Summer 201 Modules 9-1 Mastering the Fundamentals Chris Millett
Copyright 201 All rights reserved. Written permission must be secured from the author to use or reproduce any part of this book. Academic Excellence in Mathematics is a registered trademark of Chris Millett. 2
Table of Content Module 9: Principles of Fractions... 7 Section 9.1 Introduction to Fractions... 9 Introduction to Fractions... 9 Types of Fractions... 9 Simplest-Form Fraction... 9 Equivalent Fractions... 10 Reciprocal... 10 Least Common Denominator (LCD)... 10 Reducing (Simplifying) Fractions... 11 Unreducing (Unsimplifying) Fractions... 11 Converting Improper Fractions to Mixed Fractions... 11 Converting Mixed Fractions to Improper Fractions... 11 Comparing Fractions... 12 Converting Fractions to have a Common Denominator... 12 Introduction to Fractions Guided Practice... 1 Section 9.2 Fraction Arithmetic... 15 Adding and Subtracting Fractions with Like Denominators... 15 Adding and Subtracting Fractions with Unlike Denominators... 15 Adding Mixed Fractions... 16 Subtracting Mixed Fractions... 17 Multiplying Fractions... 18 Multiplying Mixed Fractions... 18 Dividing Fractions by Fractions... 18 Dividing Fractions by Whole Numbers... 19 Dividing Whole Numbers by Fractions... 19 Dividing Mixed Fractions... 20 Fraction Arithmetic Guided Practice... 21 Section 9. Advanced Fraction Principles... 22 Fractions (Tenths, Hundredths, Thousandths, and Beyond)... 22 Introduction to Complex Fractions... 2 Solving Complex Fractions... 2 Introduction to Word Problems with Fractions... 2 Solving Word Problems with Fractions... 24 Advanced Fraction Principles Guided Practice... 25 Module 9 (Principle of Fractions) Review Exercises... 29 Fraction Fundamentals... 29 Fraction Arithmetic... 1 Advanced Fraction Principles... 7 Module 10: Principles of Decimals... 41 Section 10.1 Introduction to Decimals... 4 Decimal... 4 Decimal Place Value... 4 Comparing Decimals... 44 Ordering Decimals... 44 Introduction to Decimals Guided Reinforcement... 45 Section 10.2 Rounding Decimal Numbers... 46 What is Decimal Rounding?... 46 Principles of Rounding Decimal Numbers... 46 Examples of Rounding Decimal Numbers... 47 Rounding Decimal Numbers Guided Practice... 48 Section 10. Decimal Arithmetic... 49 Common Arithmetic Operations Involving Decimals... 49 Decimal Addition... 49 Decimal Subtraction... 50
Decimal Addition and Subtraction Guided Practice... 51 Introduction to Decimal Multiplication... 52 Multiplying a Number with a Decimal by a Number without a Decimal... 52 Multiplying a Number with a Decimal by another Number with a Decimal... 5 Introduction to Decimal Division... 54 Dividing Numbers without a Decimal Point (Resulting in an Answer with a Decimal Point)... 54 Dividing a Number without a Decimal Point into a Number with a Decimal Point... 55 Dividing a Number with a Decimal Point into a Number without a Decimal Point... 56 Dividing a Number with a Decimal Point into another Number with a Decimal Point... 57 Module 10 (Principle of Decimals) Review Exercises... 59 Decimal Fundamentals... 59 Decimal Arithmetic... 61 Supplemental Decimal Arithmetic... 6 Module 11: Principles of Percentages... 67 Section 11.1 Introduction to Percentages... 69 Percentage (Definition & Examples)... 69 Converting from Percentage to Fraction... 69 Converting from Fraction to Percentage... 69 Converting from Decimal to Percentage... 69 Converting from Percentage to Decimal... 69 Section 11.2 Calculating Percentages... 70 What is a Certain Percent of a Certain Value?... 70 What Percent is a Value of another Certain Value?... 70 What Value is a Certain Percent of a Certain Value?... 70 What is a Certain Percent Increase of a Value?... 71 What is a Certain Percent Decrease of a Value?... 71 What is a Certain Percent More than a Value?... 72 What is a Certain Percent Less than a Value?... 72 Section 11. Percentage Equivalents... 7 Important Fraction, Decimal, Percentage Equivalents... 7 Section 11.4 Word Problems with a Percent... 74 Solving Word Problems with Containing Percentages... 74 Percentage Guided Practice... 75 Module 11 (Principle of Percentages) Review Exercises... 77 Percentage Fundamentals... 77 Percentage Calculations... 79 Module 12: Advanced Arithmetic Operations... 8 Section 12.1 Ratios... 85 Ratio (Definition)... 85 Ratio (Expressions)... 85 Ratio Examples... 85 Ratio Guided Practice... 86 Section 12.2 Proportion... 87 Proportion (Definition & Example)... 87 Proportion Guided Practice... 88 Section 12. Other Advanced Arithmetic Operations... 89 Principles of Exponents... 89 Square Root... 90 Order of Operations... 92 Arithmetic Word Problem Fundamentals... 92 Sequence... 9 Set... 94 Advanced Arithmetic Operations Guided Practice... 95 Section 12.4 Advanced Counting Principles... 97 Multiplication Principle of Counting... 97 Advanced Counting Principles... 97 Combination... 98 Permutation... 99 4
Advanced Counting Principles Guided Practice... 100 Module 12 (Advanced Arithmetic Operations) Review Exercises... 101 Ratio... 101 Proportion... 102 Advanced Arithmetic Operations... 10 Advanced Counting Principles... 110 Module 1: Principles of Roots and Radicals... 11 Section 1.1 Introduction to Radicals... 115 What is a Radical?... 115 Radical Terminology... 115 Learn the Following to Master Radicals... 115 Section 1.2 Rules of Radicals... 116 Product Rule of Radicals... 116 Quotient Rule of Radicals... 116 Section 1. Radical Manipulation... 117 Simplifying Radicals... 117 Radicals and Rational Exponents... 117 Positive versus Negative Rational Exponents... 117 Radical Manipulation Guided Practice... 118 Section 1.4 Radical Arithmetic... 120 Adding Radicals... 120 Subtracting Radicals... 120 Multiplying Radicals... 121 Special Rule for Multiplying Radicals... 122 Multiplying Radicals with Additional or Subtraction... 122 Radical Arithmetic Guided Practice... 12 Dividing Radicals... 125 Conjugate Pair... 125 Multiplying Conjugates... 125 Rationalizing the Denominator... 125 Dividing Radicals Guided Practice... 126 5
