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Everything You Need to Ace Math in One Big Fat Notebook: The Complete Middle School Study Guide
Everything You Need to Ace Math in One Big Fat Notebook: The Complete Middle School Study Guide
Altair Peterson
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It’s the revolutionary math study guide just for middle school students from the brains behind Brain Quest.
Everything You Need to Ace Math . . . covers everything to get a student over any math hump: fractions, decimals, and how to multiply and divide them; ratios, proportions, and percentages; geometry; statistics and probability; expressions and equations; and the coordinate plane and functions.
The BIG FAT NOTEBOOK™ series is built on a simple and irresistible conceit—borrowing the notes from the smartest kid in class. There are five books in all, and each is the only book you need for each main subject taught in middle school: Math, Science, American History, English Language Arts, and World History. Inside the reader will find every subject’s key concepts, easily digested and summarized: Critical ideas highlighted in neon colors. Definitions explained. Doodles that illuminate tricky concepts in marker. Mnemonics for memorable shortcuts. And quizzes to recap it all.
The BIG FAT NOTEBOOKS meet Common Core State Standards, Next Generation Science Standards, and state history standards, and are vetted by National and State Teacher of the Year Award–winning teachers. They make learning fun and are the perfect next step for every kid who grew up on Brain Quest.
Everything You Need to Ace Math . . . covers everything to get a student over any math hump: fractions, decimals, and how to multiply and divide them; ratios, proportions, and percentages; geometry; statistics and probability; expressions and equations; and the coordinate plane and functions.
The BIG FAT NOTEBOOK™ series is built on a simple and irresistible conceit—borrowing the notes from the smartest kid in class. There are five books in all, and each is the only book you need for each main subject taught in middle school: Math, Science, American History, English Language Arts, and World History. Inside the reader will find every subject’s key concepts, easily digested and summarized: Critical ideas highlighted in neon colors. Definitions explained. Doodles that illuminate tricky concepts in marker. Mnemonics for memorable shortcuts. And quizzes to recap it all.
The BIG FAT NOTEBOOKS meet Common Core State Standards, Next Generation Science Standards, and state history standards, and are vetted by National and State Teacher of the Year Award–winning teachers. They make learning fun and are the perfect next step for every kid who grew up on Brain Quest.
درجه (قاطیغوری(:
کال:
2016
خپرونه:
1
خپرندویه اداره:
Workman Publishing Co., Inc
ژبه:
english
صفحه:
529
ISBN 10:
0761160965
ISBN 13:
9781523504404
ISBN:
15 14 13 12 11 10 9 8 7
لړ (سلسله):
Big Fat Notebooks
فایل:
ستاسی تیګی:
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graph^{92}
angles^{88}
equation^{85}
slope^{78}
decimal^{71}
calculate^{69}
variable^{63}
coordinate^{61}
triangle^{58}
multiply^{57}
plot^{53}
probability^{50}
input^{47}
equations^{44}
output^{44}
ratio^{41}
fractions^{39}
formula^{39}
tax^{37}
coordinates^{37}
parentheses^{36}
units^{36}
multiplication^{36}
fraction^{36}
cube^{35}
divide^{34}
factors^{33}
discount^{33}
cups^{33}
in2^{32}
subtract^{32}
approximately^{32}
exponent^{31}
sales^{30}
notation^{30}
proportion^{29}
congruent^{29}
rectangle^{27}
multiplying^{27}
simplify^{27}
median^{27}
simultaneous^{27}
coordinate plane^{26}
commission^{26}
spaces^{25}
outcomes^{25}
equals^{25}
ratios^{24}
prism^{24}
vertical^{24}
divisible^{23}
convert^{23}
variables^{23}
meters^{23}
decimals^{23}
axis^{23}
inverse^{23}
radius^{22}
cylinder^{22}
subtraction^{22}
11 comments
science
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25 July 2020 (01:49)
science
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26 July 2020 (21:06)
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Thank you.
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18 June 2021 (22:02)
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16 July 2021 (06:53)
تاسی کولی شی د کتاب په اړوند نظر څرګند کړی او خپله تجربه زمونږ سره شریکه کړی، نورو لوستونکو ته به زړه راښکونکی (دلچسپه) وی چې د لوستل شوو کتابونو په اړوند ستاسی په نظر پوه شی. بدون له دی چې کتاب مو خوښ شو اویا خوش نه شو، که تاسی صادقانه په دی اړوند مفصله قصه وکړی، خلک کولی شی د ځان لپاره نوی کتابونه بیدا کړی، چې هغوی ته زړه راښکونکی (دلچسپه) دی.
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MATH Copyright © 2016 by Workman Publishing Co., Inc. By purchasing this workbook, the buyer is permitted to reproduce pages for classroom use only, but not for commercial resale. Please contact the publisher for permission to reproduce pages for an entire school or school district. With the exception of the above, no portion of this book may be reproducedmechanically, electronically, or by any other means, including photocopyingwithout written permission of the publisher. Library of Congress CataloginginPublication Data is available. ISBN 9781523504404 Writer Altair Peterson Illustrator Chris Pearce Series Designer Tim Hall Designers Gordon Whiteside, Abby Dening Art Director Colleen AF Venable Editor Nathalie Le Du Production Editor Jessica Rozler Production Manager Julie Primavera Concept by Raquel Jaramillo Workman books are available at special discounts when purchased in bulk for premiums and sales promotions as well as for fundraising or educational use. Special editions or book excerpts can also be created to specification. For details, contact the Special Sales Director at the address below, or send an email to specialmarkets@workman.com. Workman Publishing Co., Inc. 225 Varick Street New York, NY 100144381 workman.com WORKMAN, BRAIN QUEST, and BIG FAT NOTEBOOK are registered trademarks of Workman Publishing Co., Inc. Printed in China First printing August 2016 15 14 13 12 11 10 9 8 7 mailto:specialmarkets@workman.com MATH WORKMA N PUBLISHING N EW YO R K the complete middle school study guide Borrowed from the smartest kid in class Doublechecked by OUIDA NEWTON EVERYTHING YOU NEED TO KNOW TO ACE MATH HI! These are the notes from my math class. Oh, who am I? Well, some people said I was the smartest kid in class. I wrote everything you need to ace MATH, from FRACTIONS to the COORDINATE PLANE, and only the really important stuff in between you know, the stuff that ’s usually on the test! 12 I tried to keep ; everything organized, so I almost always: • Highlight vocabulary words in YELLOW. • Color in definitions in green highlighter. • Use BLUE PEN for important people, places, dates, and terms. • Doodle a pretty sweet pie chart and whatnot to visually show the big ideas. If you're not loving your textbook and you’re not so great at taking notes in class, this notebook will help. It hits all the major points. (But if your teacher spends a whole class talking about something that ’s not covered, go ahead and write that down for yourself.) Now that I’ve aced math, this notebook is YOURS. I’m done with it, so this notebook’s purpose in life is to help YOU learn and remember just what you need to ace YOUR math class. ZZZ...WHAT? MMM...PIE CONTENTS Unit 1: The NUMBER SYSTEM 1 1. Types of Numbers and the Number Line 2 2. Positive and Negative Numbers 11 3. Absolute Value 19 4. Factors and Greatest Common Factor 25 5. Multiples and Least Common Multiple 33 6. Fraction Basics: Types of Fractions, and Adding and Subtracting Fractions 39 7. Multiplying and Dividing Fractions 49 8. Adding and Subtracting Decimals 55 9. Multiplying Decimals 57 10. Dividing Decimals 61 11. Adding Positive and Negative Numbers 65 12. Subtracting Positive and Negative Numbers 71 13. Multiplying and Dividing Positive and Negative Numbers 75 14. Inequalities 79 Unit 2: RATIOS, PROPORTIONS, and PERCENTS 85 15. Ratios 86 16. Unit Rate and Unit Price 91 17. Proportions 95 18. Converting Measurements 103 19. Percent 111 20. Percent Word Problems 117 21. Taxes and Fees 123 22. Discounts and Markups 131 23. Gratuity and Commission 143 24. Simple Interest 147 25. Percent Rate of Change 155 26. Tables and Ratios 159 Unit 3: EXPRESSIONS and EQUATIONS 165 27. Expressions 166 28. Properties 173 29. Like Terms 183 30. Exponents 189 31. Order of Operations 197 32. Scientific Notation 203 33. Square and Cube Roots 209 34. Comparing Irrational Numbers 215 35. Equations 219 36. Solving for Variables 225 37. Solving Multistep Equations 231 38. Solving and Graphing Inequalities 237 39. Word Problems with Equations and Inequalities 243 Unit 4: GEOMETRY 251 40. Introduction to Geometry 252 41. Angles 267 42. Quadrilaterals and Area 277 43. Triangles and Area 287 44. The Pythagorean Theorem 295 45. Circles, Circumference, and Area 301 46. ThreeDimensional Figures 309 47. Volume 318 48. Surface Area 327 49. Angles, Triangles, and Transversal Lines 337 50. Similar Figures and Scale Drawings 345 Unit 5: STATISTICS and PROBABILITY 355 51. Introduction to Statistics 356 52. Measures of Central Tendency and Variation 365 53. Displaying Data 375 54. Probability 395 UNIT 6: The COORDINATE PLANE and FUNCTIONS 405 55. The Coordinate Plane 406 56. Relations, Lines, and Functions 417 57. Slope 431 58. Linear Equations and Functions 446 59. Simultaneous Linear Equations and Functions 456 60. Nonlinear Functions 468 61. Polygons and the Coordinate Plane 480 62. Transformations 487 63. Proportional Relationships and Graphs 508 I HEARD THERE WAS CHEESE SOMEWHERE IN THIS BOOK . . . 1 Unit The Number System 1 2 Chapter 1 TYPES of NUMBERS and the NUMBER LINE There are many different types of numbers with different names. Here are the types of numbers used most often: WHOLE NUMBERS: A number with no fractional or decimal part. Cannot be negative. EXAMPLES: 0, 1, 2, 3, 4... NATURAL NUMBERS: Whole numbers from 1 and up. Some teachers say these are all the “counting numbers.” EXAMPLES: 1, 2, 3, 4, 5... 3 INTEGERS: All whole numbers (including positive and negative whole numbers). EXAMPLES: .. . 4, 3, 2, 1, 0, 1, 2, 3, 4... RATIONAL NUMBERS: Any number that can be written by dividing one integer by anotherin plain English, any number that can be written as a fraction or ratio. (An easy way to remember this is to think of rational’s root word “ratio.”) EXAMPLES: , (which equals 0.5), 0.25 (which equals ), 7 (which equals ), 4.12 (which equals ), (which equals 0.3) IRRATIONAL NUMBERS: A number that cannot be written as a simple fraction (because the decimal goes on forever without repeating). EXAMPLES: 3.14159265... , 2 Every number has a decimal expansion. For example, 2 can be writ ten 2.000... However, you can spot an irrational number because the decimal expansion goes on forever without repeating. THE LINE OVER THE 3 MEANS THAT IT REPEATS FOREVER! (“. . .” MEANS THAT IT CONTINUES ON FOREVER) 1 4 1 2 7 1 412 100 1 3 4 REAL NUMBERS: All the numbers that can be found on a number line. Real numbers can be large or small, positive or negative, decimals, fractions, etc. EXAMPLES: 5, 17, 0.312, , π, 2 , etc. EXAMPLE: 2 is an integer, a rational number, and a real number! REAL NUMBERS IRRATIONAL NUMBERS RATIONAL NUMBERS INTEGERS WHOLE NUMBERS NATURAL NUMBERS Here’s how all the types of numbers fit together: 1 2 5 SOME OTHER EXAMPLES: 46 is natural, whole, an integer, rational, and real. 0 is whole, an integer, rational, and real. is rational and real. 6.675 is rational and real. (TERMINATING DECIMALS or decimals that end are rational.) 5 = 2.2360679775... is irrational and real. (Nonrepeating decimals that go on forever are irrational.) 1 4 6 RATIONAL NUMBERS AND THE NUMBER LINE All rational numbers can be placed on a NUMBER LINE. A number line is a line that orders and compares numbers. Smaller numbers are on the left, and larger numbers are on the right. EXAMPLE: Because 2 is larger than 1 and also larger than 0, it is placed to the right of those numbers. 2 33 2 1 0 1 3213 2 1 0 7 EXAMPLE: Similarly, because 3 is smaller than 2 and also smaller than 1, it is placed to the left of those numbers. EXAMPLE: Not only can we place integers on a number line, we can put fractions, decimals, and all other rational numbers on a number line, too: 3213 2 1 0 2.38 3 4 1 2 4 3213 2 1 0 3 18 8 STILL GOIN G! 9 For 1 through 8, classify each number in as many categories as possible. 