 4 33 5 Ordering Rational Numbers Example 6 17 11 13 Order the numbers , 1.35, 5.67, 6, , from least 2 3 4 to greatest. Graph the numbers on the number line. You may want to write improper fractions as mixed numbers. 17 2 8 6 6 13 4 4 11 3 1.35 2 0 2 4 5.67 6 8 Answer: Read the numbers graphed on the number line from 17 13 11 left to right: 2 , 6 , 4 , 1.35 , 3 , 5.67 . Copyright © Holt McDougal. All rights reserved. Chapter 5 • Pre-Algebra Notetaking Guide 87 Focus On Reasoning Use after Lesson 5.1 Inductive and Deductive Reasoning Goal: Determine whether mathematical statements are true or false. Vocabulary Inductive reasoning Conjecture Deductive reasoning Example 1 Using Inductive Reasoning Consider fractions between 0 and 1 whose denominators are powers of 3, such as 3, 9, and 27. Make a conjecture about the decimal form of such fractions. Solution Find a pattern using a few examples. 1 0.3 3 1 9 1 27 1 0.0 1 2 3 4 5 6 7 9 81 Conjecture: The decimal form of any fraction whose denominator is a power of 3 is Checkpoint Make a conjecture. 1. Make a conjecture about the difference of any two even integers. 88 Chapter 5 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Focus On Reasoning Use after Lesson 5.1 Inductive and Deductive Reasoning Goal: Determine whether mathematical statements are true or false. Vocabulary Inductive reasoning Making a conclusion based on several examples Conjecture A statement that is thought to be true but not yet shown to be true Deductive The process of starting with one or more given facts reasoning and using rules, definitions, or properties to reach a conclusion Example 1 Using Inductive Reasoning Consider fractions between 0 and 1 whose denominators are powers of 3, such as 3, 9, and 27. Make a conjecture about the decimal form of such fractions. Solution Find a pattern using a few examples. 1 0.3 3 1 9 0.1 1 27 0.0 3 7 1 0.0 1 2 3 4 5 6 7 9 81 Conjecture: The decimal form of any fraction whose denominator is a power of 3 is a repeating decimal. Checkpoint Make a conjecture. 1. Make a conjecture about the difference of any two even integers. Sample answer: The difference of any two even integers is even. 88 Chapter 5 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Example 2 Using Deductive Reasoning Give a convincing argument to show that the conjecture in Example 1 is true. Solution Since the decimal system is based on powers of form of any fraction with a a , the decimal that is a is . If the of a fraction is a factors will only consist of combinations of , its prime s and/or other cases, the long division while converting the would result in a pattern of s. In all to a that continues endlessly. So, if there are prime factors in the than or other , then the decimal form will be a . Consider fractions between 0 and 1 whose denominators are powers of 3. The in the fractions will only include s. Therefore, the decimal form of each of these fractions will be a Example 3 of these . Using a Counterexample Show the conjecture is false by finding a counterexample. Conjecture: The decimal form of any fraction whose denominator is divisible by 10 is a terminating decimal. Solution Write fractions whose denominators are divisible by 10. 1 10 Because 1 0.05 20 is a 1 30 , the conjecture is . Checkpoint Show that the conjecture is false by finding a counterexample. 2. Conjecture: The quotient of any two numbers is less than either number. Copyright © Holt McDougal. All rights reserved. Chapter 5 • Pre-Algebra Notetaking Guide 89 Using Deductive Reasoning Example 2 Give a convincing argument to show that the conjecture in Example 1 is true. Solution Since the decimal system is based on powers of 10 , the decimal form of any fraction with a denominator that is a power of 10 is a terminating decimal . If the denominator of a fraction is a power of 10 , its prime factors will only consist of combinations of 2 s and/or 5 s. In all other cases, the long division while converting the fraction to a decimal would result in a pattern of remainders that continues endlessly. So, if there are prime factors in the denominator other than 2 or 5 , then the decimal form will be a repeating decimal . Consider fractions between 0 and 1 whose denominators are powers of 3. The prime factors in the denominators of these fractions will only include 3 s. Therefore, the decimal form of each of these fractions will be a repeating decimal . Using a Counterexample Example 3 Show the conjecture is false by finding a counterexample. Conjecture: The decimal form of any fraction whose denominator is divisible by 10 is a terminating decimal. Solution Write fractions whose denominators are divisible by 10. 