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Module 9: Principles of Fractions 7
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Section 9.1 Introduction to Fractions Introduction to Fractions A comparison of the part (numerator) to the whole (denominator) Three pieces out of eight equally cut pieces of pie 8 Five free throws made out of six free throws attempted 6 5 Four complete pizzas and seven pieces out of ten equally cut pieces of pizza 4 10 7 Types of Fractions Proper The numerator is smaller than the denominator 7 is a Proper Fraction 8 Improper The numerator is larger than the denominator 8 is a Proper Fraction 7 Mixed Contains both a whole number part and a fractional part 1 8 7 is a Mixed Fraction Simplest-Form Fraction A fraction where the greatest common factor of the numerator and the denominator is 1 the greatest common factor of and 8 is 1 (this is a simplest-form fraction) 8 6 the greatest common factor of 6 and 8 is 2 (this is not a simplest-form fraction) 8 9
Section 9.1 Introduction to Fractions (continued) Equivalent Fractions Fractions with different numerators and denominators that represent the same value The value 2 1 can be represented by the fractions 4 2 and 6 Therefore, the fractions 2 1, 4 2, and 6 are Equivalent Fractions Equivalent fractions will eventually reduce to the same Simplest-Form Fraction 60 5 is an equivalent fraction to 72 42 5 Both fractions eventually reduce to 6 Reciprocal Switching the numerator and denominator of a fraction Reciprocal of 9 4 4 9 Reciprocal of 8 (or 1 8 ) 8 1 4 18 7 Reciprocal of 2 (or ) 7 7 18 Least Common Denominator (LCD) This is an extremely important concept when working with multiple fractions The LCD is the least common multiple (LCM) of the denominators of all fractions involved Example 1: what is the LCD of 2, 4 1, and 6 5 You must determine the LCM of denominators, 4, and 6 The smallest number that is a multiple of, 4, and 6 is 12 Example 2: what is the LCD of 4, 5 4, 8 5 and 10 7 You must determine the LCM of denominators 4, 5, 8, and 10 The smallest number that is a multiple of 4, 5, 8, and 10 is 40 10
Section 9.1 Introduction to Fractions (continued) Reducing (Simplifying) Fractions Determine the greatest common (GCF) factor of the numerator and denominator Divide the numerator and denominator that GCF (making the resulting GCF 1) 45 Example: Reduce 60 Determine GCF of 45 and 60 15 Divide numerator (45) and denominator (60) by the GCF (15) 45 15 60 15 4 The GCF of and 4 is 1 (therefore, the fraction has been fully reduced) Unreducing (Unsimplifying) Fractions Multiply both the numerator and denominator by the a common value Example: Unreduce 4 Determine a common value by which you multiply the numerator and denominator (i.e. 15) x15 45 4x15 60 Converting Improper Fractions to Mixed Fractions Divide the denominator into the numerator becomes the whole number part Place the remainder over the denominator becomes the fractional part Put together the whole part and the fractional part 11 Example: Convert to a mixed fraction 4 Divide the denominator into the numerator 11 4 = 2 (whole part with remainder ) Place the remainder over the denominator 4 (fractional part) Put together the whole part and the fractional part 2 4 Converting Mixed Fractions to Improper Fractions Multiply the integer by the denominator of the fraction Add this product to the numerator of the fraction Place this sum over the original denominator Example: Convert 2 4 to an improper fraction Multiply the integer by the denominator of the fraction 2 x 4 = 8 Add this product to the numerator of the fraction 8 + 11 11 Place this sum over the original denominator 4 11
Section 9.1 Introduction to Fractions (continued) Comparing Fractions Multiply the numerator of the first fraction by the denominator of the second fraction and the numerator of the second fraction by the denominator of the first fraction Compare the two multiplied values The first multiplied value corresponds to the first fraction The second multiplied value corresponds to the second fraction 7 1 Example: Compare and 8 15 Multiply 7 and 15 (corresponds to first fraction) 105 Multiply 1 and 8 (corresponds to second fraction) 104 Since the first product (7 x 15) is larger than the second product (1 x 8), the first fraction ( 8 7 ) is 1 larger than the second fraction ( ) 15 Converting Fractions to have a Common Denominator This is an extremely important concept when preparing to add or subtract fractions with different denominators Steps to converting fractions to have a common denominator First, determine the least common denominator (LCD) Then determine for each fraction what number to multiply the current denominator to reach the LCD For each fraction, multiply both the numerator and denominator by the value that will make the fraction contain the LCD, making an equivalent fraction 2 1 5 Example: Convert the following fractions to have a common denominator,, and 4 6 Determine the LCD for the fractions 12 Determine the number to multiply each fraction s denominator s to reach the LCD 2 o multiply the denominator () by 4 to become 12 1 o multiply the denominator (4) by to become 12 4 5 o multiply the denominator (6) by 2 to become 12 6 For each fraction, multiply both the numerator and denominator by the value that will make the fraction contain the LCD 2 2 4 8 o multiply the numerator and denominator by 4: x = 4 12 1 1 o multiply the numerator and denominator by : x = 4 4 12 5 5 2 10 o multiply the numerator and denominator by 2: x = 6 6 2 12 12