1. 3 2. 4.5  3. 4.89375872537653487287439843098… 4. 9.7654321 5. 1 6. 7. 2 8. 5.678 9. Is to the left or the right of 0 on a number line? 10. Is 0.001 to the left or the right of 0 on a number line? answers 145 9 3 10 1. Integer, rational, real 2. Rational, real 3. Irrational, real 4. Rational, real 5. Natural, whole, integer, rational, real 6. Integer, rational, real (because can be rewritten as 3) 7. Irrational, real 8. Rational, real 9. To the right 10. To the left 9 3 11 POSITIVE and NEGATIVE NUMBERS POSITIVE NUMBERS are used to describe quantities greater than zero, and NEGATIVE NUMBERS are used to describe quantities less than zero. Often, positive and negative numbers are used together to show quantities that have opposite directions or values. All positive numbers just look like regular numbers (+4 and 4 mean the same thing). All negative numbers have a negative sign in front of them, like this: 4. Chapter 2 REMINDER: All positive and negative whole numbers (without fractions or decimals) are integers. 12 As we know, all integers can be placed on a number line. If you put all integers on a number line, zero would be at the exact middle because zero is neither positive nor negative. Positive and negative numbers have many uses in our world, such as: NEGATIVE Debt (money that you owe) Savings (money that you keep) 3 42134 2 1 0 POSITIVE 13 Belowzero temperatures Abovezero temperatures Below sea level Above sea level Debit from a bank account Credit to your bank account Negative electric charge Positive electric charge 14 On a horizontal number line: Numbers to the left of zero are negative, and numbers to the right of zero are positive. Numbers get larger as they move to the right, and smaller as they move to the left. We draw ARROWS on each end of a number line to show that the numbers keep going (all the way to INFINITY and negative infinity!). Positive (+) and negative (−) signs are called OPPOSITES, so +5 and −5 are also called opposites. They are both the same number of spaces or the same distance from zero on the number line, but on “opposite” sides. On a vertical number line (such as a thermometer), numbers above zero are positive, and numbers below zero are negative. 120 100 80 60 40 20 0 20 40 50 40 30 20 10 0 10 20 30 40   Infinity Something that is endless, unlimited, or without bounds THE SYMBOL FOR INFINITY IS . 15 EXAMPLE: What is the opposite of 8? −8 EXAMPLE: Devin borrows $2 from his friend Stanley. Show the amount that Devin owes as an integer. −2 The OPPOSITES OF OPPOSITES PROPERTY says that the opposite of the opposite of a number is the number itself! EXAMPLE: What is the opposite of the opposite of −16? The opposite of −16 is 16. The opposite of 16 is −16. So the opposite of the opposite of −16 is −16 (which is the same as itself). 16 17 For 1 through 5, write the integer that represents each quantity. 1. A submarine is 200 feet below sea level. 2. A helicopter is 525 feet above the landing pad. 3. The temperature is 8 degrees below zero. 4. Griselda owes her friend Matty $17. 5. Matty has $1,250 in his savings account. 6. Show the location of the opposite of 2 on the number line. 7. What is the opposite of −100? 8. Draw a number line that extends from −3 to 3. 9. What is the opposite of the opposite of 79? 10. What is the opposite of the opposite of −47? answers 18 1. −200 2. +525 (or 525) 3. −8 4. −17 5. +1,250 (or 1,250) 6. 7. 100 8. 9. 79 10. −47 5 19 The ABSOLUTE VALUE of a number is its distance from zero (on the number line). Thus, the absolute value is always positive. We indicate absolute value by putting two bars around the number. EXAMPLE:   4    4  is read “the absolute value of −4.” Because −4 is 4 spaces from zero on the number line, the absolute value is 4. EXAMPLE: 9   9  is read “the absolute value of 9.” Because 9 is 9 spaces from zero on the number line, the absolute value is 9. Chapter 3 ABSOLUTE VALUE 3213 2 1 0 7654 987 6 5 49 8 4 S PACES 9 S PACES 20 Absolute value bars are also grouping symbols, so you must complete the operation inside them first, then take the absolute value. EXAMPLE:  5 3  =  2  =2 Sometimes, there are positive or negative symbols outside an absolute value bar. Think: inside, then outsidefirst take the absolute value of what is inside the bars, then apply the outside symbol. EXAMPLE:   6  =  6 (The absolute value of 6 is 6. Then we apply the negative symbol on the outside of the absolute value bars to get the answer −6.) NOW, THIS CHANGES EVERYTHING. 21 EXAMPLE:    1 6  =  1 6 (The absolute value of −16 is 16. Then we apply the negative symbol on the outside of the absolute value bars to get the answer −16.) A number in front of the absolute value bars means multiplication (like when we use parentheses). EXAMPLE: 2   4  (The absolute value of −4 is 4.) 2•4 = 8 (Once you have the value inside the absolute value bars, you can solve normally.) Multiplication can be shown in a few different ways—not just with x . All of these symbols mean multiply: 2 x 4 = 8 2 • 4 = 8 (2 ) ( 4 ) = 8 2 ( 4 ) = 8 If you use VARIABLES, you can put variables next to each other or put a number next to a variable to indicate multiplication, like so: ab = 8 3x = 1 5VARIABLE: a letter or symbol used in place of a quantity we don’t know yet BFN_MATHRPT5817cb.indd 21 5/10/17 4:50 PM 22 I integers 23answers Evaluate 1 through 8. 1.   19  2 .  49  3.  4 . 5  4.  1 5  5.  7 3  6 .  1 • 5  7.   6 5  8 .   9  9. Johanne has an account balance of −$56.50. What is the absolute value of his debt? 10. A valley is 94 feet below sea level. What is the absolute value of the elevation difference between the valley and the sea level? 24 1. 19 2. 49 3. 4.5 4. 1 5 5. 4 6. 5 7. −65 8. −9 9. 56.50 10. 94 25 FACTORS and GREATEST COMMON FACTOR Chapter 4 FACTORS are integers you multiply together to get another integer. EXAMPLE: What are the factors of 6 ? 2 and 3 are factors of 6, because 2 x 3 = 6 1 and 6 are also factors of 6, because 1 x 6 = 6 So, the factors of 6 are: 1, 2, 3, and 6. When finding the factors of a number, ask yourself, “What numbers can be multiplied together to give me this number?” Every number greater than 1 has at least two factors, because every number can be divided by 1 and itself! I AM EVERYWHERE! 26 EXAMPLE: What are the factors of 10? (Think: “What can be multiplied together to give me 10?”) 1 • 10 2 • 5 The factors of 10 are 1, 2, 5, and 10. EXAMPLE: Emilio needs to arrange chairs for a drama club meeting at his school. There are 30 students coming. What are the different ways he can arrange the chairs so that each row has the same number of chairs? 1 row of 30 chairs 2 rows of 15 chairs 3 rows of 10 chairs 5 rows of 6 chairs 30 rows of 1 chair The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The product of each pair of numbers is 30. Even though 5 x 2 also equals 10, these numbers have already been listed, so we don’t need to list them again. THIS IS THE SAME AS SAYING, “FIN D THE FACTORS OF 30.” 27 Here are some shortcuts to find an integer’s factors: An integer is divisible by 2 if it ends in an even number. EXAMPLE: 10, 92, 44, 26, and 8 are all divisible by 2 because they end in an even number. An integer is divisible by 3 if the sum of its digits is divisible by 3. EXAMPLE: 42 is divisible by 3 because 4 +2= 6 , and 6 is divisible by 3. An integer is divisible by 5 if it ends in 0 or 5. EXAMPLE: 10, 65, and 2,320 are all divisible by 5 because they end in either 0 or 5. An integer is divisible by 9 if the sum of the digits is divisible by 9. EXAMPLE: 297 is divisible by 9 because 2+9 +7 = 18, and 18 is divisible by 9. An integer is divisible by 10 if it ends in 0. EXAMPLE: 50, 110, and 31,330 are all divisible by 10 because they end in 0. 28 Prime Numbers A PRIME NUMBER is a number that has only two factors (the number itself and 1 ). Some examples of prime numbers are 2, 3, 7, and 13. Common Factors Any factors that are the same for two (or more) numbers are called COMMON FACTORS. EXAMPLE: What are the common factors of 12 and 18? The factors for 12 are 1, 2, 3, 4, 6, 12. The factors for 18 are 1, 2, 3, 6, 9, 18. The common factors of 12 and 18 (factors that both 12 and 18 have in common) are 1, 2, 3, and 6. The largest factor that both numbers share is called the GREATEST COMMON FACTOR, or GCF for short. The GCF of 12 and 18 is 6. 2 IS ALSO THE ONLY EVEN PRIME NUMBER. 29 EXAMPLE: What is the GCF of 4 and 10? Factors of 4 are 1, 2, 4. Factors of 10 are 1, 2, 5, 10. So the GCF of 4 and 10 is 2. EXAMPLE: What is the GCF of 18 and 72? Factors of 18 are 1, 2, 3, 6, 9, 18. Factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. 18 is the GCF of 18 and 72. YEAH, I KNEW HIM WHEN HE WAS JUST A PRIME NUMBER... HE'S NOT SO GREAT. 30 SO, SMART GUY, WHAT IS THIS DIVISIBLE BY? 356,724,921, 213,691,753, 611,219,398 ARGH! !! RIP! 31answers 1. What are the factors of 12? 2. What are the factors of 60? 3. Is 348 divisible by 2? 4. Is 786 divisible by 3? 5. Is 936 divisible by 9? 6. Is 3,645,211 divisible by 10? 7. Find the greatest common factor of 6 and 20. 8. Find the greatest common factor of 33 and 74. 9. Find the greatest common factor of 24 and 96. 10. Sara has 8 redcolored pens and 20 yellowcolored pens. She wants to create groups of pens such that there are the same number of redcolored pens and yellowcolored pens in each group and there are no pens left over. What is the greatest number of groups that she can create? 32 1. 1, 2, 3, 4, 6, and 12 2. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60 3. Yes, because 348 ends in an even number. 4. Yes, because 7 + 8 + 6 = 21, and 21 is divisible by 3. 5. Yes, because 9 + 3 + 6 = 18, and 18 is divisible by 9. 6. No, because it does not end in a 0. 7. 2 8. 1 9. 24 10. 4 groups. (Each group has 2 redcolored pens and 5 yellowcolored pens.) 33 When we multiply a number by any whole number (that isn’t 0), the product is a MULTIPLE of that number. Every number has an infinite list of multiples. EXAMPLE: What are the multiples of 4? 4 x 1 = 4 4 x 2= 8 4x 3 = 1 2 4x4 = 1 6 and so on…forever! The multiples of 4 are 4, 8, 12, 16… MULTIPLES AND LEAST COMMON MULTIPLE Chapter 5 34 Any multiples that are the same for two (or more) numbers are called COMMON MULTIPLES. EXAMPLE: What are the multiples of 2 and 5? The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20… The multiples of 5 are 5, 10, 15, 20… Up until this point, 2 and 5 have the multiples 10 and 20 in common. What is the smallest multiple that both 2 and 5 have in common? The smallest multiple is 10. We call this the LEAST COMMON MULTIPLE, or LCM. To find the LCM of two or more numbers, list the multiples of each number in order from least to greatest until you find the first multiple they both have in common. EXAMPLE: Find the LCM of 9 and 11. The multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 99, 108… The multiples of 11 are 11, 22, 33, 44, 55, 66, 77, 88, 99, 110… 99 is the first multiple 9 and 11 have in common, so the LCM of 9 and 11 is 99. 35 Sometimes, it ’s easier to start with the bigger number. Instead of listing all of the multiples of 9 first, start with the multiples of 1 1 , and ask yourself, “Which of these numbers is divisible by 9 ?” EXAMPLE: Susie signs up to volunteer at the animal shelter every 6 days. Luisa signs up to volunteer at the shelter every 5 days. If they both sign up to volunteer on the same day, when is the first day that Susie and Luisa will work together? Susie will work on the following days: 6th, 12th, 18th, 24th, and 30th… 30 is the first number divisible by 5, so the LCM is 30. The first day that Susie and Luisa will work together is on the 30th day. THIS IS THE SAME AS SAYING, “FIN D THE LCM FOR 5 AN D 6.” 36 1. List the first five multiples of 3. 2. List the first five multiples of 12. 3. Find the LCM of 5 and 7. 4. Find the LCM of 10 and 11. 5. Find the LCM of 4 and 6. 6. Find the LCM of 12 and 15. 7. Find the LCM of 18 and 36. 8. Kirk goes to the gym every 3 days. Deshawn goes to the gym every 4 days. If they join the gym on the same day, when is the first day that they’ll be at the gym together? 9. Betty and Jane have the same number of coins. Betty sorts her coins in groups of 6, with no coins left over. Jane sorts her coins in groups of 8, with no coins left over. What is the least possible number of coins that each of them has? BFN_MATHRPT5817cb.indd 36 5/10/17 4:50 PM 37 10. Bob and Julia have the same number of flowers. Bob sorts his flowers in bouquets of 3, with no flowers left over. Julie sorts her flowers in bouquets of 7, with no flowers left over. What is the least possible number of flowers that each of them has? answers BFN_MATHRPT5817cb.indd 37 5/10/17 4:50 PM 38 1. 