1 0.1 10 1 0.05 2 0 20 1 0.03 30 Because 1 is a counterexample , the conjecture is false . 30 Checkpoint Show that the conjecture is false by finding a counterexample. 2. Conjecture: The quotient of any two numbers is less than either number. Sample answer: 15 5 3, and 3 > 15 Copyright © Holt McDougal. All rights reserved. Chapter 5 • Pre-Algebra Notetaking Guide 89 5.2 Adding and Subtracting Like Fractions Goal: Add and subtract like fractions. Adding and Subtracting Like Fractions Words To add or subtract fractions with the same denominator, write the sum or difference of the numerators over the denominator. 4 9 1 9 9 2 11 11 11 Numbers a c 9 b c Algebra , c 0 Example 1 c a b , c 0 c c c Adding Like Fractions A survey asked 100 students ages 7 to 11 what sports apparel they prefer to wear. The circle graph at the right summarizes their responses. What fraction of the students responded either major league baseball or NBA? Apparel Survey Other 9 NFL 30 Pro wrestling 17 NBA 21 Major league baseball 23 Solution To find the fraction of the students who responded either major league baseball or NBA, find the sum of . Write sum of numerators over denominator. and Add. Then simplify. Answer: The fraction of the students who prefer either major league baseball or NBA apparel is 90 Chapter 5 • Pre-Algebra Notetaking Guide . Copyright © Holt McDougal. All rights reserved. 5.2 Adding and Subtracting Like Fractions Goal: Add and subtract like fractions. Adding and Subtracting Like Fractions Words To add or subtract fractions with the same denominator, write the sum or difference of the numerators over the denominator. Numbers 7 9 2 11 11 11 ab a b Algebra , c 0 c c c ab a b , c 0 c c c 4 9 Example 1 1 9 5 9 Adding Like Fractions A survey asked 100 students ages 7 to 11 what sports apparel they prefer to wear. The circle graph at the right summarizes their responses. What fraction of the students responded either major league baseball or NBA? Apparel Survey Other 9 NFL 30 Pro wrestling 17 NBA 21 Major league baseball 23 Solution To find the fraction of the students who responded either major 23 21 and . league baseball or NBA, find the sum of 100 100 23 21 100 100 23 21 100 Write sum of numerators over denominator. 11 44 25 100 Add. Then simplify. Answer: The fraction of the students who prefer either major 11 league baseball or NBA apparel is 2 . 5 90 Chapter 5 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Subtracting Like Fractions Example 2 When you perform operations with negative fractions, be sure to assign a negative sign in front of a fraction to the numerator of the fraction. 4 9 Write difference of numerators over denominator. 3 9 a. 2 11 Subtract. 4 11 b. 4 11 To subtract , add Write sum of numerators over denominator. Add. . Checkpoint Find the sum or difference. 2 7 4 7 3 13 1. Adding and Subtracting Mixed Numbers Example 3 3 7 6 7 a. 4 3 Write mixed numbers as improper fractions. Write sum of numerators over denominator. 3 10 Add. Then write fraction as mixed number. 9 10 b. 11 8 Write mixed numbers as improper fractions. Write difference of numerators over denominator. Copyright © Holt McDougal. All rights reserved. 8 13 2. Subtract. Then write fraction as mixed number. Chapter 5 • Pre-Algebra Notetaking Guide 91 Subtracting Like Fractions Example 2 When you perform operations with negative fractions, be sure to assign a negative sign in front of a fraction to the numerator of the fraction. 4 3 a. 9 4 9 3 9 7 9 2 4 Write difference of numerators over denominator. Subtract. 2 4 b. 1 1 11 11 11 4 4 To subtract , add 1 . 