Write the following as a fraction (questions 1 ) Introduction to Fractions Guided Practice 1. 7 points received out of 10 possible points 2. 11 pieces of pie eaten out of 15 pieces of pie cut. 17 dogs out of 2 total animals Write yes or no to indicate if the specified is a Simplest-Form Fraction (question 4 7) 4. 5 4 12 5. 21 15 6. 26 17 7. 51 Reduce the following fractions to Simplest-Form (question 8 11) 12 8. 20 16 9. 24 20 10. 5 60 11. 72 Create an Equivalent Fraction with the specified numerator or denominator (question 12 17) 12. 7 4 (numerator 20) 1. 4 (denominator 16) 14. 5 2 (numerator 12) 15. 8 7 (denominator 64) 16. 6 5 (numerator 5) 17. 9 7 (denominator 72) Convert the following fractions (Improper to Mixed or Mixed to Improper) (question 18 17) 5 18. 6 19. 4 5 20. 8 5 21. 8 7 5 1
Introduction to Fractions Guided Practice (continued) What is the reciprocal of each fraction, specified as proper or improper? (question 22 25) 18 22. 7 2. 1 4 12 24. 2 25. 5 8 7 What is the least common denominator (LCD) of each set of fractions? (question 26 28) 26. 5 2, 6 5, 15 7 27. 8, 6 1, 12 7 28. 15 7, 9 4, 5 Convert the following fractions to have a least common denominator (question 29 1) 29. 5 2, 6 5, 15 7 0. 8, 6 1, 12 7 1. 15 7, 9 4, 5 14
Section 9.2 Fraction Arithmetic Adding and Subtracting Fractions with Like Denominators Just add or subtract the numerators Keep the same denominator Reduce the final answer if possible 5 1 5 1 6 2 Example: + = = = 9 9 9 9 5 1 5 1 4 Example: = = 9 9 9 9 Adding and Subtracting Fractions with Unlike Denominators Determine the Least Common Denominator (LCD) for the two fractions Now unreduced each fraction to have that LCD Since each fraction now has the same denominator, just added the numerators Keep the same denominator Reduce the final answer if possible Example 9 4 + 6 1 Least Common Denominator (LCD) = 18 Unreduce 9 4 to have a denominator of 18 (LCD) fraction must be multiplied by 2 2 Unreduce 6 1 to have a denominator of 18 (LCD) fraction must be multiplied by 4 2 8 x = 9 2 18 4 1 Example 9 6 & 1 8 11 x = + (already reduced) 6 18 18 18 18 Least Common Denominator (LCD) = 18 Unreduce 9 4 to have a denominator of 18 (LCD) fraction must be multiplied by 2 2 Unreduce 6 1 to have a denominator of 18 (LCD) fraction must be multiplied by 4 2 8 x = 9 2 18 & 1 8 5 x = (already reduced) 6 18 18 18 18 15
Section 9.2 Fraction Arithmetic (continued) Adding Mixed Fractions Add the whole parts to one another and the fractional parts to one another If the fractions have the same denominators, use the steps for Adding Fractions with Like Denominators If the fractions have different denominators, use the steps for Adding Fractions with Unlike Denominators If the sum of the fractional parts exceeds 1, initially specify that sum as an improper fraction Convert the sum s improper fraction to the corresponding mixed fraction Add this mixed fraction to the sum of the whole parts Reduce the final answer if possible 5 1 5 1 5 1 6 2 2 Example 1: 4 + = 4 + + + = 7 + = 7 + = 7 + = 7 9 9 9 9 9 9 4 5 4 5 4 5 Example 2: 6 + 2 = 6 + 2 + + = 8 + 7 7 7 7 7 = 8 + 7 9 = 8 + 1 7 2 = 9 7 2 4 4 4 4 4 4 Example : 5 + 6 + 2 = 5 + 6 + 2 + + + = 1 + 5 5 5 5 5 5 5 11 1 1 = 1 + = 1 + 2 = 15 5 5 5 1 1 9 4 9 4 Example 4: + 2 = + 2 + + = 5 + + = 5 + 8 6 8 6 24 24 24 1 = 5 24 7 5 7 5 21 20 21 Example 5: 4 + 5 = 4 + 5 + + = 9 + + = 9 + 8 6 8 6 24 24 2420 41 17 17 = 5 = 5 + 1 = 6 24 24 24 4 5 7 4 5 7 2 25 28 85 Example 6: 2 + + 6 = 2 + + 6 + + + = 11 + + + = 11 + = 5 8 10 5 8 10 40 40 40 40 5 5 11 + 2 = 1 40 40 = 1 8 1 16
Section 9.2 Fraction Arithmetic (continued) Subtracting Mixed Fractions Subtract the whole parts from one another and the fractional parts from one another If the fractions have the same denominators, use the steps for Subtracting Fractions with Like Denominators If the fractions have different denominators, use the steps for Subtracting Fractions with Unlike Denominators If the fraction to the right of the minus sign is great than the fraction to the left of the minus sign, you will have to borrow from the whole number to the left of the minus sign Subtract 1 from the fraction to the left of the minus sign Add the denominator value on the fraction to the left of the minus sign to the numerator value to the fraction to the left of the minus sign (creating an improper fraction) Subtract the whole number to the right of the minus sign from the whole number to the left of the minus sign Subtract the fraction part of the number to the right of the minus sign from the improper fraction to the left of the minus sign Reduce the final answer if possible 5 1 5 1 5 1 4 2 2 Example 1: 6 2 = 6 2 + ( ) = 4 + = 4 + = 4 + = 4 6 6 6 6 6 6 1 5 7 5 7 5 7 5 Example 2: 6 2 = 5 2 = 5 2 + ( ) = + 6 6 6 6 6 6 6 = + 6 2 = 4 + 1 = 1 1 9 4 9 4 9 Example : 8 5 = 8 5 = 8 5 + ( ) = + 8 6 24 24 24 24 244 5 5 = + = 24 24 1 4 9 28 9 28 9 28 9 Example 4: 8 5 = 8 5 = 7 5 = 7 5 + ( ) = 2 + 6 8 24 24 24 24 24 24 24 19 2 24 19 = 2 + = 24 17