3, 6, 9, 12, 15 2. 12, 24, 36, 48, 60 3. 35 4. 110 5. 12 6. 60 7. 36 8. On the 12th day 9. 24 coins 10. 21 bouquets 39 FRACTION BASICS Fractions are real numbers that represent a part of a whole. A fraction bar separates the part from the whole like so: PART WHOLE The “part” is the NUMERATOR, and the “whole” is the DENOMINATOR. FRACTION BASICS: TYPES OF FRACTIONS, AND ADDING AND SUBTRACTING FRACTIONS Chapter 6 40 For example, suppose you cut a whole pizza into 6 pieces and eat 5 of the pieces. The “part” you have eaten is 5, and the “whole” you started with is 6. Therefore, the amount you ate is 56 of the pizza. If 3 people shared a pizza cut into 8 slices, each person would get 2 pieces, and 2 pieces would be left over. These leftover two are called the REMAINDER. REMAINDER part, quantity, or number left over after division BFN_MATHRPT5817cb.indd 40 5/10/17 4:50 PM 41 There are 3 types of fractions: 1. Proper fractions: The numerator is smaller than the denominator. EXAMPLES: , , ,  2. Improper fractions: The numerator is bigger than, or equal to, the denominator. EXAMPLES: , ,  3. Mixed numbers: There is a whole number and a fraction. EXAMPLES: 2 23 , 18 1 8 , 9 5 7 5 6 2 3 1 1,000 4 27 10 3 8 8 25 5 42 CONVERTING MIXED NUMBERS and IMPROPER FRACTIONS Remember! To CHANGE A MIXED NUMBER TO AN IMPROPER FRACTION, you will first multiply and then add. EXAMPLE: To change the mixed number 3 15 to an improper fraction, we first calculate 3 x 5 = 15 and then + 1, so that the improper fraction is 165 . To CHANGE AN IMPROPER FRACTION TO A MIXED NUMBER, you divide the numerator by the denominator. Ask yourself: “How many times does the denominator go into the numerator? What remainder do I have left over?” EXAMPLE: To change the improper fraction 23 8 to a mixed number, we calculate: 23 ÷ 8 =2R7 , so the mixed number is 2 78 . If you get an answer that is an improper fraction, always convert it into a mixed number for your final answer. Some teachers take off points if you don’t! MULTIPLY ADD “R” STAN DS FOR REMAIN DER. 43 SIMPLIFYING FRACTIONS Sometimes, the numerator and denominator will have common factors. You can SIMPLIFY them by dividing the numerator and the denominator by the greatest common factor. Some teachers call this “CROSSREDUCING,” “simplifying,” or “CANCELING.” Whatever you call it, it ’s a shortcut! EXAMPLE: 6 10 can be simplified to 3 5 because 2 is the GCF of 6 and 10. 610 = 6 ÷210÷2 = 35 EXAMPLE: 20 8 can be simplified to 5 2 because the GCF of 20 and 8 is 4. 208 = 20÷ 48÷ 4 = 52 ADDING FRACTIONS If we want to add fractions together, the denominators must be the same. EXAMPLE: 15 + 35 = 45 In the sum, the denominator stays the same and you add the numerator. For example, you have two identical candy Most teachers want you to simplify your answers if possible, so get in the habit! 44 bars, and you cut each into 5 pieces. You give your lit tle brother 1 piece from the first candy bar, and you give your sister 2 pieces from the second candy bar. How much of a whole candy bar did you give away? You gave 1 of the 5 pieces of the first candy bar to your brother = 1 5 . You gave 2 of the 5 pieces of the second candy bar to your sister = 2 5 . Now, add them together: 15 + 25 = 35 ( The denomenator stays the same, and you add the numerators.) Because both candy bars are the same size and are cut into the same number of pieces, you keep the denominator as 5 and add the numerators to get the answer of 3 5 . YOU CAN REMEMBER WITH THIS RHYME: Denominator’s the same—keep it in the game! Add up the top, simplify, and stop! COOL DENOMIN ATOR! PUT IT UP THERE! 45 SUBTRACTING FRACTIONS The same idea applies to subtractionthe denominators must be the same (both wholes must be the same size) in order to subtract. EXAMPLE: 89  7 9 = 19 (The denominator stays the same, and you subtract the numerators.) ADDING and SUBTRACTING FRACTIONS with DIFFERENT DENOMINATORS In order to add or subtract fractions with different denominators, you just have to make their denominators the same! We can do that by finding the LCM of the denominators. How to add or subtract fractions with unlike denominators: Find the LCM of both denominators. (Some teachers call this the LEAST COMMON DENOMINATOR, or LCD for short.) EXAMPLE: 25 + 14 The LCM of 5 and 4 is 20. 1. 46 Convert the numerators. (Some teachers call this RENAMING the numerators.) 2• 4 5 • 4 = 820 5 times what number equals 20? 4. So, you must also multiply the numerator by 4 to convert the numerator. 1 • 5 4 • 5 = 5 20 4 times what number equals 20? 5. So, you must also multiply the numerator by 5 to convert the numerator. Add or subtract, and simplify if necessary. 25 + 14 = 8 20 + 520 = 1320 EXAMPLE: 47  13 The LCM of 7 and 3 is 21. 4 x 3 7x3 = 12 21 1 x7 3x7 = 721 47  13 = 12 21  7 21 = 5 21 2. 3. 47 Calculate. Simplify each answer if possible. 1. 18 + 28 2. 711  411 3. 35 + 35 4. 910  410 5. 1315  415 6. 35  12 7. 45  110 8. 89  36 9. 12  38 10. 56  38 answers 48 1. 3 8 2. 3 11 3. 6 5 = 1 1 5 4. 510 = 1 2 5. 915 = 3 5 6. 1 10 7. 7 10 8. 7 18 9. 1 8 10. 11 24 49 MULTIPLYING FRACTIONS Unlike when you add and subtract fractions, the denominators do not have to be the same. To multiply fractions, first multiply the numerators. Then multiply the denominators. Simplify your answer if necessary. That’s it! EXAMPLE: 3 5 x 4 7 = 12 35 Sometimes, when multiplying fractions, you might see that a numerator and a denominator will have common factors. You can simplify them before multiplying in the same way that we simplify fractions. Some teachers call this “CROSSREDUCING” or “CANCELING.” Whatever you call it, it’s a shortcut! EXAMPLE: 1 4 • 8 9 = 29 (The GCF of 8 and 4 is 4.) Chapter 7 MULTIPLYING AND DIVIDING FRACTIONS 1 2 50 EXAMPLE: A recipe calls for 45 cup of chocolate milk, but you want to cut the recipe in half. How much chocolate milk do you need? 4 5 • 1 2 = 2 5 DIVIDING FRACTIONS To divide fractions, follow these steps: 1. Flip the second fraction to make its RECIPROCAL. 2. Change the division sign to multiplication. 3. Multiply. EXAMPLE: 3 5 ÷ 8 9 = 3 5 • 9 8 = 27 40 Don’t forget that when you are multiplying or dividing mixed numbers, you must convert them to improper fractions first! EXAMPLE: 2 13 ÷ 1 1 4 7 3 ÷ 5 4 = 7 3 x 4 5 = 28 15 = 1 13 15 1 2 A RECIPROCAL of a number is another number that, when multiplied together, their product is 1. In plain English—any number multiplied by its reciprocal equals 1. 8 1 x 1 8 = 1 2 3 x 3 2 = 1 To find the reciprocal, flip the fraction. 51 1. 3 4 • 1 2 2. 7 10 • 1 13 3. 4 5 • 1 8 4. A machine pumps 4 12 gallons of water every hour. How many gallons of water does it pump after 2 23 hours? 5. Billy jogs 45 kilometer every minute. How many kilometers does he jog after 6 18 minutes? 6. 5 7 ÷ 1 2 7. 7  8 ÷ 2 9 8. 9 12 ÷ 3 15 9. How many 34 ounce spoonfuls of sugar are in a 5 12 ounce bowl? 10. How much chocolate will each person get if 3 people share pound of chocolate equally? answers 45 52 1. 38 2. 1415 3. 110 4. 12 gallons 5. 4 910 kilometers 6. 1 37 7. 3 1516 8. 2 3132 9. 7 13 spoonfuls 10. 415 pound 5 4 3 2 1 WE NEED HALF OF THAT CHOCOLATE MILK. 5 4 3 2 1 NO PROBLEM. 53 When adding and subtracting numbers with decimals, line the decimal points up exactly on top of each other. The digits to the left of the decimal point (such as the ones, tens, and hundreds) should all line up with each other; the digits to the right of the decimal point (such as the tenths, hundredths, and thousandths) should also be aligned. Then you can add as you normally would and bring the decimal point straight down. EXAMPLE: Find the sum of 6.45 and 23.34. 6.45 +23.34 29.79 ADDING AND SUBTRACTING DECIMALS Chapter 8 54 Any time you add a whole number and a decimal, include the “invisible” decimal point to the right of the whole number. EXAMPLE: Find the sum of $5 and $3.55. 3.55 +5.00 (5 becomes 5.00.) $8.55 Do the same for subtractionalign the decimal points of each number, subtract, and drop down the decimal point. EXAMPLE: Find the difference of 14.52 and 2.4. 14.52 2.40 (2.4 becomes 2.40the value is the same.) 12.12 When adding money, everything to the left of the decimal point represents whole dollars, and everything to the right represents cents, or parts of a dollar. BFN_MATHRPT5817cb.indd 54 5/10/17 4:50 PM 55 1. $5.89 + $9.23 2. 18.1876 + 4.3215 3. 6 + 84.32 4. 1,234.56 + 8,453.234 5. 8.573 + 2.2+ 17.01 6. $67.85  $25.15 7. 100  6.781 8. 99.09  98.29 9. 14,327.81  2.6382 10. Justin goes to the mall with $120. He spends $54.67 on clothes, $13.49 on school supplies, and $8.14 on lunch. How much does he have left? answers 56 1. $15.12 2. 22.5091 3. 90.32 4. 9,687.794 5. 27.783 6. $42.70 7. 93.219 8. 0.8 9. 14,325.1718 10. $43.70 57 When multiplying decimals, you don’t need to line up the decimals. In fact, you don’t have to think about the decimal point until the very end. Steps for multiplying decimals: 1. Multiply the numbers as though they were whole numbers. 2. Include the decimal point in your answerthe number of decimal places in the answer is the same as the total number of digits to the right of the decimal point in each of the factors. MULTIPLYING DECIMALS Chapter 9 INTEGERS YOU ARE MULTIPLYING 58 EXAMPLE: 4.24 x 2.1 4.24 x2.1 424 848 8904 The total number of decimal places in 4.24 and 2.1 is 3, so the answer is 8.904. Let ’s try it again: EXAMPLE: Bruce jogs 1.2 kilometers per minute. If he jogs for 5.8 minutes, how far does he jog? 1.2 x5.8 96 60 696 The total number of decimal places in 1.2 and 5.8 is 2, so the answer is 6.96 kilometers. When counting decimal places, don’t be fooled by zeros at the end—they don’t count. 0.30 0.30 = 0.3 (only 1 decimal point) CAN’T BE COUNTED YOU DON’T NEED TO LINE UP DECIMALS! 59 1. 5.6 x 6.41 2. (3.55)(4.82) 3. 0.350 • 0.40 4. (9.8710)(3.44) 5. (1.003)(2.4) 6. 310 x 0.0002 7. 0.003 x 0.015 8. The price of fabric is $7.60 per meter. Lance bought 5.5 meters of fabric. What was the total cost? 9. Each centimeter on a map represents 3.2 meters. How many meters do 5.04 centimenters represent? 10. A gallon of gas costs $2.16. Rob buys 13.5 gallons of gas. How much did he pay? answers 60 1. 35.896 2. 17.111 3. 0.14 4. 33.95624 5. 2.4072 6. 0.062 7. 0.000045 8. $41.80 9. 16.128 meters 10. $29.16 61 You can divide decimals easily by simply making them into whole numbers. You do that by multiplying both the DIVIDEND and DIVISOR by the same power of ten. Because the new numbers are proportional to the original numbers, the answer is the same! EXAMPLE: 2.5 ÷ 0.05 = (2.5 x 100) ÷ (0.05 x 100) =250 ÷ 5 = 50 Chapter 10 DIVIDING DECIMALS CORRESPONDING IN SIZE The DIVIDEND is the number that is being divided. The DIVISOR is the number that “goes into” the dividend. The answer to a division problem is called the QUOTIENT. dividend divisor = quotient, OR dividend ÷ divisor = quotient, OR dividenddivisor quotient 62 Multiply both decimal numbers by 100, because the decimal needs to move two places in order for both the dividend and divisor to become whole numbers. Remember, every time you multiply by another power of ten, the decimal moves one more space to the right! Let ’s try another example: EXAMPLE: A car drives 21.6 miles in 2.7 hours. How many miles does it travel each hour? 21.6 2.7 = 21.6 x 10 2.7x10 = 216 27 = 8 miles Don’t be thrown off if you see decimals being divided like this: The process is the same—multiply both numbers by 10 in order for both terms to become whole numbers: 2.7 21.6 2.7 21.6 = 27 216 8 miles X 10 X 10 63 1. 7.5 ÷2.5 2. 18.4 ÷ 4.6 3. 102.84 ÷ 0.2 4. 1,250 ÷ 0.05 5. 3.98 0.4 6. 0.27 0.4 7. 1.5 3.75 8. 1.054 0.02 9. A machine pumps 8.4 gallons of water every 3.2 minutes. How many gallons does the machine pump each minute? 10. Will swims a total of 45.6 laps in 2.85 hours. How many laps does he swim each hour? answers 64 1. 3 2. 4 3. 514.2 4. 25,000 5. 9.95 6. 0.675 7. 0.4 8. 52.7 9. 2.625 gallons 10. 