Third Edition SAXON PUBLISHERS, IN. 1 ra 2 An Incremental Development John H. Saxon, Jr.
completion of this project: Author: John Saxon Editorial: Brian E. Rice, Matt Maloney, R. lint Keele Production: Adriana Maxwell, Brenda Lopez 2004 Saxon Publishers, Inc. Printed in the United States of America ISBN: 978-1-56577-149-9 ISBN: 1-56577-149-4 Manufacturing ode: 6 7 8 0868 12 11 10 Saxon Publishers gratefully acknowledges the contributions of the following individuals in the Editorial Support Services: hristopher Davey, Jenifer Sparks, Jack Day, Shelley Turner All rights reserved. No part of the material protected by this copyright may be reproduced or utilized in any form or by any means, in whole or in part, without permission in writing from the copyright owner. Requests for permission should be mailed to: opyright Permissions, Harcourt Achieve Inc., P.O. Box 27010, Austin, Texas 78755.
ontents Word Problems 48 Lesson 14 Fractions Expanding and Reducing Fractions Lesson 15 Fractions and Decimals Fractions to Decimals Rounding Repeaters Decimals to Fractions Lesson 16 Exponents 59 Lesson 17 Areas of Rectangles 61 Lesson 18 Multiplying Fractions and Whole Numbers Fractional Part of 51 55 V Preface xi Lesson 1 Whole Number Place Value Expanded Notation Reading and Writing Whole Numbers Addition 1 Lesson 2 The Number Line and Ordering Rounding Whole Numbers 7 Lesson 3 Subtraction Addition and Subtraction Patterns 11 Lesson 4 Multiplication Division Multiplication and Division Patterns 15 Lesson 5 Addition and Subtraction Word Problems 20 Lesson 6 Reading and Writing Decimal Numbers Adding and Subtracting Decimal Numbers Rounding Decimal Numbers 23 Lesson 7 Multiplying Decimal Numbers Dividing Decimal Numbers Estimation 27 Lesson 8 Multiplying and Dividing by Powers of 10 Ordering Decimal Numbers 31 Lesson 9 Points, Lines, Rays, and Line Segments Angles Perimeter 34 Lesson 10 Divisibility 39 Lesson 11 Word Problems about Equal Groups 42 Lesson 12 Prime Numbers and omposite Numbers Products of Prime Numbers 45 Lesson 13 ommon Factors and the Greatest ommon Factor Multiplication a Number 64
Lesson 20 Multiples Least ommon Multiple Lesson 21 Average Lesson 22 Multiple Fractional Factors Lesson 23 U.S. ustomary System Unit Multipliers Lesson 24 Metric System Lesson 25 Area as a Difference Lesson 46 Order of Operations with Fractions Lesson 47 Evaluation of Exponential Expressions and Radicals Lesson 48 Fractional Part of a Number Fractional Equations Lesson 49 Surface Area Notation for Numbers Between Zero and One Lesson 51 Decimal Part of a Number 70 74 76 78 81 84 152 154 157 160 163 166 vi Algebra½ Lesson 19 Symbols for Multiplication Multiplying Fractions Dividing Fractions 67 Lesson 26 Mode, Median, Mean, and Range Average in Word Problems 87 Lesson 27 Areas of Triangles 90 Lesson 28 Improper Fractions, Mixed Numbers, and Decimal Numbers 92 Lesson 29 Graphs 97 Lesson 30 Adding and Subtracting Fractions Adding and Subtracting Fractions with Unequal Denominators 100 Lesson 31 Order of Operations 104 Lesson 32 Variables and Evaluation 107 Lesson 33 Multiple Unit Multipliers onversion of Units of Area 109 Lesson 34 Adding Mixed Numbers Rate 112 Lesson 35 Subtracting Mixed Numbers 115 Lesson 36 Rate Word Problems 118 Lesson 37 Equations: Answers and Solutions 120 Lesson 38 Rectangular oordinates 123 Lesson 39 Equivalent Equations Addition-Subtraction Rule for Equations 127 Lesson 40 Reciprocals Multiplication Rule Division Rule 131 Lesson 41 Overall Average 136 Lesson 42 Symbols of Inclusion Division in Order of Operations 139 Lesson 43 Multiplying Mixed Numbers Dividing Mixed Numbers 142 Lesson 44 Roots Order of Operations with Exponents and Roots 145 Lesson 45 Volume 148 Lesson 50 Scientific Notation for Numbers Greater Than Ten Scientific
ontents Lesson 52 Fractions and Symbols of Inclusion Lesson 53 Percent Lesson 54 Ratio and Proportion pq and ~ Lesson 56 Equations with Mixed Numbers Lesson 57 Mixed Number Problems Lesson 58 The Distance Problem Lesson 59 Proportions with Fractions Lesson 60 ircles Lesson 61 Solving Equations in Two Steps Lesson 62 Fractional Part Word Problems Lesson 63 hanging Rates Lesson 64 Semicircles Similar Triangles Lesson 66 Ratio Word Problems Lesson 67 Using Ratios to ompare Lesson 69 Absolute Value Adding Signed Numbers Lesson 70 Rules for Addition of Signed Numbers Lesson 71 Powers of Fractions Roots of Fractions Lesson 72 Graphing Inequalities Lesson 73 Right ircular ylinders Lesson 7 4 Inserting Parentheses Order of Addition Lesson 75 Implied Ratios Lesson 76 Multiplication with Scientific Notation Lesson 77 Percents Greater than 100 Lesson 78 Opposites Lesson 79 Simplifying More Difficult Notations Lesson 80 Increases in Percent Lesson 81 Multiplication and Division of Signed Numbers Lesson 82 Evaluation with Signed Numbers Lesson 83 Rate Problems as Proportion Problems Properties of Equality Lesson 85 Equation of a Line Graphing a Line vii 169 171 174 177 181 183 186 189 192 195 198 200 203 206 209 212 214 219 222 225 228 231 234 237 240 243 246 249 251 254 257 260 263 266 Lesson 55 Fractions, Decimals, and Percents Reference Numbers Lesson 65 Proportions with Mixed Numbers Using Proportions with Lesson 68 Percent Word Problems Visualizing Percents Less Than 100 Lesson 84 Formats for the Addition Rule Negative oefficients