Section 9.2 Fraction Arithmetic (continued) Multiplying Fractions Reduce fractions if possible You can combine the numerator of one fraction with the denominator of another Continue the process until no further reduction is possible Multiply the numerators Multiply the denominators If you fully reduce all fractions before multiplying, the final answer will already be reduced 4 5 4 5 1 1 1 1 1 1 1 1 1 Example: x x x ( x ) x ( x ) ( x ) x ( x ) x x x ) 9 8 6 5 9 8 6 5 2 2 1 2 2 1 12 4 5 180 1 If you multiply first: x x x = (reducing takes much more effort) 9 8 6 5 2160 12 For multiplying fractions, remember this phrase Simplify before you multiply Multiplying Mixed Fractions Convert each mixed fraction to the corresponding improper fraction Reduce the fractions if possible by combining numerators and denominators (whether from the same fraction or from different fractions) Multiply the numerators Multiply the denominators If the answer results in an improper fraction, convert the final answer back to a mixed fraction 4 22 14 2 14 28 1 Example: 2 x 111 x x 9 9 11 9 1 9 9 Dividing Fractions by Fractions Multiply first fraction by the reciprocal of the second fraction Often phrased by teachers as keep it, change it, flip it Keep the first fraction as is Change the operation from division to multiplication Flip the numerator and denominator of the second fraction Do not attempt to divide fractions, but always multiply them 2 5 15 7 Example: x (or 1 ) 4 5 4 2 8 8 Keep 4 as is Change the operation from division to multiplication 2 5 Flip to 5 2 18
Section 9.2 Fraction Arithmetic (continued) Dividing Fractions by Whole Numbers Use the keep it, change it, flip it concept Keep the first fraction as is Change the operation from division to multiplication Flip the whole number to become a fraction (the reciprocal of the whole number) Now you multiply the first fraction by the second fraction See if you can reduce the fractions numerator and/or denominator prior to multiplying One all numerators and denominators have been reduced, multiply the numerators Then multiply the denominators At this point, the final answer should already be reduced 1 Example 1: 5 x 4 4 5 20 Example 2: 5 4 8 5 4 x 8 1 5 1 x 2 1 10 1 Dividing Whole Numbers by Fractions Use the keep it, change it, flip it concept Keep the first value (whole number) as is Change the operation from division to multiplication Flip the numerator and denominator of the second value (the fraction) Now you multiply the whole number by the flipped second value See if you can reduce the fractions numerator and/or denominator prior to multiplying One all numerators and denominators have been reduced, multiply the numerators Then multiply the denominators At this point, the final answer should already be reduced If the answer is an improper fraction, you can convert it to a mixed fraction 4 20 2 Example 1: 5 5 x 6 4 9 11 11 22 1 Example 2: 6 6 x 2 x 7 11 9 19
Section 9.2 Fraction Arithmetic (continued) Dividing Mixed Fractions Convert each mixed fraction to the corresponding improper fraction Use the keep it, change it, flip it concept Keep the first value (the original fraction) as is Change the operation from division to multiplication Flip the numerator and denominator of the second value (the improper fraction) Reduce the fractions if possible by combining numerators and denominators (whether from the same fraction or from different fractions) Multiply the first value by the second At this point, the final answer should already be reduced If the answer is an improper fraction, you can convert it to a mixed fraction 4 14 28 14 1 Example 1: 2 5 x x 5 5 5 5 28 5 2 10 4 28 14 28 5 2 5 2 Example 2: 5 2 x x 2 5 5 5 5 5 14 5 1 1 20
Fraction Arithmetic Guided Practice 1. What is 1 5 + 1 7? 2. What is 8 + 6 1?. What is 4 5 4 + 6 8? 4. What is 1 7 1 5? 5. What is 8 6 1?. 6. What is 6 8 4 5 4?. 7. What is 4 x 2? 8. What is 4 x 6 2? 9. What is 4 6 2? 10. Which fraction has the largest value: 5 4, 8 7, 6 5, 4? 21
Section 9. Advanced Fraction Principles Fractions (Tenths, Hundredths, Thousandths, and Beyond) Similar to integers with place value, fractions can have place value Special fraction place value occurs when the numerator is 1 and the denominator contains 1 followed by one or more zeros The fractions turn out to be the reciprocals of ten, hundred, thousand, ten thousand, etc. They have the corresponding names of tenth, hundredth, thousandth, ten thousandth, etc. Here are examples of these special fractions 1 one tenth 10 1 one hundredth 100 1 one thousandth 1000 1 one the thousandth 10000 To add or subtract these types of fractions, you must convert one or more of the fractions to have a common denominator 1 1 10 1 11 + + ( eleven hundredths ) 10 100 100 100 100 1 1 10 1 9 ( nine hundredths ) 10 100 100 100 100 1 1 100 1 101 + + ( one hundred one ten thousandths ) 100 10000 10000 10000 10000 1 1 100 1 99 ( ninety nine ten thousandths ) 100 10000 10000 10000 10000 To multiply these types of fractions, the final answer will have the sum of the number of zeros that the fractions involved in the multiplication have 1 1 1 x ( one thousandth ) 10 100 1000 1 1 1 x ( one millionth ) 100 10000 100000 To multiply these types of fractions, multiply the first fraction by the reciprocal of the second fraction 1 1 1 1000 x 10 1000 10 1 1000 100 10 1 1 1 100 1 x ( one thousandth ) 100000 100 100000 1 1000 22