16 laps 65 To add positive and negative numbers, you can use a number line or use absolute value. TECHNIQUE #1: USE a NUMBER LINE Draw a number line and begin at ZERO. For a NEGATIVE (−) number, move that many spaces to the left. For a POSITIVE (+) number, move that many spaces to the right. Wherever you end up is the answer! ADDING POSITIVE AND NEGATIVE NUMBERS Chapter 11 66 EXAMPLE: −5 + 4 Begin at zero. Because −5 is negative, move 5 spaces to the left. Because 4 is positive, move 4 spaces to the right. Where did you end up? −1 is correct! EXAMPLE: −1 + (−2) Begin at zero. Move 1 space to the left. Then move 2 more spaces to the left. Where did you end up? −3 The sum of a number and its opposite always equals zero. For example, 4 + −4 = 0. Think about it like this: if you take four steps forward, then four steps backward, you end up exactly where you began, so you’ve moved zero spaces! 67 TECHNIQUE #2: USE ABSOLUTE VALUE If you need to add larger numbers, you probably don’t want to draw a number line. So, look at the signs and decide what to do: If the signs of the numbers you are adding are the same, they are alike (they go in the same direction), so you can add those two numbers together and keep their sign. EXAMPLE: −1 + ( −2) Both −1 and −2 are negative, so they are alike. We add them together and keep their sign to get 3. If the signs of the numbers you are adding are different, subtract the absolute value of each of the two numbers. Which number had a higher absolute value? The answer will have the same sign that this number had at the beginning. To remember all this, try singing this to the tune of “Row, Row, Row Your Boat.” Same sign: keep and add! Different sign: subtract! Keep the sign of the larger amount, then you’ll be exact! 68 EXAMPLE: −10 + 4 −10 and 4 have different signs, so subtract the absolute value, like so −10  4 =1 04=6. −10 had the higher absolute value, so the answer is also negative: 6. EXAMPLE: −35 + 100 −35 + 100 = 65 (Different sign, so we have to subtract! +100 had the higher absolute value, so the answer is also positive.) EXAMPLE: The temperature in Wisconsin was −8 degrees Fahrenheit in the morning. By noon, it had risen by 22 degrees Fahrenheit. What was the temperature at noon? Use integers to solve. −8 + 22 = 14 The temperature at noon was 14 degrees Fahrenheit. 69 1. −8 + 8 2. −22 + −1 3. −14 + 19 4. 28 + (−13) 5. −12 + 3 + −8 6. −54 + −113 7. −546 + 233 8. 1,256 + (−4,450) 9. It ’s 0 degrees outside at midnight. The temperature of the air drops 20 degrees in the morning hours, then gains 3 degrees as soon as the sun comes up. What is the temperature after the sun comes up? 10. Denise owes her friend Jessica $25. She pays her back $17. How much does she still owe? answers 70 1. 0 2. −23 3. 5 4. 15 5. −17 6. −167 7. −313 8. −3,194 9. −17 degrees 10. She owes $8 (−$8). 71 NEXT UP: learning to subtract positive and negative numbers. We already know that subtraction and addition are “opposites” of each other. So, we can use this shortcut: Change a subtraction problem to an addition problem by using the additive inverse, or opposite! EXAMPLE: 5  4 The additive inverse of 4 is 4, which we can change to an addition problem, like so: 5  4 = 5 + (4). 5 + (4) = 1 SUBTRACTING POSITIVE AND NEGATIVE NUMBERS Chapter 12 72 EXAMPLE: 7  10 The additive inverse of 10 is 10. 7  10 = 7 + (10) 7 + (10) = 3 EXAMPLE: 3  (1) The additive inverse of 1 is 1. 3  (1) = 3 + 1 = 4 3 + 1 = 4 EXAMPLE: A bird is flying 42 meters above sea level. A fish is swimming 12 meters below sea level. How many meters apart are the bird and the fish? The bird’s height is 42. The fish’s height is −12 . To find the difference, we should subtract: 42  (12) = 42 + 12 = 54 Answer: They are 54 meters apart. EXAMPLE: 3  14 = 3 + (14) = 17 EXAMPLE: 4  (9) + 8 = 4 + 9 + 8 = 13 73 1. 5  (−3) 2. 16  (−6) 3. −3 9 4. −83 1 5. −14  (−6) 6. −100 (−101) 7. 11  17 8. 84 183 9. −12 (−2) + 10 10. The temperature at 2:00 p.m. is 27 degrees. At 2:00 a.m., the temperature has fallen to −4 degrees. What is the difference in temperature from 2:00 p.m. to 2:00 a.m.? answers 74 1. 8 2. 22 3. −12 4. −39 5. −8 6. 1 7. −6 8. −99 9. 0 10. 31 degrees 75 Multiply or divide the numbers, then count the number of negative signs. If there are an ODD NUMBER of negative numbers, the answer is NEGATIVE. (+) x (−) = (−) (−) ÷ (+) = (−) (+) x (+) x (−) = (−) (−) ÷ (−) ÷ (−) = (−) MULTIPLYING AND DIVIDING POSITIVE AND NEGATIVE NUMBERS Chapter 13 THERE ARE 3 NEGATIVE NUMBERS, SO THE ANSWER IS NEGATIVE. 76 If there are an EVEN NUMBER of negative numbers, the answer is POSITIVE. (−) x (−) = (+) (−) ÷ (−) = (+) (−) x (+) x (−) = (+) EXAMPLES: (−4) (−7) =28 (even number of negative numbers) −11 x4 = −44 (odd number of negative numbers) 84 4 = 21 (even number of negative numbers) 2 x 2 x 2 = 8 (even number of negative numbers) THERE ARE 2 NEGATIVE NUMBERS, SO THE ANSWER IS POSITIVE. YOU’LL NEVER CHANGE ME. BLAST!. . . =x 77 1. (−2) (−8) 2. 9 • −14 3. −20x −18 4. 100 x −12 5. Joe drops a pebble into the sea. The pebble drops 2 inches every second. How many inches below sea level does it drop after 6 seconds? 6. 66 ÷ (−3) 7. −119 ÷ −119 8. 27 3 9. 9 3 ÷ −1 10. Last week, Sal’s business lost a total of $126. If he lost the same amount of money on each of the 7 days, how much money did he lose each day? answers 78 1. 16 2. −126 3. 360 4. −1,200 5. 12 inches (or −12) 6. −22 7. 1 8. −9 9. 3 10. He lost $18 each day (or −$18). 79 An inequality is a mathematical sentence that is used to compare quantities and contains one of the following signs: a < b or “a is less than b” a > b or “a is greater than b” a b or “a is not equal to b” You can use a number line to compare quantit ies. Numbers get smaller the farther you go to the left, and larger the farther you go to the right. Whichever number is farther to the left is “less than” the number on its right. INEQUALITIES Chapter 14 OPEN SIDE > VERTEX SIDE When using an inequality sign to compare two amounts, place the sign in between the numbers with the “open” side toward the greater amount and the “vertex” side toward the lesser amount. 80 EXAMPLE: Compare −2 and 4. −2 is farther to the left than 4, so −2 < 4. We can also reverse this expression and say that 4 > −2 . −2 < 4 is the same as 4 > −2 . Remember that any negative number is always less than zero, and any positive number is always greater than zero and all negative numbers. 81 Just like when we add or subtract fractions with different denominators, we have to make the denominators the same when comparing fractions. EXAMPLE: Compare  and  The LCM of 2 and 3 is 6.  1 •3 2•3 =  3 6  1 •2 3•2 =  2 6 Compare  3 6 and  2 6 .  3 6 <  2 6 therefore  1 2 <  1 3 1 2 1 3 5 6 4 6 3 6 2 6 1 6 1 0     82 There are two other inequality symbols you should know: a ≤ b or “a is less than or equal to b” a ≥ b or “a is greater than or equal to b” EXAMPLE: x ≤ 3, which means x can equal any number less than or equal to 3. 3 and any number to the left of 3 will make this number sentence true. The value of x could be 3, 2, 1, 0, −1, and so on. But x could not be 4, 5, 6, and so on. EXAMPLE: x ≥  1 2  1 2 and any number to the right of  1 2 will make this sentence true. The value of x could be 0, 1 2 , 1, and so on. But x could not be 1, 1 1 2 , and so on. 83 1. Compare −12 and 8. 2. Compare −14 and −15. 3. Compare 0 and −8. 4. Compare 0.025 and 0.026. 5. Compare 2  5 and 4 5 . 6. Compare  2 3 and  1 2 . 7. If y ≤ −4, list 3 values that y could be. 8. If m ≥ 0, list 3 values that m could NOT be. 9. Which is warmer: −5˚C, or −8˚C ? 10. Fill in the blanks: Whichever number is farther to the left on a number line is the number on its right. answers 84 1. −12 < 8 or 8 > −12 2. −14 > −15 or −15 < −14 3. 0 > −8 or −8 < 0 4. 0.025 < 0.026 or 0.026 > 0.025 5. 4 5 > 2 5 6. − 1 2 > − 2 3 7. −4 and/or any number less than −4, such as −5, −6, etc. 8. Any number less than 0, such as −1, −2, −3, etc. 9. −5˚C 10. Less than #7 and #8 have more than one correct answer. 85 2 Unit Ratios, Proportions, and Percents 86 A RATIO is a comparison of two quantities. For example, you might use a ratio to compare the number of students who have cell phones to the number of students who don’t have cell phones. A ratio can be written a few different ways. The ratio 3 to 2 can be writ ten: 3 : 2 or 3  2 or 3 to 2 Use “a” to represent the first quantity and “b” to represent the second quantity. The ratio a to b can be writ ten: a :b or a  b or a to b RATIOS Chapter 15 A fraction can also be a ratio. 87 EXAMPLES: Five students were asked if they have a cell phone. Four said yes and one said no. What is the ratio of students who do not have cell phones to students who do? 1:4 or 1  4 or 1 to 4. (Another way to say this is, “For every 1 student who does not have a cell phone, there are 4 students who do have a cell phone.”) What is the ratio of students who have cell phones to total number of students asked? 4:5 or 4  5 or 4 to 5. EXAMPLE: Julio opens a small bag of jelly beans and counts them. He counts 10 total. Among those 10, there are 2 green jelly beans and 4 yellow jelly beans. What is the ratio of green jelly beans to yellow jelly beans? And what is the ratio of green jelly beans to total number of jelly beans? The ratio of green jelly beans to yellow jelly beans in fraction form is 24 . That can be simplified to 12 . So, for every 1 green jelly bean, there are 2 yellow jelly beans. 88 The ratio of green jelly beans to the total amount is 210 . That can be simplified to 15 . So, 1 out of every 5 jelly beans in the bag is green. Just like you simplify fractions, you can also simplify ratios! Ratios are often used to make SCALE DRAWINGS— a drawing that is similar to an actual object or place but bigger or smaller. A map shows the ratio of the distance on the map to the distance in the real world. MAP SCALE: 1 INCH = 500 MILES 89 For 1 through 6, write each ratio as a fraction. Simplify if possible. 1. 2 : 9 2. 42 : 52 3. 5 to 30 4. For every 100 apples, 22 apples are rotten. 5. 16 black cars to every 2 red cars 6. 19 : 37 For 7 through 10, write a ratio in the format of a :b to describe each situation. 7. Of the 27 people surveyed, 14 live in apartment buildings. 8. In the sixth grade, there are 8 girls to every 10 boys. 9. Exactly 84 out of every 100 homes has a computer. 10. Lucinda bought school supplies for class. She bought 8 pens, 12 pencils, and 4 highlighters. What was the ratio of pens to total items? answers 90 1. 29 2. 2126 3. 16 4. 1150 5. 81 6. 1937 7. 14:27 8. 8:10 or 4:5 9. 21:25 10. 8:24 or 1:3 91 A RATE is a special kind of ratio where the two amounts being compared have different units. For example, you might use rate to compare 3 cups of flour to 2 teaspoons of sugar. The units (cups and teaspoons) are different. A UNIT RATE is a rate that has 1 as its denominator. To find a unit rate, set up a ratio as a fraction and then divide the numerator by the denominator. EXAMPLE: A car can travel 300 miles on 15 gallons of gasoline. What is the unit rate per gallon of gasoline? 300 miles : 15 gallons = 300 miles  15 gallons = 20 miles per gallon The unit rate is 20 miles per gallon. UNIT RATE AND UNIT PRICE Chapter 16 MEANS DIVIDE This means the car can travel 20 miles on 1 gallon of gasoline. 92 EXAMPLE: An athlete can swim 12 mile every 13 hour. What is the unit rate of the athlete? 12 mile : 13 mile =  = 12 x 31 = 32 = 1 12 miles per hour When the unit rate describes a price, it is called UNIT PRICE. When you’re calculating unit price, be sure to put the price in the numerator! EXAMPLE: Jacob pays $1.60 for 2 bottles of water. What is the unit price of each bottle? $1.60:2 bottles or 1.60 2 = $0.80 The unit price is $0.80 per bottle. In plain English: How many miles per hour can the athlete swim? 12 13 93 For 1 through 10, find the unit rate or unit price. 1. My mom jogs 30 miles in 5 hours. 2. We swam 100 yards in 2 minutes. 3. Juliette bought 8 ribbons for $1.52. 4. He pumped 54 gallons in 12 minutes. 5. It costs $2,104.50 to purchase 122 soccer balls. 6. A runner sprints 12 of a mile in 115 hour. 7. Linda washes 26 bowls per 4 minutes. 8. Safira spends $42 for 12 gallons of gas. 9. Nathaniel does 240 pushups in 5 minutes. 10. A team digs 12 holes every 20 hours. Check Your Knowledge answers w 94 1. 6 miles per hour 2. 50 yards per minute 3. $0.19 per ribbon 4. 4.5 gallons per minute 5. $17.25 per soccer ball 6. 7 12 miles per hour 7. 6.5 bowls per minute 8. $3.50 per gallon of gas 9. 48 pushups per minute 10. 0.6 holes per hour 95 A PROPORTION is a number sentence where two ratios are equal. For example, someone cuts a pizza into 2 equal pieces and eats 1 piece. The ratio of pieces that person ate to the original pieces of pizza is 12 . The number 12 . is the same ratio as if that person instead cut the pizza into 4 equal pieces and ate 2 pieces. 