Principal Visual Math - See How Math Makes Sense
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visual math visual math See How Math Makes Sense Jessika Sobanski N EW YOR K Copyright © 2002 LearningExpress, LLC. All rights reserved under International and Pan-American Copyright Conventions. Published in the United States by LearningExpress, LLC, New York. Library of Congress Cataloging-in-Publication Data: Sobanski, Jessika. Visual math : see how math makes sense / by Jessika Sobanski.—1st ed. p. cm. ISBN 1-57685-404-3 1. Mathematics—Study and teaching. 2. Visual learning. I. Title. QA11.2. S63 2002 510-dc21 2001050621 Printed in the United States of America 987654321 First Edition ISBN 1-57685-404-3 For more information or to place an order, contact LearningExpress at: 900 Broadway Suite 604 New York, NY 10003 Or visit us at: www.learnatest.com contents Introduction 1 Number Concepts and Properties 19 Fractions and Decimals 53 Ratios and Proportions 85 ➧ ➧ one ➧ two three ➧ Percents 109 Algebra 137 four ➧ five ➧ Geometry and Measurement 167 Probablity and Statistics 205 ➧ six seven ➧ Tables and Charts 225 Test Your Math Skills 241 eight ➧ nine visual math Introduction this book has been designed to allow learners to “see” how math makes sense. By combining logical math concepts with pictures, previously unclear images will fade and math will suddenly click for you. Read on to see how visual learning relates to the hemispheres of your brain and how combining images and logical reasoning actually gets both sides of your brain working at the same time, in the brain-healthy, whole brain learning style. introduction 1 learning visually When we look at the percentages of how much we learn through each sense, the breakdown looks like this: ■ ■ ■ ■ ■ taste 3% smell 3% touch 6% sound 13% sight 75% Although the following discussion of left brain versus right brain and whole brain learning strategies is fascinating, the premise of this book can basically be summed up by this old adage: A picture is worth a thousand words! left brain vs. right brain Eve; rybody has a left brain and a right brain, and we all use both sides. But most people use one side more than the other. This hemispheric dominance affects the way we process information and learn. Learning with both sides helps us make the most of our brains. Incorporating whole brain learning strategies into academic endeavors will address the left-brainers and right-brainers and allow both types to use more of their brains. 2 visual math Would you rather look at this . . . . . . or read this? The processing in the left brain is linear. This means that learning occurs from part to whole. Processing in this hemisphere is also sequential. The left brain is good at processing symbols and is very logical and mathematical. The left brain also deals with verbal and written inputs and adheres to rules. Left brain processes are reality-based. The processing in the right brain is holistic. This means that learning occurs by first envisioning the whole picture. Processing in this hemisphere is random. The right brain is also color-sensitive. This hemisphere is good at processing the concrete: things that can be seen, touched, and felt. The right brain is very intuitive and non-verbal. Right brain processes are fantasy-oriented. introduction 3 which half of your brain is dominant? The following questions are merely a survey that may help you better understand which side of the brain you emphasize while thinking, acting, learning, and so on. There are no wrong answers. Read through the choices given to you and pick the one that best fits your personality. Remember, accuracy depends on honesty. You’ll find the answers at the end of the chapter. 1. When learning something new, you would rather A. learn by demonstration. B. learn by explanation. C. learn by reading the directions. 2. Personally, you are more inclined to learn A. a second language. B. sign language. C. Neither would be more or less difficult for me. 3. Which presentation of statistical data is more understandable? A. visual data, such as a graph or chart B. numerical data C. Both ways are just as understandable to me. 4. Which courses did you/do you enjoy most in school? A. philosophy/creative writing B. mathematics/science C. I was not partial to any particular course. 5. When choosing a movie to watch, you are more likely to enjoy A. a non-fiction documentary. B. a realistic “whodunnit” mystery film. C. a science fiction horror film. 6. The ideal activity on your night off is A. hanging out with a few close friends. B. sitting at home enjoying your favorite hobby. C. going dancing and meeting new people. 4 visual math 7. If choosing a vacation, you would A. choose a place you’ve never been to. B. choose the same place you went last year. C. choose a place similar to one you’ve gone to before. 