viii Lesson 86 Algebraic Phrases Lesson 87 Properties of Algebra Lesson 88 Surface Area of a Right Solid Lesson 89 Trichotomy Symbols of Negation Lesson 90 Algebraic Sentences Algebra½ Lesson 91 Order of Operations with Signed Numbers and Symbols of Inclusion 286 Lesson 92 Estimating Roots Lesson 93 Fraction Bars as Symbols of Inclusion Lesson 94 Terms Adding Like Terms, Part 1 293 Lesson 95 Variables on Both Sides 296 Lesson 96 Multiple-Term Equations 298 Lesson 97 Two-Step Problems 301 Lesson 99 Exponents and Signed Numbers 308 Lesson 100 Advanced Ratio Problems 310 Lesson 101 Multiplication of Exponential Expressions Variable Bases 314 Lesson 102 Adding Like Terms, Part 2 317 Lesson 103 Distributive Property Lesson 104 lassifying Triangles Angles in Triangles Lesson 105 Evaluating Powers of Negative Bases Lesson 106 Roots of Negative Numbers Negative Exponents Zero Exponents Lesson 107 Roman Numerals Lesson 108 Fractional Percents Lesson 109 Simple Interest ompound Interest Lesson 110 Markup and Markdown Lesson 111 ommission Profit Lesson 112 Probability, Part 1 Lesson 113 Inch Scale Metric Scale Lesson 114 Probability, Part 2: Independent Events 355 Lesson 115 Polygons ongruence and Transformation 358 Lesson 116 Area of Parallelograms and Trapezoids 363 Lesson 117 Equations with x 2 Pythagorean Theorem Demonstration of the Lesson 118 English Volume onversions 372 269 272 275 279 283 288 290 319 322 326 329 333 336 338 342 345 347 351 Lesson 98 Adjacent Angles omplementary and Supplementary Angles Measuring Angles 303 Pythagorean Theorem 367
ontents Lesson 119 Metric Volume onversions Lesson 120 Volume of Pyramids, ones, and Spheres Surface Area of Pyramids and ones Lesson 121 Forming Solids Symmetry Lesson 122 Permutations Lesson 123 Numerals and Numbers The Subsets of the Real Numbers Appendix - Additional Topics Topic A Geometric onstructions Topic B Representing Data Topic Arithmetic in Base 2 Topic D Theorems About Angles and ircular Arcs Topic E Approximating Roots Topic F Polynomials Topic G Transformational Geometry Topic H Advanced Graphing Topic J Basic Trigonometry Glossary Index Answers to Odd-Numbered Problems ix 375 378 383 388 391 399 403 408 416 419 424 429 435 437 441 447 453 463 Topic I Slope
1.A whole number place value Numbers Addition 0, l, 2,3, 4, 5,6, 7, 8,9 numbers. We may show the set of counting numbers this way: {l, 2, 3, 4, 5,... } of counting numbers, then we form the set of whole numbers. we can leave it off. Both of these after the 7. {0, l,2, 3,4,... } 427. 427 Whole Number Place Values en en en 1l ~ en "O en en.e en 0 o E en :c en 1l en "O ~....5... "O.Q "O.Q (I) en (I) en (I) - en "O :::i "O (I) 0 1l (I) - :::i 0 :::i :::i :::i :::i cil c (I).c 2 E.c 2 :c.c 2 E.c 2.c 2 :::i "O, -, -, - -, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;:! ;:! 0 0 0 0 0 0 0 0 0 0 0 o_ c5 c5 c5 c5 c5 c5 c5 c5 c5 c5 c5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o _ ;:! c5 c5 c5 c5 c5 c5 c5 c5 0 0 0 0 0 0 0 0 0 0 o _ ;:! c5 c5 c5 c5 c5 0 0 0 0 0 o _ ;:! c5 c5 ;:! 1 LESSON 1 Whole Number Place Value Expanded Notation Reading and Writing Whole We use the Hindu-Arabic system to write our numbers. This system is a base 10 system and thus has ten different symbols. The symbols are called digits, or numerals, and they are The numbers we say when we count are called counting numbers, or natural The three dots, called an ellipsis, mean that the list continues without end. The symbols, { }, are called braces and are sometimes used to designate a set. If we include zero with the set When we write whole numbers, we can write the decimal point at the end of the number, or represent the same number. In the right-hand number, the decimal point is assumed to be The value of a digit in a number depends on where the digit appears in the number. The first place to the left of the decimal point is the ones' place. We also call this place the units' place, which has a place value of 1. The next place to the left of the units' place is the tens' place, with a place value of 10, followed by the hundreds' place, with a place value of 100, and then the thousands' place, with a place value of 1,000. Each place to the left has one more zero. g g :::i "O c en "O en Q. :c ' E ' en ' E "O.Q "O _Q.Q :::i en.el 13