1  2 = 2  4 Chapter 17 PROPORTIONS 96 You can check if two ratios form a proportion by using cross products. To find cross products, set the two ratios next to each other, then multiply diagonally. If both products are equal to each other, then the two ratios are equal and form a proportion. 1  2 2  4 1 x 4 = 4 2x 2 = 4 4 = 4 The cross products are equal, so 1  2 = 2  4 . EXAMPLE: Are 3  5 and 9  15 proportional? 3  5 9  15 3 x 15 = 45 9 x 5 = 45 45 = 45 3  5 and 9  15 ARE proportionaltheir cross products are equal. Two ratios that form a proportion are called EQUIVALENT FRACTIONS. SOMETIMES, TEACHERS ALSO CALL THIS CROSS MULTIPLICATION. 97 You can also use a proportion to FIND AN UNKNOWN QUANTITY. For example, you are making lemonade, and the recipe says to use 5 cups of water for every lemon you squeeze. How many cups of water do you need if you have 6 lemons? First, set up a ratio: 5 cups1 lemon Second, set up a ratio for what you are trying to figure out. Because you don’t know how many cups are required for 6 lemons, use x for the amount of water. x cups 6 lemons Third, set up a proportion by setting the ratios equal to each other: 5 cups 1 lemon x cups 6 lemons Last, use cross products to find the missing number! 1•x = 5 x 6 1•x = 30 (Divide both sides by 1 so you can get x alone.) x = 30 You need 30 cups for 6 lemons! NOTICE THAT THE UNITS ACROSS FROM EACH OTHER MATCH. 98 EXAMPLE: You drive 150 miles in 3 hours. At this rate, how far would you travel in 7 hours? 150 miles 3 hours = x miles 7 hours 150•7 = 3 •x 1,050 = 3x (Divide both sides by 3 so you can get x alone.) 350 = x You’ll travel 350 miles in 7 hours. Sometimes, a proportion stays the same, even in different scenarios. For example, Tim runs 12 a mile, and then he drinks 1 cup of water. If Tim runs 1 mile, he needs 2 cups of water. If Tim runs 1.5 miles, he needs 3 cups of water (and so on). The proportion stays the same, and we multiply by the same number in each scenario (in this case, we multiply by 2). This is known as the CONSTANT OF PROPORTIONALITY or the CONSTANT OF VARIATION and is closely related to unit rate (or unit price). Whenever you see “at this rate,” set up a proportion! 99 EXAMPLE: A recipe requires 6 cups of water for 2 pitchers of fruit punch. The same recipe requires 15 cups of water for 5 pitchers of fruit punch. How many cups of water are required to make 1 pitcher of fruit punch? We set up a proportion: 6 cups 2 pitchers = x cups 1 pitcher or 15 cups 5 pitchers = x cups 1 pitcher By solving for x in both cases, we find out that the answer is always 3 cups. We can also see unit rate by using a table. With the data from the table, we can set up a proportion: EXAMPLE: Daphne often walks laps at the track. The table below describes how much time she walks and how many laps she finishes. How many minutes does Daphne walk per lap? Total minutes walking 28 42 Total number of laps 4 6 28 minutes 4 laps = x minutes 1 lap or 42 minutes 6 laps = x minutes 1 lap Solving for x, we find out that the answer is 7 minutes. 100 1. Do the ratios 3 4 and 6 8 form a proportion? Show why or why not with cross products. 2. Do the ratios 4 9 and 6 11 form a proportion? Show why or why not with cross products. 3. Do the ratios 4 5 and 12 20 form a proportion? Show why or why not with cross products. 4. Solve for the unknown: 3 15 = 9 x . 5. Solve for the unknown: 8 5 = y  19 . Answer in decimal form. 6. Solve for the unknown: m 6.5 = 11 4 . Answer in decimal form. 7. In order to make the color pink, a painter mixes 2 cups of white paint with 5 cups of red. If the painter wants to use 4 cups of white paint, how many cups of red paint will she need to make the same color pink? 101 8. Four cookies cost $7. At this rate, how much will 9 cookies cost? 9. Three bagels cost $2.67. At this rate, how much will 10 bagels cost? 10. It rained 3.75 inches in 15 hours. At this rate, how much will it rain in 35 hours? Answer in decimal form. answers 102 1. Yes, because 2. No, because 3. No, because 4. x = 45 5. y = 30.4 6. m = 17.875 7. 10 cups 8. $15.75 9. $8.90 10. 8.75 inches 3 4 6 8 3 x 8 = 24 6 x 4 = 24 24 = 24 4 9 6 11 4 x 11= 44 6 x 9 = 54 44 = 54 4 5 12 20 4 x 20= 80 12 x 5 = 60 80 = 60 103 Sometimes, we want to change one type of measurement unit (such as inches) to another unit (such as feet). This is called CONVERTING MEASUREMENTS. STANDARD SYSTEM of MEASUREMENT In the U.S., we use the STANDARD SYSTEM of measurement. Here are some standard system measurements and their equivalent units: Length 12 inches (in) = 1 foot (ft) 3 feet (ft) = 1 yard (yd) 1,760 yards (yd) = 1 mile (mi) CONVERTING MEASUREMENTS Chapter 18 104 Weight 1 pound (lb) = 16 ounces (oz) 1 ton (t) = 2,000 pounds (lb) Capacity 1 tablespoon (tbsp) = 3 teaspoons (tsp) 1 fluid ounce (oz) = 2 tablespoons (tbsp) 1 cup (c) = 8 fluid ounces (oz) 1 pint (pt) = 2 cups (c) 1 quart (qt) = 2 pints (pt) 1 gallon (gal) = 4 quarts (qt) When converting between measurements, set up a proportion and solve. EXAMPLE: How many quarts are there in 10 pints? We already know that 1 quart is the same as 2 pints, so we use this ratio: x quarts 10 pints = 1 quart 2 pints We cross multiply to find the answer is 5 quarts. 105 EXAMPLE: How many pints are there in 64 fluid ounces? We can use ratios and proportions, and repeat this process until we end up with the right units. We already know that there are 8 fluid ounces in 1 cup, so we change from fluid ounces to cups first. x cups 64 fluid ounces = 1 cup 8 fluid ounces We cross multiply to find the answer is 8 cups. Next, we change 8 cups to pints. We already know that there are 2 cups in 1 pint, so we set up another proportion: x pints 8 cups = 1 pint 2 cups We cross multiply to find the answer is 4 pints. MAKE SURE YOUR UNITS ALWAYS MATCH HORIZONTALLY. 106 METRIC SYSTEM of MEASUREMENT Most other countries use the METRIC SYSTEM of measurement. Here are some metric system measurements and their equivalent units: Length 10 millimeters (mm) = 1 centimeter (cm) 100 centimeters (cm) = 1 meter (m) 1,000 meters (m) = 1 kilometer (km) Weight 1,000 milligrams (mg) = 1 gram (g) 1,000 grams (g) = 1 kilogram (kg) When converting between measurements, set up a proportion and solve. EXAMPLE: How many centimeters are there in 2 kilometers? We can use ratios and proportions because we already know that there are 1,000 meters in 1 kilometer: x meters 2 kilometers = 1,000 meters 1 kilometer WE ALSO USE THE METRIC SYSTEM IN SCIENCE CLASS! 107 We cross multiply to find the answer is 2,000 meters. Next, we change 2,000 meters to centimeters. We already know that there are 100 centimeters in 1 meter, so we set up another proportion: x centimeters 2,000 meters = 100 centimeters 1 meter We cross multiply to find the answer is 200,000 cm. CONVERTING BETWEEN MEASUREMENT SYSTEMS Sometimes, we want to change one type of measurement unit (such as inches) to another unit (such as centimeters). When we change units from the standard system to the metric system or vice versa, we are CONVERTING BETWEEN MEASUREMENT SYSTEMS. Here are some of the COMMON CONVERSIONS OF STANDARD TO METRIC: Length 1 inch (in) = 2.54 centimeters (cm) 3.28 feet (ft) = 1 meter (m) (approximately) 1 yard (yd) = 0.9144 meter (m) 1 mile (mi) = 1.61 kilometers (km) (approximately) 108 Weight 1 ounce (oz) = 28.349 grams (g) (approximately) 1 pound (lb) = 453.592 grams (g) (approximately) 1 pound (lb) = 0.454 kilograms (kg) (approximately) Capacity 1 fluid ounce (fl oz) = 29.574 milliliters (ml) (approximately) 1 pint (pt) = 473.177 milliliters (ml) (approximately) 1 pint (pt) = 0.473 liters (l) (approximately) 1 gallon (gal) = 3.785 liters (l) (approximately) When converting between measurement systems, just set up a proportion and solve. EXAMPLE: How many gallons are in 12 liters? First, set up a proportion with the unknown quantity as x. 1 gallon 3.785 liters = x gallons 12 liters 3.785x = 12 x = approximately 3.17 gallons So, there are roughly 3 gallons in 12 liters! (Divide both sides by 3.785 to isolate x on one side of the equal sign.) Next, use cross products to find the missing number. 109 For 1 through 8, fill in the blanks. 1. 26 feet = _____ inches 2. _____ gallons = 24 quarts 3. 30 teaspoons = _____ fluid ounces 4. _____ millimeters = 0.08 kilometers 5. 30 centimeters = _____ inches 6. 4.5 miles = _____ feet 7. _____ grams = 36 ounces 8. 5.25 pints = _____ liters 9. While hiking a trail that is 7 miles long, you see a sign that says, “Distance you’ve traveled: 10,000 feet.” How many feet remain in the hike? 10. Mount Everest, on the border of Nepal, is 8,848 meters tall, while Chimborazo in Ecuador is 6,310 meters tall. What is the difference in elevation between the two mountains in feet? answers 110 1. 312 2. 6 3. 5 4. 80,000 5. Approximately 11.81 6. 23,760 7. Approximately 1,020.564 8. Approximately 2.48325 9. 26,960 10. Approximately 8,325.64 111 PERCENT means “per hundred.” Percentages are ratios that compare a quantity to 100. For example, 33% means “33 per hundred” and can also be writ ten 33 100 or 0.33. EXAMPLES of a percent as a fraction: 3% = 3 100 25% = 25 100 = 1 4 EXAMPLES of a fraction as a percent: 11 100 = 11% 1 5 = 20 100 = 20% PERCENT Chapter 19 SHORTCUT: Any time you have a percent, you can put the number over 100 and get rid of the % sign. Don’t forget to simplify the fraction if possible! THIS IS A PROPORTION! 112 EXAMPLES of a percent as a decimal: 65% = 65 100 = 0.65 6.5% = 6.5 100 = 0.065 To turn a fraction into a percent, divide the NUMERATOR (top of the fraction) by the DENOMINATOR (bottom of the fraction). EXAMPLE: 14 50 = 14 ÷ 50 = 0.28 = 28% (Once you get the decimal form of the answer, move the decimal two spaces to the right, then include the % sign at the end.) SHORTCUT: When dividing by 100, just move the decimal point two spaces to the left! REMEMBER: Any number that doesn’t have a decimal point has an “invisible” decimal point at the far right of the number: 14 is the same as 14.0. RIGHT, BOSS!WE ARE INVISIBLE! WE LURK IN THE SHADOWS! RIGHT, ZERO? 113 LET’S TRY IT AGAIN: Five out of every eight albums that Latrell owns are jazz. What percentage of his music collection is jazz? 5 8 = 5 ÷ 8 = 0.625 (Move the decimal two spaces to the right and include a percent sign.) Jazz makes up 62.5% of Latrell’s music collection. Alternative method: You can also solve problems like this by setting up a proportion, like this: 5 8 x 100 8• x = 5• 100 8• x = 500 (Divide both sides by 8 so you can get x alone.) x = 62.5 62.5% of Latrell’s music is jazz. 114 OH, BROTHER. BOSS . . . I’M AFRAID OF THE DARK . . . 115 1. Write 45% as a fraction. 2. Write 68% as a fraction. 3. Write 275% as a fraction. 4. Write 8% as a decimal. 5. Write 95.4% as a decimal. 6. Write 0.003% as a decimal. 7. 6 20 is what percent? 8. 15 80 is what percent? 9. In the school election, Tammy received 3 out of every 7 votes. What percent of the votes was this (approximate to the nearest percent)? 10. If you get 17 out of 20 questions correct on your next test, what percent of the test did you answer incorrectly? answers YOU CAN WRITE YOUR ANSWER AS AN IMPROPER FRACTION OR A MIXED NUMBER. 116 1. 45 100 = 9 20 2. 68 100 = 17 25 3. 275 100 = 11 4 or 2 3 4 4. 0.08 5. 0.954 6. 0.00003 7. 30% 8. 18.75% 9. Approximately 43% 10. 15% 117 The key to solving percent word problems is to translate the word problem into mathematical symbols first. Remember these steps, and solving them becomes much easier: STEP 1: Find the word “is.” Put an equal sign above it. This becomes the center of your equation. STEP 2: Everything that comes before the word “is” can be changed into math symbols and writ ten to the left of the equal sign. Everything that comes after the word “is” should be writ ten to the right of the = sign. PERCENT WORD PROBLEMS Chapter 20 118 STEP 3: Look for key words: “What” or “What number” means an unknown number. Represent the unknown number with a variable like x. “Of” means “multiply.” Percents can be represented as decimals, so if you see % move the decimal two spaces to the left and get rid of the percent sign. STEP 4: Now you have your number sentence, so do the math! EXAMPLE: What is 75% of 45? x = 0.75•45 x = 33.75 So, 33.75 is 75% of 45. USE X FOR “WHAT.” USE THE EQUAL SIGN FOR “IS.” USE MULTIPLICATION SYMBOL FOR “OF .” CONVERT 75% TO 0.75. 