8. Outside of special occasions, which best describes your wardrobe? A. relaxed, with your own personal sense of style B. neat and similar to that of others C. not interested in what other people think 9. When meeting new people, which personality trait most appeals to you? A. humor B. modesty C. intelligence 10. When planning a recreational activity, you would rather A. make long-term plans B. spontaneous plans C. It doesn’t bother me either way. 11. When debating a subject you are passionate about, you A. let your emotions control the conversation. B. keep cool and collected, controlling your emotions. C. don’t let your emotions play a factor. 12. When faced with a difficult decision, you A. make a decision influenced by a similar experience. B. make a decision based on instinct. C. find out all the info and make the best decision. 13. When it comes to workspace, which best describes you? A. a completely cluttered mess B. slightly messy but generally organized C. neat and organized 14. When engaged in a conversation, you tend to interpret participants’ responses A. purely by the words they are saying. B. by body language only. C. both factors combined. introduction 5 15. Immersed in thought while lying in bed ready to go to sleep, you are more likely to A. think about what you want to dream about. B. analyze the day’s events. C. plan ahead for tomorrow. 16. While driving home from a new job you realize there may be other routes to take. Which best describes you? A. you’d stray from the usual path to find the most convenient route B. you’d consult a map and take a new route the next day C. you’d stay on the familiar route 17. If your boss at work gave you an unfamiliar task, would you A. get ideas from someone who is familiar with the task and improve on them? B. ask someone how best to get it done and follow his/her instructions? C. develop your own technique? 18. When you go to a museum, which exhibits interest you the most? A. artistic exhibits (paintings, sculptures) B. antique exhibits (architecture, armor, relics) C. prehistoric exhibits (extinct animals, prehistoric man, dinosaurs) 19. At a job interview, you would prefer the interviewer to ask A. questions open for discussion. B. questions requiring short specific answers. C. questions that have short answers but allow you to add detail and substance. 20. Which adjective best describes you? A. focused B. independent C. social D. spacey After completing the survey consult the Brain Dominance Survey Key (located on page 17 at the end of this chapter) to discover your brain dominance. 6 visual math whole-brain learning strategies No matter which hemisphere of your brain is dominant, keeping both hemispheres actively involved in the learning process will help you make the most of your brain. Here are some tips on creating a whole brain learning environment for yourself: ■ ■ ■ ■ ■ ■ Learn in a relaxed environment. The best recall occurs when brain wave patterns show a relaxed state. Learn in a multi-sensory environment by involving visual, auditory, and kinesthetic activities. Use color! This stimulates the right brain and helps recall. Make sure you take breaks every hour. Try to relate what you are learning to a bigger picture. Reinforce what you have learned through practice and review. The following section contains “Brain Games” that are really good for your brain. Solving these puzzles requires the use of your whole brain. The concept is that you are using the logical (left) and visual (right) portions of your brain at the same time. You can solve a few now, and come back and try some more at a later time. Many standardized tests include questions that are based on logic, patterns, and sequences. These puzzles will help you foster those skills. In addition, as you read through this book, you will see that the logical concepts being presented are demonstrated in a visual manner. In other words, this book is good for your whole brain! brain games 1. Bill, the neighborhood ice cream truck driver, is making his daily rounds. Today is extremely hot and muggy, and the kids around town are anxious for Bill to bring them ice cream. There is one problem: The hot and muggy weather has caused most of Bill’s freezers to break down, leaving him with room for only 20 ice cream bars. Since Bill’s house is just around the corner, he can always restock with more. The children usually wait in introduction 7 the same spot for Bill to come around, but he does not want to pass them without any ice cream because he fears lost sales. Help Bill plan the best route to serve all the children and restock when he needs to without passing them by. Use the least number of trips because those kids are hot! A number in a circle represents each group of kids. Each child will buy one ice cream bar and then leave. The arrows indicate one-way streets. 