1.B expanded notation 46,235. (5 X 1000) + (2 X 10) solution Standard notation is our usual way of writing numbers. The number has four ten thousands, solution There are six millions, five thousands, (5 X 1000) (6 X 1,000,000) If we add them all together, we get one hundred, (1 X 100) three hundred thousands, (3 X 100,000) two tens, (2 X 10) and six ones. (6 X 1) 2 Algebra ½ Lesson 1 To find the value of a digit in a number, multiply the digit times the place value. For example, the 5 in the left-hand number below 415,623 701,586 731,235 has a value of 5 x 1000, or 5000, because it is in the thousands' place. The value of the 5 in the center number is 5 x 100, or 500, because it is in the hundreds' place. The value of the 5 in the right-hand number is 5 x 1, or 5, because it is in the units' (ones') place. example 1.1 In the number 46,235 : (a) What is the value of the digit 5? (b) What is the value of the digit 2? ( c) What is the value of the digit 4? solution First we write the decimal point at the end of the number. (a) The 5 is one place to the left of the decimal point. This is the units' place. This digit has a value of 5 x 1, or 5. (b) The 2 is three places to the left of the decimal point. This is the hundreds' place. This digit has a value of 2 x 100, or 200. (c) The 4 is five places to the left of the decimal point. This is the ten-thousands' place. This digit has a value of 4 x 10,000, or 40,000. Writing a number in expanded notation is a good way to practice the idea of place value. When we write a number in expanded notation, we consider the value of every digit in the number individually. To write a number in expanded notation, we write each of the nonzero digits multiplied by the place value of the digit. We use parentheses to enclose each of these multiplications and put a plus sign between each set of parentheses. To write 5020 in expanded notation, we write because this number contains five thousands and two tens. example 1.2 Write the following number in standard notation: (4 x 10,000) + (6 x 100) + (5 x 1) no thousands, six hundreds, no tens, and five ones. The number is 40,605. example 1.3 Write the number 6,305,126 in expanded notation. (6 X 1,000,000) + (3 X 100,000) + (5 X 1000) + (1 X 100) + (2 X 10) + (6 X 1)
1. reading and writing whole numbers The words 23 is written twenty-three 35 is written thirty-five 42 is written forty-two 51 is written fifty-one ten thousand ten-thousands' place, the words are hyphenated. Other examples of this rule are hundred-millions' digit ten-billions' place hundred-thousands' place 64 is written sixty-four 79 is written seventy-nine 86 is written eighty-six The word and is not used when we write out whole numbers. ~ ~ :J 501 is written five hundred one 98 is written ninety-eight 422 is written four hundred twenty-two (1) (1) (1) Place Value (1) J: 0 J: 0 J: 0 J: 0 J: 0 4,125,678,942. Then we read the number, beginning with the leftmost group. First we read the number in the group, and then we read the name of the group. Then we move to the right and repeat the procedure. four billion, one hundred twenty-five million, six hundred seventy-eight thousand, nine hundred forty-two. 1. reading and writing whole numbers 3 We begin by noting that all numbers between 20 and 100 that do not end in zero are hyphenated words when we write them out. The hyphen is also used in whole numbers when the whole number is used as a modifier. are not hyphenated. But when we use these words as a modifier, as when we say not five hundred and one 370 is written three hundred seventy not three hundred and seventy not four hundred and twenty-two Do not think of this as a useless exercise! Knowing how to correctly and accurately write numbers is necessary when writing a check, for example. Before we read whole numbers, we place a comma after every third digit beginning at the decimal point and moving to the left. t The commas divide the numbers into groups of three digits. Units E Trillions Billions Millions Thousands (Ones) 5 en en en en en cii "O "O "O "O "O E ~ ~ c:; "O en en "O en en 'O en en "O en en "O en en (1) (1) ~ :J ~ :J ~ :J ~ :J ~ (1) 0 To read the number 4125678942, we begin on the right-hand end, write a decimal point, and separate the number into groups of three by writing commas. tit is our convention to usually write four-digit whole numbers without commas.