Section 9. Advanced Fraction Principles (continued) Introduction to Complex Fractions A complex fraction contains a fraction within a fraction A fraction in the numerator. A Fraction in the denominator A fraction in both the numerator and the denominator. Examples of complex fractions 1 8 5 2 5 7 4 5 Solving Complex Fractions Complex fractions must be converted to simple fractions where there is neither a fraction in the numerator nor a fraction in the denominator Treat the division bar of the main fraction as a division symbol between the main fractions numerator and denominator 1 8 1 1 1 1 8 ( Dividing Fractions by Whole Numbers ) x 8 24 5 5 5 2 ( Dividing Whole Numbers by Fractions ) 5 x 2 5 2 25 12 2 1 2 5 7 4 5 4 5 15 ( Dividing Fractions by Fractions ) x 7 5 7 4 28 Introduction to Word Problems with Fractions Word problems that cause any of the following fraction operations Specifying fractions Reducing (simplifying) fractions Adding fractions Subtracting fractions Multiplying fractions Dividing fractions 2
Section 9. Advanced Fraction Principles (continued) Solving Word Problems with Fractions Understand what the word problem is telling you Understand what the word problem is asking you Example 1: John had 10 marbles in his pocket. He lost marbles because of a hole in his pocket. Specify the fractions of marbles lost. Total marbles that John had 10 Total marbles lost by John Fraction of marbles lost 10 Example 2: John had 10 marbles in his pocket. He lost marbles because of a hole in his pocket. Specify the fractions of marbles that John still has. Total marbles that John had 10 Total marbles lost by John Total marbles that John still has 10 = 7 Fraction of marbles John still has 10 7 24
Advanced Fraction Principles Guided Practice 1 1. What is 1000 1 +? 10000 1 2. What is 10000 1 +? 1000000 1 1. What is +? 10 1000 1 4. What is 100000 1 +? 1000000 1 1 5. What is +? 10 1000000 1 4 6. Simplify 6 1 2 7. Simplify 9 1 8. Simplify 7 4 9. Simplify 4 5 10. Simplify 8 25
Advanced Fraction Principles Guided Practice (continued) 11. Simplify 8 4 12. Simplify 10 2 5 1. Simplify 1 2 14. Simplify 16 1 5 15. Simplify 9 9 10 16. Simplify 2 5 4 17. Simplify 4 2 5 18. Simplify 5 8 1 2 19. Simplify 5 8 1 2 20. Simplify 5 7 2 5 26
Advanced Fraction Principles Guided Practice (continued) 21. Mary was planning to bake 12 pies for the family reunion. She ran out of time and only baked 10 pies. Specify the pies baked as a simplified fraction of the pies planned. 22. In the state basketball tournament, Bill shot 15 free throws. He made 12 and missed. Specify the free throws made as a simplified fraction of the free throws attempted. 2. In the state basketball tournament, Robert shot 18 free throws. He made 14 and missed 4. Specify the free throws missed as a simplified fraction of the free throws attempted. 24. In the state basketball tournament, Henry shot 14 free throws. He made 9 and missed 5. Specify the fraction of free throws missed to free throw made. 27
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Module 9 (Principle of Fractions) Review Exercises Fraction Fundamentals Write the following as a fraction (questions 1 ) 1. 20 cars sold out of 27 cars available 2. 4 book read out of 5 books assigned. 21 green marbles out of 25 total marbles Write yes or no to indicate if the specified is a Simplest-Form Fraction (question 4 7) 24 4. 45 1 5. 5 9 6. 57 49 7. 84 Reduce the following fractions to Simplest-Form (question 8 11) 15 8. 27 18 9. 45 5 10. 6 9 11. 65 Create an Equivalent Fraction with the specified numerator or denominator (question 12 17) 12. 7 4 (numerator 28) 1. 4 (denominator 24) 14. 5 2 (numerator 16) 15. 8 7 (denominator 72) 16. 6 5 (numerator 55) 17. 9 7 (denominator 108) 29
Module 9 (Principles of Fractions) Review Exercises (continued) Fractional Fundamentals (continued) Convert the following fractions (Improper to Mixed or Mixed to Improper) (question 18 17) 18. 8 5 19. 7 5 20. 9 5 21. 9 7 4 What is the reciprocal of each fraction, specified as proper or improper? (question 22 25) 22. 7 26 2. 4 4 16 24. 27 25. 8 9 7 What is the least common denominator (LCD) of each set of fractions? (question 26 28) 26. 5 2, 6 5, 4 27. 8, 6 1, 16 28. 8, 6 1, 16 Convert the following fractions to have a least common denominator (question 29 1) 29. 5 2, 6 5, 4 0. 8, 6 1, 16 1. 8, 6 1, 16 0
Fraction Arithmetic Module 9 (Principles of Fractions) Review Exercises (continued) Fractional answers should be fully reduced unless otherwise specified. 12 1. Fully reduce the fraction. 20 6 2. Fully reduce the fraction. 48 6. Fully reduce the fraction. 60 48 4. Fully reduce the fraction. 72 80 5. Fully reduce the fraction. 96 6. Unreduce the fraction 6 5 to create an Equivalent Fraction (numerator 40). 7. Unreduce the fraction 5 to create an Equivalent Fraction (denominator 45). 8. Unreduce the fraction 8 7 to create an Equivalent Fraction (numerator 48). 9. Unreduce the fraction 10 9 to create an Equivalent Fraction (denominator 70). 10. Unreduce the fraction 12 7 to create an Equivalent Fraction (denominator 84). 1
Module 9 (Principles of Fractions) Review Exercises (continued) Fractional Arithmetic (continued) 11. Convert the fraction 5 to Mixed. 12. Convert the fraction 6 4 to Mixed. 55 1. Convert the fraction to Mixed. 6 14. Convert the fraction 7 55 to Mixed. 67 15. Convert the fraction to Mixed. 8 16. Convert the fraction 8 5 to Improper. 17. Convert the fraction 5 4 to Improper. 18. Convert the fraction 9 8 7 to Improper. 19. Convert the fraction 12 58 to Improper. 20. Convert the fraction 15 4 1 to Improper. 2
Module 9 (Principles of Fractions) Review Exercises (continued) Fractional Arithmetic (continued) 21. What is the reciprocal of 9 5?. 1 22. What is the reciprocal of?. 7 2. What is the reciprocal of 2 5 4?. 24. What is the reciprocal of 7 6 5?. 25. What is the reciprocal of 11 5 2?. 26. What is 7 + 7 2? 27. What is 8 1 + 8 5? 28. What is 5 7 1 + 4 7 4? 29. What is 5 5 4 + 7 5? 0. What is 12 9 7 + 9 9 8?