1 11 24 11 Write sum of numerators over denominator. 6 1 1 Add. Checkpoint Find the sum or difference. 2 7 4 7 3 13 1. 6 7 5 13 Adding and Subtracting Mixed Numbers Example 3 3 6 31 27 a. 4 3 7 7 7 7 3 Write mixed numbers as improper fractions. 31 27 7 Write sum of numerators over denominator. 58 2 7 8 7 Add. Then write fraction as mixed number. 9 113 89 1 b. 11 8 1 0 0 10 10 Copyright © Holt McDougal. All rights reserved. 8 13 2. Write mixed numbers as improper fractions. 113 89 10 Write difference of numerators over denominator. 24 2 2 1 0 5 Subtract. Then write fraction as mixed number. Chapter 5 • Pre-Algebra Notetaking Guide 91 Simplifying Variable Expressions Example 4 4a 21 Write sum of numerators over denominator. 10a 21 a. Add. Simplify. 9 5b 4 5b b. 4 5b To subtract , add Write sum of numerators over denominator. Add. Simplify. . Checkpoint Find the sum or difference. 2 11 4 11 3. 3 5 2a 25 8a 25 5. 92 Chapter 5 • Pre-Algebra Notetaking Guide 5 13 6 13 4. 4 3 17 3c 5 3c 6. Copyright © Holt McDougal. All rights reserved. Simplifying Variable Expressions Example 4 4a 10a 4a 10a a. 21 21 21 Write sum of numerators over denominator. 14a 2 1 Add. a 2 Simplify. 3 9 9 4 4 b. 5b 5b 5b 5b 4 4 To subtract , add 5b . 5b 9 4 5b Write sum of numerators over denominator. 5 Add. 5b 1 b Simplify. Checkpoint Find the sum or difference. 2 11 4 11 5 13 3. 3 5 6 13 4. 4 3 11 13 6 11 7 8 2a 25 17 3c 8a 25 5. 2a 5 92 Chapter 5 • Pre-Algebra Notetaking Guide 5 3c 6. 4 c Copyright © Holt McDougal. All rights reserved. 5.3 Adding and Subtracting Unlike Fractions Goal: Add and subtract unlike fractions. Adding and Subtracting Fractions Example 1 7 15 1 5 1 5 7 15 a. 2 3 Write using LCD. Write sum of numerators over denominator. Add. Simplify. 3 4 b. Write fractions using LCD. Write difference of numerators over denominator. Subtract. Write fraction as a mixed number. Checkpoint Find the sum or difference. 3 7 5 21 1. Copyright © Holt McDougal. All rights reserved. 1 4 3 10 2. Chapter 5 • Pre-Algebra Notetaking Guide 93 5.3 Adding and Subtracting Unlike Fractions Goal: Add and subtract unlike fractions. Adding and Subtracting Fractions Example 1 1 7 7 3 a. 1 5 5 15 15 73 15 Write sum of numerators over denominator. 0 1 Add. 2 Simplify. 15 3 8 9 b. 12 1 2 3 4 2 1 5 Write using LCD. 3 Write fractions using LCD. 8 9 1 2 Write difference of numerators over denominator. 17 Subtract. 12 5 11 2 Write fraction as a mixed number. Checkpoint Find the sum or difference. 3 7 1 4 5 21 1. 2 3 Copyright © Holt McDougal. All rights reserved. 3 10 2. 1 20 Chapter 5 • Pre-Algebra Notetaking Guide 93 Adding Mixed Numbers Example 2 1 6 3 10 5 2 Write mixed numbers as improper fractions. Write fractions using LCD. Write sum of numerators over denominator. Add. Then write fraction as a mixed number. Subtracting Mixed Numbers Example 3 1 5 1 2 You are hiking a 12 -mile trail. You have already hiked 6 miles. How many more miles do you have to hike before reaching the end of the trail? Solution Your total hiking distance is . You have already hiked . To find the remaining distance, subtract. 1 5 1 2 12 6 Write mixed numbers as improper fractions. Write fractions using LCD. Write difference of numerators over denominator. Subtract. Then write fraction as a mixed number. Answer: You need to hike 94 Chapter 5 • Pre-Algebra Notetaking Guide miles. Copyright © Holt McDougal. All rights reserved. Adding Mixed Numbers Example 2 10 31 3 1 5 2 6 6 310 60 1203 138 60 Write mixed numbers as improper fractions. Write fractions using LCD. 310 (138) Write sum of numerators over denominator. 448 7 60 7 1 5 Add. Then write fraction as a mixed number. 60 Subtracting Mixed Numbers Example 3 1 5 1 2 You are hiking a 12 -mile trail. You have already hiked 6 miles. How many more miles do you have to hike before reaching the end of the trail? Solution 1 Your total hiking distance is 12 5 miles . You have already hiked 1 6 miles . To find the remaining distance, subtract. 