8 visual math 2. Can you place each labeled piece of the puzzle in the correct position? introduction 9 3. Little Harry finds his way to a curiosity shop and finds interest in a few knick-knacks. Among them is an old kaleidoscope, which Harry seems to like very much. The owner of the shop encourages Harry to take a look through the “scope” and to give it a few turns. Doing so, Harry sees a pattern of various shapes and colors and decides that he would like to purchase the “scope” from the owner. The owner acknowledges Harry’s interest and tells Harry if he can guess what the pattern is after turning the kaleidoscope twice, he can have the kaleidoscope for free. Help Harry. 10 visual math Judging from the choices given and the patterns in the kaleidoscope before and after each turn, pick the pattern that is most likely to appear when Harry turns the kaleidoscope again. introduction 11 4. Margaret was working on an art project for school. The right side of the dashed line in the figure below should be symmetrical to the left side. However, 5 circles are not symmetrical. Can you find them? 12 visual math 5. In the math puzzle below, there is a specific relationship between the numbers in the squares. Can you figure out the pattern and fill in the missing piece? introduction 13 solutions to chapter exercises 1. 2. 14 visual math 3. The correct answer is c. Notice the central shapes: Also, if you notice the shape in the “12 o’clock” position of each figure, it goes from black circle white square black triangle, and thus you might expect the next “12 o’clock” shape to be white. A white circle would also make sense, as the pattern would go from circles in the periphery to squares, to triangles. introduction 15 4. 5. 16 visual math brain dominance survey key Add up your score from the Brain Dominance Survey on pages 4–6 based on the following point values for each answer choice 1. A. 4 B. 1 C. 2 10. A. 1 B. 5 C. 3 2. A. 1 B. 5 C. 3 11. A. 5 B. 3 C. 1 3. A. 4 B. 1 C. 3 12. A. 4 B. 5 C. 1 4. A. 5 B. 1 C. 3 13. A. 4 B. 5 C. 1 5. A. 5 B. 1 C. 3 14. A. 1 B. 5 C. 3 6. A. 2 B. 1 C. 5 15. A. 5 B. 2 C. 1 7. A. 5 B. 2 C. 4 16. A. 4 B. 2 C. 1 8. A. 4 B. 2 C. 5 17. A. 1 B. 2 C. 5 9. A. 5 B. 3 C. 1 18. A. 5 B. 3 C. 1 introduction 17 19. A. 5 B. 2 C. 4 20. A. B. C. D. 1 1 5 5 If your score is between the numbers 23–32, your brain functions predominantly on the left side. If your score is between the numbers 88–97, your brain functions predominantly on the right side. If your score is between the numbers 33–54, your brain functions mostly on the left side but you have adapted, in some ways, to include some right brain attributes. If your score is between the numbers 66–87, your brain functions mostly on the right side, but you have adapted, in some ways, to include some left brain attributes. If your score is between the numbers 55–65, your brain functions on a bilateral level with equal right and left side attributes. 18 visual math chapter one Number Concepts and Properties getting number savvy Numbers, numbers, numbers. Before we begin to explore mathematical concepts and properties, let’s discuss number terminology. The counting numbers: 0, 1, 2, 3, 4, 5, 6, and so on, are also known as the whole numbers. No fractions or decimals are allowed in the world of whole numbers. What a wonderful world, you say. No pesky fractions and bothersome decimals. But, as we leave the tranquil world of whole numbers and enter into the realm of integers, we are still free of fractions and decimals, but are sub- number concepts and properties 19 jected to the negative counterparts of all those whole numbers that we hold so dear. The set of integers would be: . . . . −3, −2, −1, 0, 1, 2, 3 . . . . The real numbers include any number that you can think of that is not imaginary. You may have seen the imaginary number i, or maybe you haven’t. The point is, you don’t have to worry about it. Just know that imaginary numbers are not allowed in the set of real numbers. No pink elephants either! Numbers that are included in the real numbers are fractions, decimals, zero, π, negatives, and positives. A special subset of the integers is the prime numbers. A prime number has just two positive factors: one and itself. It saddens many to realize that 1 is not prime by definition. Examples of prime numbers include: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Note that the opposite (negative version) of the above numbers are also prime. For example, the factors of −23 are 1, −23, −1, and 23. Thus, −23 is prime because it has exactly two positive factors: 1 and 23. Not nearly as popular as the prime numbers are the composite numbers. Composite numbers have more than two factors. Note that 1 isn’t composite either. 