119 EXAMPLE: 13 is what percent of 25? 13 = x•25 (Divide both sides by 25 to get x alone.) 0.52 = x (To convert 0.52 to a percent, move the decimal two spaces to the right and include the % sign.) 52% = x So, 13 is 52% of 25. EXAMPLE: 4 is 40% of what number? 4 = 0.40•x (Divide both sides by 0.4 to get x alone.) 10 = x So, 4 is 40% of 10. Don’t forget to doublecheck your math, read through the word problem again, and think about whether your answer makes sense. 120 EXAMPLE: What percentage of 5 is 1.25? x•5 = 1.25 x = 0.25 So, 25% of 5 is 1.25. 121 1. What is 45% of 60? 2. What is 15% of 250? 3. What is 3% of 97? 4. 11 is what percent of 20? 5. 2 is what percent of 20? 6. 17 is what percent of 25? 7. 35 is 10% of what number? 8. 40 is 80% of what number? 9. 102,000 is 8% of what number? 10. George wants to buy a new bike, which costs $280. So far, he has earned $56. What percent of the total price has he already earned? answers 122 1. 27 2. 37.5 3. 2.91 4. 55% 5. 10% 6. 68% 7. 350 8. 50 9. 1,275,000 10. George has already earned 20% of the total price. 123 TAXES TAXES are fees charged by the government to pay for creating and taking care of things that we all share, like roads and parks. SALES TAX is a fee charged on something purchased. The amount of sales tax we pay is usually determined by a percentage. For example, an 8% sales tax means we pay an extra 8 cents for every 100 cents ($1 ) we spend. Eight percent can also be writ ten as a ratio (8:100) or fraction ( ).8 100 The tax rate stays the same, even as the price of things change. So the more something costs, the more taxes we have to pay. That’s a proportion! Sales taxes are charged by your state and city so that they can provide their own services to the people like you who live in your state. Sales tax rates vary from state to state. TAXES AND FEES Chapter 21 124 EXAMPLE: You want to buy a sweater that costs $40, and your state’s sales tax is 8%. How much will the tax be? (There are three different ways to figure out how much you will pay.) Method 1: Multiply the cost of the sweater by the percent to find the tax. STEP 1: Change 8% to a decimal. 8% = 0.08 STEP 2: Multiply 0.08 and 40. 40 x 0.08 = 3.2 So, the tax will be $3.20. Don’t forget to include a dollar sign and use standard dollar notation when writing your final answer. $40 125 Method 2: Set up a proportion and solve to find the tax. STEP 1: 8% = 8 100 STEP 2: Set your tax equal to the proportional ratio with the unknown quantity. 8 100 = x 40 STEP 3: Cross multiply to solve. 100x = 320 x = 3.2 So, the tax will be $3.20. Method 3: Create an equation to find the answer. STEP 1: Make a question: “What is 8% of $40?” STEP 2: Translate the word problem into mathematical symbols. x = 0.08 x 40 x = 3.2 So, the tax will be $3.20. 126 Finding the Original Price You can also find the original price if you know the final price and the percent of tax. EXAMPLE: You bought new headphones. The receipt says that the total cost of headphones is $53.99, including an 8% sales tax. What was the original price of the headphones without the tax? STEP 1: Add the percent of the cost of the headphones and the percent of the tax to get the total cost percent. 100% + 8% tax = 108% STEP 2: Convert the percent to a decimal. 108% = 1.08 STEP 3: Solve for the original price. 53.99 = 1.08 • x (Divide both sides by 1.08 to get x alone.) x = 49.99 (rounding to the nearest cent) The original cost of the headphones was $49.99. YOU PAID FULL PRICE, SO THE COST OF THE HEAD PHONES IS 100% OF THE ORIGINAL PRICE. 127 FEES Other types of fees can work like a taxthe amount of the fee can be determined by a percentage of something else. EXAMPLE: A bike rental company charges a 17% late fee whenever a bike is returned late. If the regular rental fee is $65, but you return the bike late, what is the late fee, and what is the total that you have to pay? (Let ’s use Method 1 from before.) 17% = 0.17 65 x 0.17 = 11.05 So, the late fee is $11.05. To get the total that you have to pay, you add the late fee to the original rental price. $11.05 + $65 = $76.05 So, you have to pay $76.05. Finding the Original Price You can also find the original price if you know the final price and the percent of the fee. I WAS . . . ONLY A FEW . . . MINUTES LATE! 128 EXAMPLE: You rent a snowboard for the day, but have such a blast that you lose track of time and return the board late. The receipt says that the total cost of the rental was $66.08 including a 12% late fee. What was the original price of the snowboard rental without the fee? STEP 1: Add the percent of the cost of the rental and the percent of the fee to get the total cost percent: 100% + 12% tax = 112% STEP 2: Convert the percent to a decimal. 112% = 1.12 STEP 3: Solve for the original price. 66.08 = 1.12 • x x = 59 The original cost of the snowboard rental was $59.00. YOU PAID FULL PRICE, SO THE COST OF THE SNOWBOARD RENTAL IS 100% OF THE ORIGINAL PRICE. 129 1. Complete the following table. Round answers to the nearest cent. 2. You buy your favorite band’s new album. The receipt says that the total cost of the album is $11 .65, including a 6% sales tax. What was the original price of the album without the tax? answers 8% Sales Tax 8.5% Sales Tax 9.25% Sales Tax Book $12.00 Total Price (with tax) Board game $27.50 Total Price (with tax) Television $234.25 Total Price (with tax) 130 8% Sales Tax 8.5% Sales Tax 9.25% Sales Tax Book $12.00 $0.96 $1.02 $1.11 Total Price (with tax) $12.96 $13.02 $13.11 Board game $27.50 $2.20 $2.34 $2.54 Total Price (with tax) $29.70 $29.84 $30.04 Television $234.25 $18.74 $19.91 $21.67 Total Price (with tax) $252.99 $254.16 $255.92 2. $10.99 1. 131 DISCOUNTS Stores use DISCOUNTS to get us to buy their products. In any mall or store, you will often see signs such as But don’t be swayed by signs and commercials that promise to save you money. Calculate how much you will save to decide for yourself whether it ’s a good deal or not. Calculating a discount is like calculating tax, but because you are saving money, you subtract it from the original price. DISCOUNTS AND MARKUPS Other words and phrases that mean you will save money (and that you subtract the discount from the original price): savings, price reduction, markdown, sale, clearance. Chapter 22 132 EXAMPLE: A new hat costs $12.50. A sign in the window at the store says, “All items 20O/O off.” What is the discount off of the hat, and what is the new price of the hat? Method 1: Find out the value of the discount and subtract it from the original price. STEP 1: Change the percent discount to a decimal. 20% = 0.20 STEP 2: Multiply the decimal by the original amount to get the discount. 0.20 x $12.50 = $2.50 STEP 3: Subtract the discount from the original price. $12.50 − $2.50 = $10 The new price of the hat is $10.00. Method 2: Create an equation to find the answer. STEP 1: Write a question: “What is 20% of $12.50?” STEP 2: Translate the word problem into mathematical symbols. x = 0.20 • 12.50 x = 2.5 133 STEP 3: Subtract the discount from the original price. $12.50 − $2.50 = $10 The new price of the hat is $10.00. What if you are lucky enough to get an additional discount after the first? Just deal with one discount at a time! EXAMPLE: Valery’s Videos is selling all games at a 25% discount. However, you also have a membership card to the store, which gives you an additional 15% off. What will you end up paying for $100 worth of video games? Let ’s deal with the first discount: 25% = 0.25 0.25 x $100 = $25 So, the first discount is $25. $100 − $25 = $75 The first discounted price is $75. Now, we can calculate the additional 15% discount from the membership card. 134 15% = 0.15 0.15 x $75 = $11.25 So, the second discount is $11.25. $75 − $ 11.25 = $63.75 The final price is $63.75. That ’s a pretty good deal! Finding the Original Price You can also find the original price if you know the final price and the discount. EXAMPLE: A video game is on sale for 30% off of the regular price. If the sale price is $41.99, what was the original price? STEP 1: Subtract the percent of the discount from the percent of the original cost: 100%  30% = 70% STEP 2: Convert the percent to a decimal. 70% = 0.7 STEP 3: Solve for the original price. 41.99 = 0.7 • x (Divide both sides by 0.7 to get x alone.) (DON’T FORGET THAT THE SECOND DISCOUNT IS ADDITIONAL, SO IT’S CALCULATED BASED ON THE FIRST DISCOUNTED PRICE—NOT THE ORIGINAL PRICE.) UNLIKE THE EXAMPLES IN THE LAST CHAPTER, YOU DID NOT PAY FULL PRICE—YOU PAID ONLY 70% OF THE ORIGINAL PRICE. SWEET DEAL! 135 x = 59.99 (rounding to the nearest cent) The original price of the video game was $59.99. Finding the Percent Discount Similarly, you can also find the percent discount if you know the final price and the original price. EXAMPLE: Julie paid $35 for a shirt that is on sale. The original price was $50. What was the percent discount? 35 = x• 50 (Divide both sides by 50 to get x alone.) x = 0.7 (This tells us Julie paid 70% of the original price for the shirt.) 1  0.7 = 0.3 (We need to subtract the percent paid from the original price to find the percent discount.) The discount was 30% off of the original price. MARKUPS Stores often offer discounts during sales. But if they did that all the time, they would probably go out of business. In fact, stores and manufacturers usually increase the price of their products to make a profit. These increases are known as MARKUPS. 136 EXAMPLE: A video game costs $40 to make. To make a profit, a manufacturer marks it up 20%. What is the markup amount? What is the new price of the game? Method 1: Find out the value of the markup. STEP 1: Change the percent discount to a decimal. 20% = 0.20 STEP 2: Multiply the decimal by the original cost. This is the markup. 0.20 x $40 = $8 STEP 3: Add the markup price to the original cost. $40 + $8 = $48 The new price of the game is $48. Method 2: Create an equation to find the answer. STEP 1: Write a question: “What is 20% of $40?” STEP 2: Translate the word problem into mathematical symbols. x = 0.20 • 40 x = 8 STEP 3: Add the markup price to the original cost. $40 + $8 = $48 The new price of the game is $48. 137 Finding the Original Cost Just like when you calculate for tax and fees, you can also find the original cost if you know the final price and the markup. EXAMPLE: A bakery charges $5.08 for a cake. In order to make a profit, the store marks up its goods by 70%. What is the original cost of the cake? STEP 1: Add the percent of the original cost of the cake and the percent of the markup to get the total cost percent: 100% + 70% = 170% STEP 2: Convert the percent to a decimal. 170% = 1.7 STEP 3: Solve for the original cost. 5.08 = 1.7 • x x = 2.99 (rounding to the nearest cent) The original cost of the cake was $2.99. YOU PAID THE FULL ORIGINAL COST PLUS THE STORE’S MARKUP, SO THE COST OF THE CAKE IS ACTUALLY 170% OF THE ORIGINAL COST. 138 1. A computer has a price tag of $300. The store is giving you a 15% discount for the computer. Find the discount and final price of the computer. 2. Find the discount and final price when you receive 20% off a pair of pants that costs $48.00. 3. A bike is on sale for 45% off of the regular price. If the sale price is $299.75, what was the original price? 4. At a clothing store, a sign in the window says, “Clearance sale: 15% off all items.” You find a shirt you like with an original price of $30.00; however, a sticker on the tag says, “Take an additional 10% off the final price.” How much will this shirt cost after the discounts are taken? 5. You want to buy a new truck. At dealership A, the truck you want costs $14,500, but they offer you a 10% discount. You find the same truck at dealership B, where it costs $16,000, but they offer you a 14% discount. Which dealership is offering you a better deal? 139 6. A manufacturer makes a bookshelf that costs $50. The price at the store is increased by a markup of 8%. Find the markup amount and the new price. 7. A bike mechanic makes a bike for $350. A bike shop then marks it up by 15%. What is the markup amount? What is the new price? 