Module 9 (Principles of Fractions) Review Exercises (continued) Fractional Arithmetic (continued) 1. What is 5 1 + 8 5? 2. What is 8 + 6 1?. What is 5 4 + 8 7? 4. What is 4 2 + 7 8 5? 5. What is 9 5 1 + 4 8 7? 6. What is 7 7 2? 7. What is 8 1 8 5? 8. What is 5 7 1 4 7 4? 9. What is 5 5 4 7 5? 40. What is 12 9 7 9 9 8? 4
Module 9 (Principles of Fractions) Review Exercises (continued) Fractional Arithmetic (continued) 41. What is 5 1 8 5? 42. What is 8 6 1? 4. What is 5 4 8 7? 44. What is 4 2 7 8 5? 45. What is 9 5 1 4 8 7? 46. What is 8 5 x 5? 8 15 47. What is x? 5 16 48. What is 2 1 x 5 4 1? 49. What is 4 8 5 x 2 2? 50. What is 6 8 5 x 1 1? 5
Module 9 (Principles of Fractions) Review Exercises (continued) Fractional Arithmetic (continued) 51. What is 4 4 1? 52. What is 4 1 1? 5. What is 2 5 1 5 2? 54. What is 5 6 5 2 1? 55. What is 8 8 1 8 7? 56. Which fraction has the largest value: 8, 1, 10, 5 2? 57. Which fraction has the largest value: 8, 10 4, 1, 6 1? 58. Which fraction has the largest value: 2 1, 8 5, 5, 2? 59. Which fraction has the largest value: 4, 2, 8 5, 5? 60. Which fraction has the largest value: 8 7, 5 4, 6 5, 4? 6
Module 9 (Principles of Fractions) Review Exercises (continued) Advanced Fraction Principles 1 1 61. What is +? 10 100 1 1 62. What is +? 10 1000 1 1 6. What is +? 10 10000 1 1 64. What is +? 10 100000 1 1 65. What is +? 10 1000000 1 1 66. What is +? 100 1000 1 1 67. What is +? 100 10000 1 1 68. What is +? 100 100000 1 69. What is 1000 1 +? 100000 1 70. What is 1000 1 +? 1000000 7
Module 9 (Principles of Fractions) Review Exercises (continued) Advanced Fraction Principles (continued) 1 5 71. Simplify 12 1 6 72. Simplify 12 1 8 7. Simplify 12 4 5 74. Simplify 12 5 6 75. Simplify 12 8 76. Simplify 12 77. Simplify 78. Simplify 79. Simplify 12 1 5 12 4 5 12 8 12 80. Simplify 8 8
Module 9 (Principles of Fractions) Review Exercises (continued) Advanced Fraction Principles (continued) 81. Simplify 82. Simplify 8. Simplify 84. Simplify 85. Simplify 7 8 4 5 6 2 5 6 2 1 4 9 10 6 5 5 The following scenario applies to questions 85 90. During target practice with a bow and arrow, John had the following results: 1) Hit the bulls eye times 2) Hit the target in an area other than the bulls eye 7 times ) Missed the target completely 5 times 86. Write the fraction of bulls eyes made to total shots attempted. 87. Write the fraction of bulls eyes made to total missed target. 88. Write the fraction of target hit (not in the bulls eye area) to total shots attempted. 89. Write the fraction of target hit (in any area) to total shots attempted. 90. Write the fraction of target hit (in any area) to target missed completely. 9
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Module 10: Principles of Decimals 41
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Section 10.1 Introduction to Decimals Decimal Another way of expressing a fraction (divide denominator into numerator) The fraction 8 7 is the decimal of 7 8 0.875 Decimal Place Value Decimals have place value to the right of the decimal point just as whole numbers have place value to the left of the decimal point Common decimal place values are Tenths : 0.1 (one tenth) Hundredths : 0.01 (one hundredth) Thousandths : 0.001 (one thousandth) Ten Thousandths : 0.0001 (one ten thousandth) Hundred Thousandths : 0.00001 (one hundred thousandth) Millionths : 0.000001 (one millionth) In stating the value of a decimal, use the rightmost non-zero digit to determine the place value name 0. three tenths 0.45 forty-five hundredths 0.17 three hundred seventeen thousandths 0.4208 four thousand two hundred eight ten thousandths 0.185 thirteen thousand eight hundred fifty three hundred thousandths 0.426712 four hundred twenty six thousand seven hundred twelve millionths 4
Section 10.1 Introduction to Decimals (continued) Comparing Decimals Starting from the left most position, the decimal containing the largest corresponding digit has the largest value Starting from the left most position, the decimal containing the smallest corresponding digit has the smallest value Example: What are the largest and smallest numbers: 1.254, 1.87, 2.08, and 1.099, 1.987 For the largest value, first look at the digits to the left of the decimal point o The values of the corresponding numbers are : first (1), second (1), third (2), fourth(1), fifth (1) o Since the third number has a 2, it is the largest number, regardless of the digits to the right of the decimal point For the smallest value, first look at the digits to the left of the decimal point o The values of the corresponding numbers are : first (1), second (1), third (2), fourth(1), fifth (1) o Since the first, second, fourth, and fifth numbers contain 1, you must now look at their digits to the right of the decimal point o The values digits immediately to the right of the decimal point of the corresponding number are: first (2), second (), fourth (0), and fifth (9) o Since the fourth number contains the smallest value (0) in that position, it is the smallest number, regardless of the digits to the right of that place value Ordering Decimals To order decimal values (such as smallest to largest or largest to smallest), perform the steps above for comparing decimals Based upon the results of comparing the decimals, place the decimals in the proper order o If ordering the decimals from smallest to largest, list the smallest value first, then the next smallest, and proceed with this pattern to include all the values o If ordering the decimals from largest to smallest, list the largest value first, then the next largest, and proceed with this pattern to include all the values Example: Order from smallest to largest the following numbers: 1.254, 1.87, 2.08, and 1.099, 1.987 Determine the smallest value o Starting with the leftmost (ones) digit of each number, comparing the corresponding number, and proceeding with the corresponding digits to the right, the smallest value is 1.099 o Following the same process for the remaining numbers, the second smallest value is 1.254 o The third smallest value is 1.87 o The fourth smallest vale is 1.987 o The fifth smallest (largest) value is 2.08 o Order the values 1.099, 1.254, 1.87, 1.987, 2.08 44
Introduction to Decimals Guided Reinforcement Name the following decimals [Example: 0.7 thirty-seven hundredths] (questions 1 6) 1. 0.7 2. 0.14. 0.212 4. 0.478 5. 0.51617 6. 0.000045 Answer the following decimal comparisons (questions 7 8) 7. Which number is the smallest:.407,.047,.704,.074,.470,.740? 8. Which number is the largest:.407,.047,.704,.074,.470,.740? Order the following numbers as specified (questions 9 10) 9..407,.047,.704,.074,.470,.740 (smallest to largest)? 10..407,.047,.704,.074,.470,.740 (largest to smallest)? 45