20 visual math examples of different types of numbers So where do irrational and rational numbers fit into all this? Here’s how it works. The first five letters of RATIONAL are R-A-T-I-O. Rational numbers can be represented as a ratio of two integers. In other words, it can be written as a decimal that either ends or repeats. Irrational numbers can’t be represented as a ratio, because their decimal extensions go on and on forever without repeating. π is the famous irrational number. Other irrational numbers are 2 and 11 . Your turn! When you finish, you can find the answers at the end of the chapter, starting on page 48. Exercise 1: Fill in the diagram below using the numbers 2, π, 212, 17, −13, .675, 79, 6, 1, −13, −555, and 8,700. Note that the real numbers encompass all of the integers and whole numbers. Thus, if you place a number into the gray area labeled “whole numbers” you are also categorizing it as a real number and an integer. number concepts and properties 21 dealing with negatives When working with negative values, it is helpful to think about a number line, a thermometer, or money. The following visual depicts addition and subtraction from different points on the number line. addition and subtraction 22 visual math When multiplying and dividing signed numbers, you should be familiar with the rules below, where is a positive integer, and is a negative integer. multiplication and division Another operation to consider for signed numbers is the effect of raising these numbers to different powers. The number in concern is the base and the power to which it is raised is the exponent. For example, when looking at 45, we call 4 the base and 5 the exponent. number concepts and properties 23 45 = 4 × 4 × 4 × 4 × 4 When raising signed numbers to different powers, it is important to bear the following rules in mind. Here, represents a positive or negative number. exponents Some examples: 50 = 1 72 = 7 × 7 = 49 (−8)2 = 64 (−3)3 = −3 × −3 × −3 = 9 × −3 = −27 24 visual math 1 When dealing with negative exponents, remember that a−n = an . For example: Some examples: 4−2 = 412 = 116 −2−3 = −123 = −18 = −18 Rules for operations with exponents: When multiplying powers of the same base, add the exponents. 42 • 43 = 42+3 = 45 Notice that this rule works because 42 • 43 = (4 • 4) • (4 • 4 • 4), which is the same as 45. number concepts and properties 25 When dividing powers of the same base, subtract the exponents. 58 ÷ 52 = 58-2 = 56 58 5 This rule works because 2 = We can cancel: 5•5•5•5•5•5•5•5 5•5 5•5•5•5•5•5•5•5 5•5 = 56 When raising a power to a power, multiply the exponents. (63)4 = 63•4 = 612 This rule works because (63)4 = (6 • 6 • 6)4 = (6 • 6 • 6)(6 • 6 • 6)(6 • 6 • 6) (6 • 6 • 6), which is the same as 612. Tips when dealing with exponents: Always raise everything inside the parentheses by the power outside of the parentheses. (7 • 63)4 = 74 • 63•4 = 74 • 612 (5a 3)2 = 52 • a 3•2 = 25a 6 When you do not have the same base, try to convert to the same base: 254 • 512 = (52)4 • 512 = 58 • 512 = 520 Exercise 2: Simplify the following. 32 • 34 • 3 = 710 = 75 (23b7)2 = 52 ÷ 5−2 = 42 • 25 = 4−2 ÷ 4−5 = 26 visual math place value Each place that a digit occupies within a number has a name. We typically see numbers in base 10. In base 10 there are ten possible digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. When you have more than 9, you need to add another spot, namely, the tens place. 10 represents no ones and 1 ten. Below are all the places: The number above can be represented in expanded notation as (1 × 1,000,000) + (2 × 100,000) + (3 × 10,000) + (4 × 1,000) + (5 × 100) + (6 × 10) + (7 × 1). Note that in base 10, ones represent 100 = 1, the tens represent 101 = 10, the hundreds represent 102 = 100, and so forth. 106 105 1,000,000 100,000 ones tens hundreds thousands ten thousands millions hundred thousands Base 10: 104 103 102 101 100 10,000 1000 100 10 1 number concepts and properties 27 Base 2 uses only 2 digits: 0 and 1. If you’ve been around computers, you have probably heard the term binary. Even the most complex functions that a computer provides boils down to a data stream consisting of zeros and ones (base 2). The 1 stands for on and the 0 stands for off. Here are the places in base two with the base ten equivalents noted underneath: Base 2 27 26 25 24 23 22 21 20 Base 10 128 64 32 16 8 4 2 1 A solo 0 or a 1 is called a bit and 8 bits in a row is termed a byte. Consider the following byte: 10110101 To figure out the base 10 equivalent of the byte above, we first have to see where each bit falls according to its place value: 27 26 25 24 23 22 21 20 128 64 32 16 8 4 2 1 1 0 1 1 0 1 0 1 Next, we add up the values that are indicated: 27 26 25 24 23 22 21 20 128 64 32 16 8 4 2 1 0 1 1 0 0 1 0 