8. A supermarket charges $3.24 for a carton of milk. They mark up the milk by 35% in order to make a profit. What is the original cost of the milk? 9. Phoebe wants to buy a TV. Store #1 sells the TV for $300. Store #2 has a TV that costs $250, but marks up the price by 25%. From which store should Phoebe buy the TV? 10. A furniture store has a bed that costs $200 in stock. It decreased the price by 30%. It then marked up the price by 20%. What is the new price of the bed? answers 140 1. Discount = $45; New Price = $255 2. Discount = $9.60; New Price = $38.40 3. Original price = $545 4. $22.95 5. Dealership A’s truck will cost $13,050. Dealership B’s truck will cost $13,760. Dealership A is the better deal. 6. Markup = $4; New Price = $54 7. Markup = $52.50; New Price = $402.50 8. Original Price = $2.40 9. Store #1 = $300; Store #2 = $312.50. Phoebe should buy the TV from Store #1. 10. Original price = $200; Discount Amount = $60; New price after discount = $140. Markup Amount = $28; New price after the markup = $168 141 A GRATUITY is a “tip”a gift, usually in the form of money that you give someone in return for his or her service. We usually talk about tips and gratuity in regard to servers at restaurants. A COMMISSION is a fee paid to someone for his or her services in helping to sell something to a customer. We usually talk about commissions in regard to salespeople at stores. In both cases, how much you pay usually depends on the total cost of the meal or item you purchased. You can calculate gratuity and commission just like sales tax. GRATUITY AND COMMISSION Chapter 23 142 EXAMPLE OF GRATUITY: At the end of a meal, your server brings the final bill, which is $25. You want to leave a 15% gratuity. How much is the tip in dollars, and how much should you leave in total? 15% = 0.15 $25 x 0.15 = $3.75 The tip is $3.75. $25 + 3.75 = $28.75 The total you should leave is $28.75. Again, the more your bill is, the more the gratuity or commission will be— they have a proportional relationship. UHOH... 143 EXAMPLE OF COMMISSION: My sister got a summer job working at her favorite clothing store at the mall. Her boss agreed to pay 12% commission on her total sales. At the end of her first week, her sales totaled $3,500. How much did she earn in commission? 12% = 0.12 $3,500 x 0.12 = $420.00 She earned $420. Alternative method: You can also solve these problems by setting up proportions, like this: 12 100 = x 3,500 100x = 42,000 x = $420 144 1. The Lee family eats dinner at a restaurant for a total bill of $45. They decide to give a tip of 18%. How much tip will they give? 2. A saleswoman will receive 35% commission of her total sales. She makes a total of $6,000. What is the commission that she will receive? 3. A business pays a catering company $875 for a special event. The business decides to give the catering company a tip of 25%. How much is the tip, and how much does the business pay in total to the catering company? 4. Mr. and Mrs. Smith pay their babysit ter a total of $70. They also decide to give a tip of 32%. How much is the tip, and how much do Mr. and Mrs. Smith pay the babysit ter? 5. If you give your hairdresser a 10% tip on a $25 haircut, how much will the total cost be? 6. The bill for dinner at Zolo’s Restaurant is $32.75. You decide to leave a 17% gratuity. What is the total amount of money that you will pay? 145 7. Julio gets a job selling motor scooters and is paid 8% commission on all his sales. At the end of the week, Julio’s sales are $5,450. How much has he earned in commission? 8. Amber’s boss tells her that she can choose whether she wants to be paid 12% commission or a flat fee (onetime payment) of $500. Her total sales for the period are $3,950. Which should she select? 9. Mauricio and Judith are salespeople at different stores, and both are paid on commission. Mauricio earns 8% commission on his total sales, and Judith earns 9.5% commission. Last month, Mauricio sold $25,000, while Judith sold $22,000. Who earned more? 10. Luke is a waiter at a restaurant. He receives an 18% tip from a group whose bill is $236. Mary is an electronics salesperson next door. She receives a 12% commission from selling a total of $380 worth of electronics equipment. Who received more money? answers 146 1. $8.10 2. $2,100 3. Tip = $218.75; Total = $1,093.75 4. Tip = $22.40; Total = $92.40 5. $27.50 6. $38.32 7. $436 8. Amber’s commission would be $474, so she should choose the $500 flat fee. 9. Mauricio earned $2,000 in commission, and Judith earned $2,090 in commission. Judith earned more. 10. Luke received a tip of $42.48. Mary received a commission of $45.60. So, Mary received more money than Luke. 147 INTEREST is a fee that someone pays in order to borrow money. Interest functions in two ways: 1. A bank may pay you interest if you put your money into a savings account. Depositing your money in the bank makes the bank stronger and allows them to lend money to other people, so they pay you interest for that service. 2. You may pay interest to a bank if you borrow money from themit’s a fee they charge so that you can use somebody else’s money before you have your own. SIMPLE INTEREST Chapter 24 148 You need to know three things to determine the amount of interest that must be paid (if you are the BORROWER) or earned (if you are the LENDER): 1. PRINCIPAL: The amount of money that is being borrowed or loaned 2. INTEREST RATE: The percentage that will be paid for every year the money is borrowed or loaned 3. TIME: The amount of time that money will be borrowed or loaned Once you have determined the principal, rate, and time, you can use this SIMPLE INTEREST FORMULA: interest = principal × interest rate × time I = P • R • T If you are given weeks, months, or days, write a fraction to calculate interest in terms of years. Examples: 9 months = 9— 12 year 80 days = 80— 365 year 10 weeks = 10— 52 year 149 EXAMPLE: You deposit $200 into a savings account that offers a 5% interest rate. How much interest will you have earned at the end of 3 years? Principal (P) = $200 Rate (R) = 5% = 0.05 Time (T ) = 3 years Now, substitute these numbers into the formula, and solve! I = P • R • T I = ($200) (0.05) (3) I = $30 After 3 years, you would earn an extra $30. Not bad for just letting your money sit in a bank for a few years! BALANCE is the total amount when you add the interest and beginning principal together. Simple interest can also be thought of like a ratio. 5% interest = 5 — 100 So, for every $100 you deposit, the bank will pay you $5 each year. Then you multiply $5 by the number of years. ALWAYS CHANGE A PERCENT TO A DECIMAL WHEN CALCULATING! INTERESTING . . . BFN_MATHRPT5817cb.indd 149 5/10/17 4:50 PM 150 EXAMPLE: In order to purchase your first used car, you need to borrow $11,000. Your bank agrees to loan you the money for 5 years if you pay 3.25% interest each year. How much interest will you have paid after the 5 years? What will be the total cost of the car? P = $11,000 R = 3.25% = 0.0325 T = 5 years I = P • R • T I = ($11,000) (0.0325) (5) I = $1,787.50 You’ll have to pay $1,787.50 in interest alone! With this in mind, what will be the total price of the car? $11,000 + $1,787.50 = $12,787.50 The car will cost $12,787.50 in total. 151 EXAMPLE: Joey has $3,000. He deposits it in a bank that offers an annual interest rate of 4%. How long does he need to leave it in the bank in order to earn $600 in interest? I = $600 P = $3,000 R = 4% (use .04) T = x I = P • R • T $600 = $3,000 (.04)T $600 = $120T 5 = T So, Joey will earn $600 after 5 years. (In this case, we know what the interest will be, but we don’t know the length of time. We use x to represent time and fill in all the other information we know.) (Divide both sides by 120 to get T by itself.) BANK HAS IT BEEN 5 YEARS YET? IT’S BEEN 2 HOURS. 152 For 1 through 5: Enrique deposits $750 into a savings account that pays 4.25% annual interest. He plans to leave the money in the bank for 3 years. 1. What is the principal? 2. What is the interest rate? (Write your answer as a decimal.) 3. What is the time? 4. How much interest will Enrique earn after 3 years? (Round up to the nearest cent.) 5. What will be Enrique’s balance after 3 years? For 6 through 9: Sabrina gets a car loan for $7,500 at 6% interest for 3 years. 6. How much interest will she pay over the 3 years? 153 7. Mario also gets a car loan for $7,500; however, his interest rate is 6% for 5 years. How much interest will Mario pay over the 5 years? 8. How much more interest does Mario pay than Sabrina in order to borrow the same amount of money at the same interest rate over 5 years instead of 3? 9. What does your answer for #8 tell you about borrowing money? 10. Complete the following chart: INTEREST PRINCIPAL INTEREST RATE TIME $2,574.50 5.5% 2 years $2,976.00 $6,200.00 12% answers 154 1. $750 2. 0.0425 3. 3 years 4. $95.63 5. $845.63 6. $1,350 7. $2,250 8. $900 9. The longer you borrow money, the more interest you must pay. Interest Principal Interest Rate Time $283.20 $2,574.50 5.5% 2 years $2,976.00 $6,200.00 12% 4 years 10. 155 Chapter 25 Sometimes, it is difficult to tell whether a change in the amount of something is a big deal or not. We use PERCENT RATE OF CHANGE to show how much an amount has changed in relation to the original amount. Another way to think about it is simply as the rate of change expressed as a percent. To calculate the percent rate of change: First, set up this ratio: CHANGE IN QUANTITY ORIGINAL QUANTITY (The “change in quantity” is the difference between the original and new quantity.) PERCENT RATE OF CHANGE When the original amount goes up, we calculate percent increase. When the original amount goes down, we calculate percent decrease. 156 Second , divide. Last, move the decimal two spaces to the right and add your % symbol. EXAMPLE: A store purchases Tshirts from a factory for $20 each and sells them to customers for $23. What is the percent increase in price? 23  20  20 = 3  20 = 0.15 = 15% increase EXAMPLE: On your first history test, you get 14 questions correct. On your second test, you don’t study as much, so you get only 10 questions correct. What is the percent decrease from your first to your second test? 14  10  14 = 4  14 = 2  7 = 0.29 = 29% decrease Remember to reduce fractions whenever possible to make your calculations easier. FOR PERCENTAGES, ROUND TO THE NEAREST HUNDREDTH PLACE. 157 For 1 through 5: Are the following rates of change percent increases or decreases? 1. 7% to 17% 3. 5.0025% to 5.0021% 2. 87.5% to 36.2% 4. 92 1  2 % to 92 1  5 % 5. 31.5% to 75% 6. Find the percent increase or decrease from 8 to 18. 7. Find the percent increase or decrease from 0.05 to 0.03. 8. Find the percent increase or decrease from 2 to 2,222. 9. A bike store purchases mountain bikes from the manufacturer for $250 each. They then sell the bikes to their customers for $625. What percent of change is this? 10. While working at the taco shop, Gerard noticed that on Sunday they sold 135 tacos. However, the next day, they sold only 108. What percent of change happened from Sunday to Monday? answers 158 1. Increase 2. Decrease 3. Decrease 4. Decrease 5. Increase 6. 125% increase 7. 40% decrease 8. 111,000% increase 9. 150% increase 10. 20% decrease 159 We can use tables to compare ratios and proportions. For example, Sue runs laps around a track. Her coach records the time below: NUMBER OF LAPS TOTAL MINUTES RUN 2 6 minutes 5 15 minutes What if Sue’s coach wanted to find out how long it would take her to run 1 lap? If her speed remains constant, this is easy to calculate because we have already learned how to find unit rate! We can set up this proportion: 1  x = 2  6 Another option is to set up this proportion: 1  x = 5  15 The answer is 3 minutes. TABLES AND RATIOS Chapter 26 160 EXAMPLE: Linda and Tim are racing around a track. Their coach records their times below. Linda NUMBER OF LAPS TOTAL MINUTES RUN 1 ? 2 8 minutes 6 24 minutes Tim NUMBER OF LAPS TOTAL MINUTES RUN 1 ? 3 15 minutes 4 20 minutes caution! We can use tables only if rates are PROPORTIONAL! Otherwise, there is no ratio or proportion to extrapolate from or base our calculations on. 161 If each runner’s speed stays constant, how would their coach find out who runs faster? Their coach must complete the table and find out how much time it would take Tim to run 1 lap and how much time it would take Linda to run 1 lap, and then compare them. The coach can find out the missing times with proportions: LINDA: 1  x = 2  8 x = 4 So, it takes Linda 4 minutes to run 1 lap. TIM: 1  x = 3  15 x = 5 So, it takes Tim 5 minutes to run 1 lap. Linda runs faster than Tim! WOOHOO! 162 Nathalie, Patty, Mary, and Mino are picking coconuts. They record their times in the table below. Fill in the missing numbers (assuming their rates ater proportional). 