Section 10.2 Rounding Decimal Numbers What is Decimal Rounding? As numbers get smaller (tenths, hundredths, thousandths), you may often not be concerned with the precise value You may want a ballpark value Rounding decimal numbers allows you to eliminate some of the precision of the number and just deal with a ballpark value of the decimal number instead Principles of Rounding Decimal Numbers You are normally required to round a number to a certain place value, such as Round to the nearest tenths Round to the nearest hundredths Round to the nearest hundred thousandths Notice the phrasing round to the nearest... Your final (rounded) answer will contain no more precision than the place value stated Your final (rounded) answer will contain zeros for all digits to the right of the place value stated o When those zeros appear after the decimal point, the do not have to be written o Round to the nearest tenths zeros in the hundredths place value and beyond do not have to be written o Round to the nearest hundredths zeros in the thousandths place value and beyond do not have to be written o Round to the nearest thousandths zeros in the ten thousandths place value and beyond do not have to be written Rounding to a certain place value tells you what value your original value is closer to using the stated place value Steps in Decimal Rounding Look at the place value which was stated in the rounding requirement Now look at the digit immediately to the right place of the place value which was stated in the rounding requirement o If that digit is less than five (0, 1, 2,, or 4), keep the digit stated in the rounding requirement the same and make all digits to the right of it become zeros o If that digit is five or more (5, 6, 7, 8, or 9), increase by one the digit stated in the rounding requirement and make all digits to the right of it become zeros o Zeros after the required decimal place value do not have to be written 46
Section 10.2 Rounding Decimal Numbers (continued) Examples of Rounding Decimal Numbers Round 24.87 to the nearest tenths Look at the place value of the tenths position Look at the digit immediately to the right of the tenths position 8 is in the hundredths position Since the value in the hundredths position (8) is five or larger, increase the tenths position value by 1 from to 4 All digits to the right of the tenths position will now be ignored (and not written) 24.87 rounded to the nearest tenths 24.4 The number 24.87 is closer to 24.4 than it is to 24. Round 1.56268 to the nearest hundredths Look at the place value of the hundredths position 6 Look at the digit immediately to the right of the hundredths position 2 in the thousandths position Since the value of the thousandths position (2) is less than five, keep the digit in the hundredths position (6) the same All digits to the right of the tenths position will now be ignored (and not written) 1.56268 rounded to the nearest hundredths 1.56 The number 1.56268 is closer to 1.56 than it is to 1.57 Round 1.56268 to the nearest thousandths Look at the place value of the thousandths position 2 Look at the digit immediately to the right of the thousandths position 6 is in the ten thousandths position Since the value in the ten thousandths position (6) is five or larger, increase the thousandths position value by 1 from 2 to All digits to the right of the thousandths position will now be ignored (and not written) 1.56268 rounded to the nearest thousandths 1.56 The number 1.56258 is closer to 1.56 than it is to 1.562 47
Rounding Decimal Numbers Guided Practice Write the correct value from the specified rounding 1) 18.47584 Rounded to the nearest tenth 2) 18.47584 Rounded to the nearest hundredth ) 18.47584 Rounded to the nearest hundred thousandth 4) 18.42649 Rounded to the nearest tenth 5) 18.42649 Rounded to the nearest hundredth 6) 18.42649 Rounded to the nearest thousandth 7) 18.42649 Rounded to the nearest ten thousandth 8) 278.42649 Rounded to the nearest unit (ones) 9) 278.72649 Rounded to the unit nearest (ones) 10) 278.72649 Rounded to the nearest ten 48
Section 10. Decimal Arithmetic Common Arithmetic Operations Involving Decimals Addition Subtraction Multiplication Multiplying a number with a decimal by a number without a decimal Multiplying a number with a decimal by another number with a decimal Division Dividing Decimals by Whole Numbers Dividing Whole Numbers by Decimals Dividing Decimals by Decimals Dividing Decimals with Zeros in the Quotient Decimal Addition To add one or more numbers with a decimal point, perform the following steps Line up the decimal points of all values to be added o Remember that if a number does not have a decimal point, you can place a decimal point just to the right of the rightmost digit o You can also add zeros to the right of the rightmost digit after the decimal point o Examples: 257 257. 85 85.000 415 415.00000 Add the digits of the numbers, starting from the right and proceeding to the left o The same rules apply for carrying and regrouping with a decimal point as apply for carrying and regrouping without a decimal point The number of digits after the decimal in the final answer will be the same as the number of digits in whichever original number has the most digits after the decimal Example: 5.5 +.475 Rewrite the numbers vertically, and line up the decimal points of the values You can also add zeros to the right of the rightmost digit after the decimal point to make all values have the same number of digits after the decimal point o 5.5 can be written as 5.50 to have three digits after the decimal point (like.475 has) Add the digits of the numbers starting from the right and proceeding to the left Since.475 has three digits to the right of the decimal point, the final answer will also have three digits to the right of the decimal point 5.50 +.475 ------------- 9.005 49
Section 10. Decimal Arithmetic (continued) Decimal Subtraction To subtract one or more numbers with a decimal point, perform the following steps Line up the decimal points of all values to be subtracted o Just like with decimal addition, if a number does not have a decimal point, you can place a decimal point just to the right of the rightmost digit and add zeros if necessary Subtract the digits of the numbers, starting from the right and proceeding to the left o The same rules apply for borrowing and regrouping with a decimal point as apply for borrowing and regrouping without a decimal point The number of digits after the decimal in the final answer will be the same as the number of digits in whichever original number has the most digits after the decimal Example: 5.5.475 Rewrite the numbers vertically, and line up the decimal points of the values You can also add zeros to the right of the rightmost digit after the decimal point to make all values have the same number of digits after the decimal point o 5.5 can be written as 5.50 to have three digits after the decimal point (like.475 has) Subtract the digits of the numbers starting from the right and proceeding to the left Since.475 has three digits to the right of the decimal point, the final answer will also have three digits to the right of the decimal point 5.50.475 ------------- 1.055 50
Decimal Addition and Subtraction Guided Practice 1. What is 6.57 + 7.? 2. What is 6.57 + 7.8?. What is 6.57 + 7.86? 4. What is 18.865 12.9? 5. What is 18.865 12.98? 6. What is 18.865 12.987? 51
Section 10. Decimal Arithmetic (continued) Introduction to Decimal Multiplication You will typically perform one of the following for decimal multiplication Multiplying a number with a decimal by a number without a decimal Multiplying a number with a decimal by another number with a decimal In multiplying numbers with a decimal, you will use the same rules that apply for multiplying numbers without a decimal Multiplying a Number with a Decimal by a Number without a Decimal To multiply a number with a decimal by a number without a decimal, perform the following steps Write the numbers vertically o Place the number with the greater number of digits on top o Place the number with the fewer number of digits on the bottom Multiply the digits of the bottom number by each of the digits of the top number o The same rules apply for carrying and regrouping with a decimal point as apply for carrying and regrouping without a decimal point The number of digits after the decimal in the final answer will be the same as the number of digits after the decimal point of the original number that contained a decimal point Example: 2.7 x 21 Rewrite the numbers vertically o 2.7 has three digits place this number on top o 21 has two digits place this number on the bottom Multiply the digits of the bottom number by each of the digits of the top number Since the number with a decimal point (2.7) contains two digits to the right of the decimal point, the final answer will contain two digits to the right of the decimal point 2.7 x 21 ----------- 27 4740 ----------- 49.77 52