2 61 1 1 13 12 6 5 2 5 2 122 65 1 0 10 122 65 10 57 7 5 1 0 10 Write mixed numbers as improper fractions. Write fractions using LCD. Write difference of numerators over denominator. Subtract. Then write fraction as a mixed number. Answer: You need to hike 5 7 miles. 10 94 Chapter 5 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Checkpoint Find the sum or difference. 5 6 4 9 1 3 3. 4 2 3 7 4. 2 3 Simplifying an Expression Example 4 a 4 a 8 Simplify the expression . a a a p 4 8 4 a 8 a a Write using LCD. 8 4 Multiply. Write difference of numerators over denominator. Subtract. Checkpoint Find the sum or difference. b 3 b 8 5. Copyright © Holt McDougal. All rights reserved. c 5 c 7 6. Chapter 5 • Pre-Algebra Notetaking Guide 95 Checkpoint Find the sum or difference. 5 6 4 9 1 3 3. 4 2 3 7 4. 2 3 16 21 5 18 5 7 Simplifying an Expression Example 4 a 4 a 8 Simplify the expression . 2 a a a p 2 4 8 4 a 8 a a Write using LCD. 8 4 2a 8 Multiply. 2a a Write difference of numerators over denominator. a Subtract. 8 8 Checkpoint Find the sum or difference. b 3 b 8 c 5 5. 11b 24 Copyright © Holt McDougal. All rights reserved. c 7 6. 2c 35 Chapter 5 • Pre-Algebra Notetaking Guide 95 5.4 Multiplying Fractions Goal: Multiply fractions and mixed numbers. Multiplying Fractions Words The product of two or more fractions is equal to the product of the numerators over the product of the denominators. 3 5 4 7 Numbers p a b c d , where b 0 and d 0 Algebra p Multiplying Fractions Example 1 5 3 5 p p 12 20 12 Assign negative sign to numerator. Use rule for multiplying fractions. Divide out common factors. Multiply. Checkpoint Find the product. 7 16 5 14 1. p 96 Chapter 5 • Pre-Algebra Notetaking Guide 2 15 5 18 2. p Copyright © Holt McDougal. All rights reserved. 5.4 Multiplying Fractions Goal: Multiply fractions and mixed numbers. Multiplying Fractions Words The product of two or more fractions is equal to the product of the numerators over the product of the denominators. 3 5 4 7 3p4 12 5p7 35 Numbers p a b ac c d , where b 0 and d 0 Algebra p Multiplying Fractions Example 1 bd 5 3 5 p p 12 20 12 203 5 p (3) 12 p 20 Assign negative sign to numerator. Use rule for multiplying fractions. 1 1 p ( 5 3) p2 12 0 4 Divide out common factors. 4 1 1 16 1 6 Multiply. Checkpoint Find the product. 7 16 5 14 2 15 1. p 5 32 96 Chapter 5 • Pre-Algebra Notetaking Guide 5 18 2. p 1 27 Copyright © Holt McDougal. All rights reserved. Multiplying a Mixed Number and an Integer Example 2 1 2 Water Use The showerhead in your home uses 2 gallons of water per minute. If you take a 7-minute shower, how many gallons of water do you use? Solution Number of Gallons per Gallons p minutes minute used p Substitute values. Write numbers as improper fractions. p Use rule for multiplying fractions. Multiply. Write fraction as a mixed number. Answer: You use Multiplying Mixed Numbers Example 3 1 5 1 6 3 p 4 Copyright © Holt McDougal. All rights reserved. gallons of water. p Write mixed numbers as improper fractions. Use rule for multiplying fractions. Divide out common factors. Multiply. Write fraction as a mixed number. Chapter 5 • Pre-Algebra Notetaking Guide 97 Multiplying a Mixed Number and an Integer Example 2 1 2 Water Use The showerhead in your home uses 2 gallons of water per minute. If you take a 7-minute shower, how many gallons of water do you use? Solution Number of Gallons per Gallons p minutes minute used 2 1 p 7 2 Substitute values. 5 p 7 Write numbers as improper fractions. 5p7 Use rule for multiplying fractions. 5 3 Multiply. 171 2 Write fraction as a mixed number. 2 1 2p1 2 Answer: You use 171 gallons of water. 2 Multiplying Mixed Numbers Example 3 1 1 16 25 3 p 4 5 p 6 5 6 8 5 1 6 p 25 p6 5 1 Use rule for multiplying fractions. Divide out common factors. 3 40 Multiply. 3 1 13 3 Copyright © Holt McDougal. All rights reserved. Write mixed numbers as improper fractions. Write fraction as a mixed number. Chapter 5 • Pre-Algebra Notetaking Guide 97 Checkpoint Find the product. 2 9 3 4 3. 5 p 6 Example 4 1 3 4. 2 p 5 Simplifying Expressions Simplify the expression. Use rule for multiplying fractions. Divide out common factor. m 10 a. p 4 7 n4 9n2 b. p 12 10 Multiply. Use rule for multiplying fractions. Divide out common factor. Product of powers property Add exponents. Checkpoint Simplify the expression. 