1. Nathalie NUMBER OF COCONUTS MINUTES 1 5 30 48 2. Patty NUMBER OF COCONUTS MINUTES 1 2 14 6 3. Mary NUMBER OF COCONUTS MINUTES 1 4 8 16 163 4. Mino NUMBER OF COCONUTS MINUTES 1 20 9 36 40 5. Who picked 1 coconut in the least amount of time? answers 164 Nathalie NUMBER OF COCONUTS MINUTES 1 6 5 30 8 48 Patty NUMBER OF COCONUTS MINUTES 1 7 2 14 6 42 Mary NUMBER OF COCONUTS MINUTES 1 2 2 4 8 16 Mino NUMBER OF COCONUTS MINUTES 1 4 5 20 9 36 10 40 5. Mary picked 1 coconut in the least amount of time: 2 minutes. 1. 2. 3. 4. 165 Expressions and Equations 3 Unit 166 In math, an EXPRESSION is a mathematical phrase that contains numbers, VARIABLES (let ters or symbols used in place of a quantity we don’t know yet), and/or operators (such as + and ). EXAMPLES: x + 5 3m  z a  b 44k 59 + −3 Sometimes, an expression allows us to do calculations to find out what quantity the variable is. EXAMPLE: When Georgia runs, she runs a 6mile loop each day. We don’t know how many days she runs, so we’ll call that number “d.” So, now we can say that Georgia runs 6d miles. (In other words, 6d is the expression that represents how much Georgia runs each week.) EXPRESSIONS Chapter 27 167 When a number is attached to a variable, like 6d, you multiply the number and the variable. Any number that is used to multiply a variable (in this case 6) is called the COEFFICIENT. A CONSTANT is a number that stays fixed in an expression (it stays “constant”). For example, in the expression 6x + 4, the constant is 4. An expression is made up of one or more TERMSa number by itself or the product of a number and variable (or more than one variable). Each term is separated by an addition calculation symbol. In the expression 6x + 4, there are two terms: 6x and 4. TERM a number by itself or the product of a number and variable(s). Terms in a math sentence are separated by a + or  symbol. 168 EXAMPLE: Name the variable, terms, coefficient, and constant of 8y  2. The variable is y . The terms are 8y and 2. The coefficient is 8. The constant is 2. Operators tell us what to do. Addition (+), subtraction (), multiplication (x), and division (÷) are the most common operators. Word problems that deal with expressions use words instead of operators. Here’s a quick translation: OPERATION OPERATOR KEYWORDS sum + greater than more than plus added to increased by Huh? You might have thought terms were always separated by an addition symbol . . . BUT if you’re adding a negative number, the + becomes a  ! Keep an eye out for + and  when looking for terms in an expression. 169 difference  less than decreased by subtracted from fewer product x times multiplied by of quotient ÷ divided by per EXAMPLE: “14 increased by g” = 14 + g EXAMPLE: “17 less than h” = h  17 (Be careful! Anytime you are translating “less than,” the second number in the word problem is written first in the expression!) EXAMPLE: “The product of −7 and x ” = −7 • x This can also be writ ten (−7)(x) or −7(x) or −7x. EXAMPLE: “The quotient of 99 and w ” = 99 ÷w This can also be writ ten 99w . 170 For 1 through 3, name the variable(s), coefficient(s), and/or constant(s), if applicable. 1. 3y 2. 5x + 11 3. −52m + 6y  22 For 4 and 5, list the terms. 4. 2,500 + 1 1 t  3w 5. 17 + d (−4) For 6 through 10, write the expression. 6. 19 less than y 7. The quotient of 44 and 11 8. The product of −13 and k 171 9. Katherine drives 27 miles to work each day. Last Wednesday, she had to run some errands and drove a few extra miles. Write an expression that shows how many miles she drove on Wednesday. (Use x as your variable.) 10. There is a hiphop dance contest on Saturday nights at a club. Because there was a popular DJ playing, the organizers expected 2 times the amount of people. The organizers also invited an extra 30 people from out of town. Write an expression that shows how many people they can expect to come to the event. (Use x as your variable.) answers 172 1. Variable: y; Coefficient: 3; No constants 2. Variable: x; Coefficient: 5; Constant: 11 3. Variables: m, y; Coefficients: −52, 6; Constant: −22 4. 2,500, 11 t, 3w 5. 17, d (−4) 6. y  19 7. 44 ÷ 11 or 4411 8. −13k 9. 27 + x 10. 2x + 30 BFN_MATHRPT5817cb.indd 172 5/10/17 4:51 PM 173 Properties are like a set of math rules that are always true. They often help us solve equations. Here are some important ones: The IDENTITY PROPERTY OF ADDITION looks like this: a + 0 = a. It says that if you add zero to any number, that number stays the same. EXAMPLE: 5 + 0 = 5 The IDENTITY PROPERTY OF MULTIPLICATION looks like this: a x 1 = a. It says that if you multiply any number by 1, that number stays the same. EXAMPLE: 7 x 1 = 7 PROPERTIES Chapter 28 174 The COMMUTATIVE PROPERTY OF ADDITION looks like this: a + b = b + a. It says that when adding two (or more) numbers, you can add them in any order and the answer will be the same. EXAMPLE: 3 + 11 = 11 + 3 (Both expressions equal 14.) The COMMUTATIVE PROPERTY OF MULTIPLICATION looks like this: a • b = b • a. It says that when multiplying two (or more) numbers, you can multiply them in any order and the answer will be the same. EXAMPLE: −5 • 4 = 4 • −5 (Both expressions equal −20.) When talking about properties, your teacher or textbook may use the term EQUIVALENT EXPRESSIONS, which simply means that the math sentences have equal value. For example, 3 + 11 = 11 + 3. (They are equivalent expressions.) DON’T FORGET: The commutative properties only work with addition and multiplication; they do NOT work with subtraction and division! 175 The ASSOCIATIVE PROPERTY OF ADDITION looks like this: (a + b) + c = a + (b + c). It says that when adding three different numbers, you can change the order that you add them by moving the parentheses and the answer will still be the same. EXAMPLE: (2 + 5) + 8 = 2 + (5 + 8) (Both expressions equal 15.) The ASSOCIATIVE PROPERTY OF MULTIPLICATION looks like this: (a • b) • c = a • (b • c). It says that when multiplying 3 different numbers, you can change the order that you multiply them by moving the parentheses and the answer will still be the same. EXAMPLE: (2 • 5) • 8 = 2 • (5 • 8) (Both expressions equal 80.) DON’T FORGET: The associative properties only work with addition and multiplication; they do NOT work with subtraction and division! 176 The DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION looks like this: a(b + c) = ab + ac. It says that adding two numbers inside parentheses, then multiplying that sum by a number outside the parentheses is equal to first multiplying the number outside the parentheses by each of the numbers inside the parentheses and then adding the two products together. EXAMPLE: 2 (4 + 6) = 2 • 4 + 2 • 6 (You “distribute” the “2 •” across the terms inside the parentheses. Both expressions equal 20.) EXAMPLE: 7 (x + 8) = (x + 8) = 7 (x) + 7 (8) = 7x + 56 Think about catapulting the number outside the parentheses inside to simplify. The DISTRIBUTIVE PROPERTY allows us to simplify an expression by taking out the parentheses. 177 The DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER SUBTRACTION looks like this a(b  c) = ab  ac. It says that subtracting two numbers inside parentheses, then multiplying that difference times a number outside the parentheses is equal to first multiplying the number outside the parentheses by each of the numbers inside the parentheses and then subtracting the two products. EXAMPLE: 9(5  3) = 9(5)  9(3) (Both expressions equal 18.) EXAMPLE: 6 (x  8) = (x  8) = 6 (x)  6 (8) = 6x  48 178 FACTORING is the reverse of the distributive property. Instead of getting rid of parentheses, factoring allows us to include parentheses (because sometimes it ’s simpler to work with an expression that has parentheses). EXAMPLE: Factor 15y + 12. STEP 1: Ask yourself, “What is the greatest common factor of both terms?” In the above case, the GCF of 15y and 12 is 3. ( 15y = 3 • 5 • y and 12 = 3 • 4 ) STEP 2: Divide all terms by the GCF and put the GCF on the outside of the parentheses. 15y + 12 = 3(5y + 4) EXAMPLE: Factor 12a + 18. The GCF of 12a and 18 is 6. So, we divide all terms by 6 and put it outside of the parentheses. 12a + 18 = 6(2a + 3) You can always check your answer by using the DISTRIBUTIVE PROPERTY. Your answer should match the expression you started with! 179 I LOVE THE DISTRIBUTIV E PROPERT Y ! WOO HOO ! 180 In each blank space below, use the property listed to write an equivalent expression. PROPERTY EXPRESSION EQUIVALENT EXPRESSION Identity Property of Addition 6 Identity Property of Multiplication y Commutative Property of Addition 6 + 14 Commutative Property of Multiplication 8 • m Associative Property of Addition (x + 4) + 9 Associative Property of Multiplication 7•( r • 11 ) Distributive Property of Multiplication over Addition 5 (v + 22) Distributive Property of Multiplication over Subtraction 8(7  w) Factor 18x + 6 Factor 14  35z 181 1. Distribute 3(x + 2y  5). 2. Distribute 1  2 (4a  3b  c). 3. Factor 6x + 10y + 18. 4. Factor 3g  12h  99j. 5. Mr. Smith asks Johnny to solve (12  8)  1. Johnny says that he can use the Associative Property and rewrite the problem as 12  (8  1). Do you agree with Johnny? Why or why not? answers 182 1. 3x + 6y  15 2. 2a 3 2 b  1 2 c 3. 2(3x + 5y + 9) 4. 3(g  4h  33j) 5. PROPERTY EXPRESSION EQUIVALENT EXPRESSION Identity Property of Addition 6 6 + 0 Identity Property of Multiplication y y • 1 or 1y Commutative Property of Addition 6 + 14 14 + 6 Commutative Property of Multiplication 8 • m m • 8 Associative Property of Addition (x + 4) + 9 x + (4 + 9) Associative Property of Multiplication 7•(r • 11) (7 • r ) • 11 Distributive Property of Multiplication over Addition 5(v + 22) 5v + 110 Distributive Property of Multiplication over Subtraction 8(7  w) 56  8w Factor 18x + 6 6(3x + 1) Factor 14  35z 7(2  5z) No, Johnny is wrong because the Associative Property does not work with subtraction the order in which you subtract matters. 183 A term is a number by itself or the product of a number and variable (or more than one variable). EXAMPLES: 5 (a number by itself) x (a variable) 7y (a number and a variable) 16mn2 (a number and more than one variable) In an expression, terms are separated by an addition calculation, which may appear as a positive or negative sign. EXAMPLES: 5x + 3y + 12 (The terms are 5x, 3y , and 12.) 3g2 + 47h  19 (The terms are 3g2, 47h, and 19.) LIKE TERMS Chapter 29 ALTHOUGH THIS MAY LOOK LIKE A SUBTRACTION SYMBOL, YOU’RE ACTUALLY ADDING A NEGATIVE NUMBER. BFN_MATHRPT5817cb.indd 183 5/10/17 4:51 PM 184 We COLLECT LIKE TERMS (also called COMBINING LIKE TERMS) to simplify an expressionmeaning, we rewrite the expression so that it contains fewer numbers, variables, and operations. Basically, you make it look more “simple.” EXAMPLE: Denise has 6 apples in her basket. Let ’s call each apple “a.” We could express this as a + a + a + a + a + a, but it would be much simpler to write 6a. When we put a + a + a + a + a + a together to get 6a, we are collecting like terms. (Each term is the variable a, so we can combine them with the coefficient of 6, which tells us how many a’s we have.) When combining terms with the same variable, add the coefficients. EXAMPLE: Denise now has 6 apples in her pink basket, 1 apple in her purple basket, and 7 apples in her white basket. 185 We could express this as 6a + a + 7a but it would be much simpler to write 14a. EXAMPLE: 9x  3x + 5x (When there is a “ ” sign in front of the term, we have to subtract.) 9x  3x + 5x = 11x If two terms do NOT have the exact same variable, they cannot be combined. EXAMPLE: 7m + 3y  2m + y + 8 (The 7m and 2m combine to make 5m, the 3y and y combine to make 4y , and the constant 8 does not combine with anything.) 7m + 3y  2m + y + 8 = 5m + 4y + 8 A variable without a coefficient actually has a coefficient of 1. So “m” really means “1m” and “k3” really means “1k3.” (Remember the identity property of multiplication!) REMEMBER: A term with a variable cannot be combined with a constant. 3ab can combine with 4ba, because the commutative property of multiplication tells us that ab and ba are equivalent! SORRY— WE’RE JUST NOT A GOOD COMBO. 186 When simplifying, we often put the term with the greatest exponent first, and we put the constant last. This is called DESCENDING ORDER. EXAMPLE: 7m2 + 2m  6 Sometimes, we need to use the distributive property first and then collect like terms. EXAMPLE: 3x + 4(x + 3)  1 3x + 4(x + 3)  1 Fi