Section 10. Decimal Arithmetic (continued) Multiplying a Number with a Decimal by another Number with a Decimal To multiply a number with a decimal by another number with a decimal, perform the following steps Write the numbers vertically o Place the number with the greater number of digits on top o Place the number with the fewer number of digits on the bottom Multiply the digits of the bottom number by each of the digits of the top number o The same rules apply for carrying and regrouping with a decimal point as apply for carrying and regrouping without a decimal point Add the number of digits after the decimal point of the first number to the number f digits after the decimal point to the second number o The number of digits after the decimal point in the final answer will be this sum Example: 2.7 x 2.1 Rewrite the numbers vertically o 2.7 has three digits place this number on top o 2.1 has two digits place this number on the bottom Multiply the digits of the bottom number by each of the digits of the top number Since the first number (2.7) contains two digits to the right of the decimal point and the second number (2.1) contains one digit to the right of the decimal point, the final answer will contain three (2 + 1) digits to the right of the decimal point 2.7 x 2.1 ----------- 27 4740 ----------- 4.977 The number of digits to the right of the decimal in the final answer is always the sum of the number of digits to the right of the decimal of each numbers being multiplied Even when multiplying a number with a decimal by a number without a decimal, the number of digits to the right of the decimal in the final answer will be the sum of the number digits to the right of the decimal of each number In the example on the previous page (2.7 x 21), the number of digits to the right of the decimal in the final answer was the sum of the number of digits to the right of the decimal of the first number 2.7 (2 digit) and the number of digits to the right of the decimal of the second number 21 (0 digits) o 49.77 contains two digits to the right of the decimal 5
Section 10. Decimal Arithmetic (continued) Introduction to Decimal Division You will typically perform one of the following for decimal division Dividing numbers without a decimal, but the answer contains a decimal Dividing a number without a decimal point into a number with a decimal point Dividing a number with a decimal point into a number without a decimal point Dividing a number with a decimal point into another number with a decimal point In dividing numbers with a decimal Place the decimal point in the answer above the decimal point in the original problem Write the answer in the proper place value position Dividing Numbers without a Decimal Point (Resulting in an Answer with a Decimal Point) Write the division fact out the normal way Place the decimal point in the original number after the last digit, and add one or more 0 s Place the decimal point for the answer immediately above the decimal point in the original number Example 14 40 Write 14 40 the normal way and place the decimal point after. 40 ) 14.000 Now begin the normal steps of division, and place each answer in the proper place value position How many times will 40 divide into 14 0 o You do not have to write the 0 How many times will 40 divide into 140 o Place the above the 0 (from 140), just to the right of the decimal point. 40 ) 14.000-12 0 ------- 200 How many times will 40 divide into 200 5 o Place the 5 to the right of the.5 40 ) 14.000-12 0 ------- 200-200 ------- 0 With remainder 9, you are done: 40 140 =.5 54
Section 10. Decimal Arithmetic (continued) Dividing a Number without a Decimal Point into a Number with a Decimal Point Write the division fact out the normal way Place the decimal point for the answer immediately above the decimal point in the original number Example 48.6 27 Write 48.6 27 the normal way and place the decimal point for the answer immediately above the decimal point of the original number. 27 ) 48.6 Now begin the normal steps of division, and place each answer in the proper place value position How many times will 27 divide into 48 1 o Place the 1 above the 8 from 48 (just to the left of the decimal point) 1. 27 ) 48.6-27 ------- 21 6 How many times will 27 divide into 216 8 o Place the 8 above the 6 from 48.6 (just to the right of the decimal point) 1.8 27 ) 48.6-27 ------- 216-216 ------ 0 With remainder 0, you are done: 48.6 27 = 1.8 55
Section 10. Decimal Arithmetic (continued) Dividing a Number with a Decimal Point into a Number without a Decimal Point Write the division fact out the normal way Move the decimal point of the divided by number to the right of the last digit Now move the decimal point of the divided into number the same number of places, adding 0 s if necessary Divide these modified numbers the normal way Example 256 -.16 Write 256.16 the normal way.16 ) 256 Since.16 contains two digits after the decimal point, move the decimal point two places to the right (becoming 16) o Because you moved the decimal point two places to the right in.16, you must now move the decimal point two places to the right in 256 (becoming 25600).16 ) 256 16 ) 25600 How many times will 16 divide into 25 1 o Place the 1 above the 5 from 25 1 16 ) 25600-16 ------- 96 How many times will 16 divide into 96 6 o Place the 6 above the 6 from 256 (just to the right of the decimal point) 16 16 ) 25600-16 ------- 96-96 ------ 0 As you bring down the remaining two 0 s, 16 will divide into them 0 times 1600 16 ) 25600 Your final answer is 1600 o.16 divided into 256 gives you the same answer as 16 divided into 25600 56
Section 10. Decimal Arithmetic (continued) Dividing a Number with a Decimal Point into another Number with a Decimal Point Write the division fact out the normal way Move the decimal point of the divided by number to the right of the last digit Now move the decimal point of the divided into number the same number of places, adding 0 s if necessary Divide these modified numbers the normal way Example 5.78 1.7 Write 5.78 1.7 the normal way 1.7 ) 5.78 Since 1.7 contains one digit after the decimal point, move the decimal point one place to the right (becoming 17) o Because you moved the decimal point one places to the right in 1.7, you must now move the decimal point one place to the right in 5.78 (becoming 57.8) 1.7 ) 5.78 17 ) 57.8 How many times will 17 divide into 57 o Place the above the 7 from 57 (just to the left of the decimal point). 17 ) 57.8-51 ------- 68 How many times will 17 divide into 68 4 o Place the 4 above the 4 from 57.8 (just to the right of the decimal point).4 17 ) 57.8-51 ------- 68-68 ------ 0 With remainder 0, you are done o Your final answer is.4 o 1.7 divided into 5.78 gives you the same answer as 17 divided into 57.8 57