2x 5 3x 2 8 5. p 98 Chapter 5 • Pre-Algebra Notetaking Guide 4y3 15 5y 6 16 6. p Copyright © Holt McDougal. All rights reserved. Checkpoint Find the product. 2 9 3 4 3. 5 p 6 1 3 2 3 14 31 Example 4 1 3 4. 2 p 5 Simplifying Expressions Simplify the expression. 5 m p () 10 m 10 a. p 4 7 p7 4 Use rule for multiplying fractions. Divide out common factor. 2 5m 5m 14 14 Multiply. 3 n4 p 9n2 n4 9n2 b. p 12 10 2 p 10 1 Use rule for multiplying fractions. Divide out common factor. 4 3n4 2 Product of powers property 3n6 Add exponents. 40 40 Checkpoint Simplify the expression. 2x 5 4y3 15 3x 2 8 3x 3 20 98 Chapter 5 • Pre-Algebra Notetaking Guide 5y 6 16 6. p 5. p y9 12 Copyright © Holt McDougal. All rights reserved. Focus On Measurement Converting Temperatures Use after Lesson 5.4 Goal: Convert temperatures between degrees Celsius and degrees Fahrenheit. Temperature Conversions Fahrenheit to Celsius The Celsius scale is also known as the centigrade scale because there are 100 degrees between the freezing point of water (0°C) and the boiling point of water (100°C). To convert degrees Fahrenheit to degrees Celsius, use the formula C . Celsius to Fahrenheit To convert degrees Celsius to degrees Fahrenheit, use the formula F . Converting a Temperature to Degrees Celsius Example 1 Cooking The internal temperature of a baked chicken is 167° F. Convert this temperature to degrees Celsius. Solution C 5 ( 9 5 ( 9 5 9 ( Write formula for degrees Celsius. ) ) Substitute ) for . . Use rule for multiplying fractions. Divide out common factor. 5 p (13 5) 9 p1 . Answer: In degrees Celsius, the internal temperature of the baked chicken is Copyright © Holt McDougal. All rights reserved. . Chapter 5 • Pre-Algebra Notetaking Guide 99 Focus On Measurement Converting Temperatures Use after Lesson 5.4 Goal: Convert temperatures between degrees Celsius and degrees Fahrenheit. Temperature Conversions Fahrenheit to Celsius The Celsius scale is also known as the centigrade scale because there are 100 degrees between the freezing point of water (0°C) and the boiling point of water (100°C). To convert degrees Fahrenheit to degrees Celsius, use the formula C 5 (F 32) . 9 Celsius to Fahrenheit To convert degrees Celsius to degrees Fahrenheit, use the formula 9 F C 32 . 5 Converting a Temperature to Degrees Celsius Example 1 Cooking The internal temperature of a baked chicken is 167° F. Convert this temperature to degrees Celsius. Solution C 5 ( F 32 ) 9 5 ( 167 32 ) 9 5 ( 135 ) 9 15 5 p (13 5) 1 9 p1 75 Write formula for degrees Celsius. Substitute 167 for F . Subtract . Use rule for multiplying fractions. Divide out common factor. Multiply . Answer: In degrees Celsius, the internal temperature of the baked chicken is 75°C . Copyright © Holt McDougal. All rights reserved. Chapter 5 • Pre-Algebra Notetaking Guide 99 Checkpoint Convert the temperature to degrees Celsius. 1. 95° F 2. 5° F Converting a Temperature to Degrees Fahrenheit Example 2 Walking Mrs. Hochstetler likes to go walking when the temperature is around 25°C. Convert this temperature to degrees Fahrenheit. Solution F 9 5 9 5 ( Write formula for degrees Fahrenheit. ) 9 p 25 5 p1 Substitute for . Use rule for multiplying fractions. Divide out common factor. Answer: In degrees Fahrenheit, Mrs. Hochstetler likes to go walking at . Checkpoint Convert the temperature to degrees Fahrenheit. 3. 68° C 100 Chapter 5 • Pre-Algebra Notetaking Guide 4. 37° C Copyright © Holt McDougal. All rights reserved. Checkpoint Convert the temperature to degrees Celsius. 1. 95° F 2. 5° F 35° C 15° C Converting a Temperature to Degrees Fahrenheit Example 2 Walking Mrs. Hochstetler likes to go walking when the temperature is around 25°C. Convert this temperature to degrees Fahrenheit. Solution F 9 C 32 5 9 Write formula for degrees Fahrenheit. ( 25 ) 32 5 Substitute 25 for C . 9 p 25 32 5p Use rule for multiplying fractions. Divide out common factor. 45 32 Multiply. 77 Add. 5 1 1 Answer: In degrees Fahrenheit, Mrs. Hochstetler likes to go walking at 77° F . Checkpoint Convert the temperature to degrees Fahrenheit. 