Bolts 2 = - Ratio Total = 13 TR 13 Now we cover the middle line and see that what remains concerns nuts and ratio total. Nuts = 11 l 0 - = NR 11 TR 13 (a) (b) 310 Algebra ½ Lesson 100 Simplify: 22. -2 2 - (-3) 2-3 4 (99) 24. -2 2 (2 2-3 2) - 3(2 3-2 2 ) (99) 23. -3 2 + (-2) 2 (99) 25. (93) 3-2(3-2) + 3 2 4(-2 + 5) 26. (93,99) -3-2(3 2-1) + 2 2 2 2 ( - 3-2 2 ) 27. 3.!_ X 2,!_ (43) 2 3 1 4 1 6 28. Sketch a rectangular coordinate system, and graph the line x = 2y. (85) Evaluate: 29. abc + ab (82) 30. -ab + be (82) if a = -2, b = -3, and c = -4 if a = 2, b = -3, and c = -4 LESSON 100 Advanced Ratio Problems In this lesson we will consider advanced ratio problems. We begin with an example. example 100.1 The ratio of nuts to bolts was 11 to 2. If there were 260 nuts and bolts in the pile, how many were bolts? solution We begin confidently by writing N 11 = B 2 Now where does the 260 go? What do we do now? The answer is that this problem is more difficult than the ratio problems we have been working, and we have begun the wrong way. This problem has nuts and bolts but also has the total. There is a trick to working these problems. The trick is to begin by listing the given parts of the ratio, including the ratio total. Nuts = 11 Bolts = 2 Ratio Total = 13 There are three ratios hidden here. We can see the ratios if we cover up the lines one at a time with a finger. First we cover the top line. We see that what remains concerns bolts and ratio total. We will use the subscript R to represent ratio values and the subscript A to represent actual values. j\ I\, 0 = BR 2 Ratio Total = 13
the actual total. BA (a).!i_ = R 9 WA Nuts = 11 Bolts = 2 ----- (c) 2 260 ratio (a) replaced TA with 260 multiplied both sides by 260 simplified = WR proportion TA TR WA 7 = - 640 16 (c) R = 9 replaced TA with 640 l6wa = 7 640 cross multiplied MWA 7 640 =.l-6 16 WA = 280 acres simplified divided both sides by 16 100 Advanced Ratio Problems 311 Now we cover the bottom line and see that what remains concerns nuts and bolts. We see that the given information was enough to write three ratios. Ratio (a) is between bolts and ratio total. Ratio (b) is between nuts and ratio total. Ratio (c) is between nuts and bolts. We were told that we had a total of 260 and were asked for the number of bolts. So we want to use Ratio (a). BR= 3_ TR 13 Ratio (a) is the reduced form of the actual number of bolts to the actual total. Thus we can In ratio problems that discuss the total, it is helpful to begin by writing all three ratios. Then write a proportion showing that ratio (a) is equal to the ratio of the actual number of bolts to example 100.2 Farmer Dunncowski wanted to plant his farm with wheat and corn in the ratio of 7 acres to 9 acres. If he had 640 acres, how many should be planted in wheat? We were given the total and were asked for wheat, so we will use (b). TR 16 If he plants 280 acres with wheat, then 640-280 = 360 acres will be planted in corn. Now we can substitute and solve. BA 2 = - 260 13 = 13 BA = 40 bolts reread the problem to see which ratio should be used. solution First we write down the figures given and the ratio total. WR = 7 R = 9 TR = 16 With this information we can write three ratios. w 7