3. 68° C 4. 37° C 154.4° F 100 Chapter 5 • Pre-Algebra Notetaking Guide 98.6° F Copyright © Holt McDougal. All rights reserved. 5.5 Dividing Fractions Goal: Divide fractions and mixed numbers. Vocabulary Reciprocals: Using Reciprocals to Divide Words To divide by any nonzero number, multiply by its reciprocal. 2 9 3 7 2 9 Numbers p a b c d a b Algebra p , where b 0, c 0, and d 0 Dividing a Fraction by a Fraction Example 1 3 7 6 11 p Use rule for multiplying fractions. Divide out common factor. Multiply by reciprocal. Multiply. Check: To check, multiply the quotient by the divisor to see if you get the dividend: 6 11 p Copyright © Holt McDougal. All rights reserved. . Chapter 5 • Pre-Algebra Notetaking Guide 101 5.5 Dividing Fractions Goal: Divide fractions and mixed numbers. Vocabulary Reciprocals: Two nonzero numbers whose product is 1 are reciprocals. Using Reciprocals to Divide Words To divide by any nonzero number, multiply by its reciprocal. 2 3 2 14 7 Numbers p 3 2 7 9 7 9 a a c d ad Algebra p c b , where b 0, c 0, and d 0 c b b d Example 1 Dividing a Fraction by a Fraction 3 6 3 11 7 p 6 7 11 Multiply by reciprocal. 1 3 p 11 7p6 Use rule for multiplying fractions. Divide out common factor. 2 11 11 14 14 Multiply. Check: To check, multiply the quotient by the divisor to see if you get the dividend: 6 11 3 p 14 7 11 Copyright © Holt McDougal. All rights reserved. Solution checks . Chapter 5 • Pre-Algebra Notetaking Guide 101 Dividing a Mixed Number by a Mixed Number Example 2 1 2 3 4 2 3 Write mixed numbers as improper fractions. p Multiply by reciprocal. Use rule for multiplying fractions. Divide out common factors. Multiply. Checkpoint Find the quotient. 8 21 9 14 5 24 2. 3 5 4. 3 5 7 10 3. 4 1 102 5 12 1. Chapter 5 • Pre-Algebra Notetaking Guide 1 4 1 2 Copyright © Holt McDougal. All rights reserved. Dividing a Mixed Number by a Mixed Number Example 2 1 3 145 5 2 3 2 2 4 145 5 2 p 1 2 5 p ( ) 4 p 15 2 1 Write mixed numbers as improper fractions. Multiply by reciprocal. Use rule for multiplying fractions. Divide out common factors. 3 2 2 3 3 Multiply. Checkpoint Find the quotient. 8 21 9 14 5 12 1. 5 24 2. 16 27 3 5 1 4 7 10 3. 4 1 12 17 Chapter 5 • Pre-Algebra Notetaking Guide 1 2 4. 3 5 2 102 2 13 22 Copyright © Holt McDougal. All rights reserved. Example 3 Dividing a Whole Number by a Mixed Number 1 5 Dogs You have two dogs that eat about 1 pounds of dog food per day. How many whole days will a 5-pound bag of dog food last? Solution Divide to find how long the bag of dog food will last. Number of pounds Number of Number eaten per day pounds in bag of days Substitute values. Write numbers as improper fractions. p Multiply by reciprocal. Use rule for multiplying fractions. Multiply. Write fraction as a mixed number. Answer: A 5-pound bag of dog food will last Copyright © Holt McDougal. All rights reserved. . Chapter 5 • Pre-Algebra Notetaking Guide 103 Example 3 Dividing a Whole Number by a Mixed Number 1 5 Dogs You have two dogs that eat about 1 pounds of dog food per day. How many whole days will a 5-pound bag of dog food last? Solution Divide to find how long the bag of dog food will last. Number of pounds Number of Number eaten per day pounds in bag of days 5 1 1 5 Substitute values. 5 6 Write numbers as improper fractions. 5 p 5 1 Multiply by reciprocal. 5p5 Use rule for multiplying fractions. 5 2 Multiply. 4 1 6 Write fraction as a mixed number. 1 5 6 1p6 6 Answer: A 5-pound bag of dog food will last 4 whole days . Copyright © Holt McDougal. All rights reserved. Chapter 5 • Pre-Algebra Notetaking Guide 103 5.6 Using Multiplicative Inverses to Solve Equations Goal: Use multiplicative inverses to solve equations. Vocabulary Multiplicative inverse: Multiplicative Inverse Property Words The product of a number and its multiplicative inverse is 1. 3 5 5 3 Numbers p 1 a b b a Algebra p 1, where a 0, b 0 Example 1 Solving a One-Step Equation 3 x 15 5 Original equation Multiply each side by multiplicative 3 x 5 (15) x (15) x inverse of . Multiplicative inverse property Multiply. Answer: The solution is . Checkpoint Solve the equation. Check your solution. 6 11 1. x 18 104 Chapter 5 • Pre-Algebra Notetaking Guide 7 13 2. x 28 Copyright © Holt McDougal. All rights reserved. 5.6 Using Multiplicative Inverses to Solve Equations Goal: Use multiplicative inverses to solve equations. Vocabulary Multiplicative inverse: The multiplicative inverse of a nonzero number is the number’s reciprocal. Multiplicative Inverse Property Words The product of a number and its multiplicative inverse is 1. 3 5 5 3 Numbers p 1 a b b a Algebra p 1, where a 0, b 0 Example 1 Solving a One-Step Equation 3 x 15 5 5 3 3 x 5 (15) 5 3 Original equation Multiply each side by multiplicative inverse of 3 . 5 1 x 5 (15) 3 x 25 Multiplicative inverse property Multiply. Answer: The solution is 25 . Checkpoint Solve the equation. Check your solution. 6 11 7 13 1. x 18 2. x 28 33 104 Chapter 5 • Pre-Algebra Notetaking Guide 52 Copyright © Holt McDougal. All rights reserved. Solving a Two-Step Equation Example 2 3 4 7 12 1 2 Original equation 1 2 Subtract x 7 12 3 4 x 7 12 x Write fractions using LCD. 7 12 x 7 12 Subtract. x x Example 3 from each side. Multiply each side by multiplicative inverse of . Multiply. Writing and Solving a Two-Step Equation Tree Growth The height of a certain Norway Spruce is 10 feet. 1 2 If the tree’s height grows 2 feet per year, find how long it will take the tree to reach a height of 25 feet. Solution Number of New Growth Current p years height rate height 1 2 10 2 x 25 Write equation. 1 2 Subtract 10 2 x 25 Simplify. Write mixed number as improper fraction. x x x ( ) Multiply each side by multiplicative inverse of . Multiply. Answer: The tree will be 25 feet tall after Copyright © Holt McDougal. All rights reserved. from each side. years. Chapter 5 • Pre-Algebra Notetaking Guide 105 Example 2 Solving a Two-Step Equation 3 4 7 12 1 2 x Original equation 7 3 1 3 3 x 4 4 12 4 3 Subtract 4 from each side. 2 7 3 2 x 4 4 12 7 Write fractions using LCD. 1 x 4 12 12 7 12 x 7 7 12 Subtract. 1 4 x 3 7 Example 3 Multiply each side by 7 multiplicative inverse of 1 . 2 Multiply. Writing and Solving a Two-Step Equation Tree Growth The height of a certain Norway Spruce is 10 feet. 1 2 If the tree’s height grows 2 feet per year, find how long it will take the tree to reach a height of 25 feet. Solution Number of New Growth Current p years height rate height 1 2 10 2 x 25 Write equation. 1 2 Subtract 10 from each side. 10 2 x 10 25 10 Simplify. Write mixed number as improper fraction. 5 x 15 2 2 5 5 2 2 x 5 x 6 ( 15 ) Multiply each side by 5 multiplicative inverse of 2 . Multiply. Answer: The tree will be 25 feet tall after 6 years. Copyright © Holt McDougal. All rights reserved. Chapter 5 • Pre-Algebra Notetaking Guide 105 5.7 Equations and Inequalities with Rational Numbers Goal: Use the LCD to solve equations and inequalities. Solving an Equation by Clearing Fractions Example 1 1 2 3 x 4 5 10 Original equation 14 x 130 25 Multiply each side by LCD of fractions. 25 Use distributive property. Simplify. Subtract each side. Simplify. from Divide each side by . x Simplify. Checkpoint Solve the equation by first clearing the fractions. 1 3 5 6 7 9 1. x 106 Chapter 5 • Pre-Algebra Notetaking Guide 3 10 7 15 2 3 2. x Copyright © Holt McDougal. All rights reserved. 5.7 Equations and Inequalities with Rational Numbers Goal: Use the LCD to solve equations and inequalities. Solving an Equation by Clearing Fractions Example 1 1 2 3 x 4 5 10 Original equation Multiply each side by LCD of fractions. 25 Use distributive property. 1 2 3 20 x 20 4 5 10 20 1 x 4 20 3 10 20 5x 6 8 Simplify. 5x 6 6 8 6 5x 2 5x 2 5 5 Subtract 6 from each side. Simplify. Divide each side by 5 . 2 x 5 Simplify. Checkpoint Solve the equation by first clearing the fractions. 1 3 5 6 7 9 3 10 1. x 1 6 106 Chapter 5 • Pre-Algebra Notetaking Guide 7 15 2 3 2. x 11 14 Copyright © Holt McDougal. All rights reserved. Example 2 Solving an Equation by Clearing Decimals Solve the equation 2.75 6.15 0.4m. Because the greatest number of decimal places in any of the terms with decimals is , multiply each side of the equation by , or . 2.75 6.15 0.4m (2.75) (6.15 0.4m) m Copyright © Holt McDougal. All rights reserved. Original equation Multiply each side by . Use distributive property. Simplify. Subtract each side. from Simplify. Divide each side by . Simplify. Chapter 5 • Pre-Algebra Notetaking Guide 107 Example 2 Solving an Equation by Clearing Decimals Solve the equation 2.75 6.15 0.4m. Because the greatest number of decimal places in any of the terms with decimals is 2 , multiply each side of the equation by 10 2 , or 100 . 2.75 6.15 0.4m 100 (2.75) 100 (6.15 0.4m) Use distributive property. Simplify. 275 615 40m 275 615 615 40m 615 340 40m 340 40m 40 40 8.5 m Copyright © Holt McDougal. All rights reserved. Original equation Multiply each side by 100 . Subtract 615 from each side. Simplify. Divide each side by 40 . Simplify. Chapter 5 • Pre-Algebra Notetaking Guide 107 Solving an Inequality with Fractions Example 3 Geometry Describe the possible values of x if the area of the rectangle is at least 24 square inches. 6 2 x 5 Solution Length p Width p ≥ Area ≥ Substitute. ≥ Use distributive property. ≥ Subtract ≥ ≥ ( ) Answer: The possible values of x are Chapter 5 • Pre-Algebra Notetaking Guide from each side. Simplify. ≥ 108 2 Multiply each side by multiplicative inverse of . Simplify. . Copyright © Holt McDougal. All rights reserved. Solving an Inequality with Fractions Example 3 Geometry Describe the possible values of x if the area of the rectangle is at least 24 square inches. 6 2 x 5 Solution Length p Width 25 x 2 12 x 5 12 x 5 ≥ Area p 6 ≥ 24 Substitute. 12 ≥ 24 Use distributive property. 12 12 ≥ 24 12 12 x ≥ 12 5 5 12 12 x 5 ≥ 5 12 x ≥ 5 2 Subtract 12 from each side. Simplify. ( 12 ) Multiply each side by 12 multiplicative inverse of 5 . Simplify. Answer: The possible values of x are 5 or more . 108 Chapter 5 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. Example 4 Solving an Inequality by Clearing Fractions 1 6 5 6 5 12 m ≤ 16 m 152 ≤ ≤ ≤ ≤ Original inequality 56 Multiply each side by LCD of fractions. Use distributive property. 5 6 Simplify. Add ≤ to each side. Simplify. Divide each side by . the inequality symbol. Simplify. m Checkpoint Solve the inequality by first clearing the fractions. 4 11 2 3 3. x 1 < Copyright © Holt McDougal. All rights reserved. 3 7 1 4 1 2 4. x < Chapter 5 • Pre-Algebra Notetaking Guide 109 Solving an Inequality by Clearing Fractions Example 4 1 6 5 6 5 12 m ≤ Original inequality Multiply each side by LCD of fractions. Use distributive property. 1 5 5 12 m ≤ 12 6 6 12 12 1 6 m 12 5 12 5 6 ≤ 12 2m 5 ≤ 10 Simplify. 2m 5 5 ≤ 10 5 Add 5 to each side. 2m ≤ 5 Simplify. Divide each side by 2 . Reverse the inequality symbol. 2m 5 ≥ 2 2 m ≥ 5 2 Simplify. Checkpoint Solve the inequality by first clearing the fractions. 4 11 2 3 3 7 3. x 1 < 11 12 x < Copyright © Holt McDougal. All rights reserved. 1 4 1 2 4. x < 7 12 x < Chapter 5 • Pre-Algebra Notetaking Guide 109 5 Words to Review Give an example of the vocabulary word. Rational number Terminating decimal Repeating decimal Inductive reasoning Deductive reasoning Counterexample Reciprocal Multiplicative inverse Review your notes and Chapter 5 by using the Chapter Review on pages 264–267 of your textbook. 110 Chapter 5 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved. 5 Words to Review Give an example of the vocabulary word. Rational number Terminating decimal 3 4 3 0.75 4 Repeating decimal Inductive reasoning 1 0.3 3 Examples: 2 4 6, 4 6 10, 6 8 14 Conjecture: The sum of two even integers is even. Deductive reasoning Counterexample Two even integers can be represented by 2a and 2b, where a and b are any integers. The sum 2a + 2b can be written as 2(a + b). This sum is even because it is a multiple of 2. Therefore, the sum of two integers is even. Reciprocal Conjecture: An integer multiplied by a negative integer is negative. Counterexample: 0 • –4 = 0 Multiplicative inverse 3 4 4 3 The reciprocal of is . 3 4 p 1 4 3 Review your notes and Chapter 5 by using the Chapter Review on pages 264–267 of your textbook. 110 Chapter 5 • Pre-Algebra Notetaking Guide Copyright © Holt McDougal. All rights reserved.