1 128 + 64 + 32 +4 128 + 64 + 32 + 4 + 1 = 229, so 10110101two = 229ten. 28 visual math +1 Exercise 3: Express 6,871,235 in expanded notation: roots 1 1 What is the value of 42? How about 273? Looks complicated, huh? Those two 3 questions could actually be rewritten as 4 and 27 . Let’s take a closer look at roots and how to convert icky fractional exponents into pleasant little radicals. Some examples: 2 252 = 25 1 3 83 = 8 1 2 162 = 16 1 number concepts and properties 29 2 Note that is the same as . The radical symbol indicates the root to be taken (the index). If there is no index labeled, take the square root. Let’s say you are presented with the question: “What is 49 ?” Here, you need to figure out what number squared equals 49. We know that −72 = 49 and 72 = 49, so your answer is ±7. However, you will mostly be asked to find the principal square root, which is always positive. 3 If you had to find , 27 you would be looking for the number which, when 3 cubed, would yield 27; 3 × 3 × 3 = 27, so 27 = 3. Exercise 4: Complete the chart below. exponential form 1 1253 1 1212 1 646 −83 1 30 visual math rewrite as a radical solve ways to manipulate radicals 1. You can express the number under the radical as the product of other numbers. You can then equate the root of the product of those numbers as the product of separate roots of those numbers. 12 = 4 • 3 = 4 • 3 = 23 2. When multiplying two roots (with the same index) you can combine them under the same radical. 8 • 2 = 8•2 = 16 =4 3. When dividing two roots (with the same index) you can combine them under the same radical. 242 ÷ 2 = 242 2 ÷ = 121 = 11 4. You can take a “division problem” out from under the radical and place each “piece” under its own radical. 145 = 15 ÷ 4 = 15 ÷2 5. You can only add roots if they have the same index and radicand. Recall: 23 + 33 = 53 6. Also, you can only subtract roots if they have the same index and radicand. 85 − 25 = 35 number concepts and properties 31 order of operations When you get a messy mathematical expression involving every operation under the sun, it is important to perform the operations in the correct order. The order of operations is: 1. parentheses 2. exponents 3. multiplication/division 4. addition/subtraction Many people say, “Please excuse my dear Aunt Sally,” or “PEMDAS” in order to remember the correct order of operations. Should we bother with order of operations? Does order matter? If we took a problem like 6 + 8 × 2 − 3 × 5 and just took each operation in order of appearance, we’d get: 6+8×2−3×5 14 × 2 − 3 × 5 28 5 × 5 23 × 5 115 Is this right? No. Let’s proceed in the correct order. There are no parentheses or exponents, so we need to do any multiplication or division first (in the order in which they occur) . . . 6+8×2−3×5 6 + 16 − 3 × 5 32 visual math We still have a multiplication to take care of . . . 6 + 16 − 3 × 5 22 − 15 7 Notice that when you threw caution to the wind you got an answer that wasn’t even close to the actual value! Take your time and carry out each operation in the correct order! Let’s go through a harder question step by step. 5 + 22 × 8 − (5 × 32) 5 + 22 × 8 − (5 × 32) When we look inside the parentheses, we first deal with the exponent rather than the multiplication because the E comes first in P E MDAS: Parentheses 5 + 22 × 8 − (5 × 9) Next, we multiply: 5 + 24 × 8 − (45) 5 + 22 × 8 − 45 Exponents 5 + 4 × 8 − 45 Multiplication/Division 5 + 4 × 8 − 45 5 + 32 − 45 Addition/Subtraction 5 + 32 − 45 37 − 45 −8 number concepts and properties 33 Exercise 5: Use the chart below to sequentially follow the steps in PEMDAS. 23 − 7 × (5 − 8) ÷ 3 Parentheses Exponents Multiplication/Division Addition/Subtraction logarithms Logarithms, or logs, can be to different bases. log2 denotes a log to the base 2, and log10 denotes a log to the base 10. Logarithms are exponents. When you solve a log, you are actually calculating the exponent that the base was raised to. Look at this problem: log2 4 = ? Logs can be tackled easily by making a spiral right through the problem . . . . 34 visual math You spiral your way through the 2, then the “?” and end on the 4. “2 to what power is 4?” 2 to the second power is 4, so, log2 4 = 2. log10 is so common that if you see a question like log 100 = ?, you are taking the log to the base 10. Thus log 100 = log10 100. To solve, make a spiral . . . . The number 10 to the second power is 100, so log10 100 = 2. Are logs just fun with spirals? You ask. Logs are used in many branches of math and science. For example, in chemistry you study pH. Well, pH is just the negative log of the concentration of hydronium ions. (That’s water with an extra number concepts and properties 35 positive charge added, or H30+.) Let’s say you had a mystery solution that had a hydronium ion concentration of 1 × 10−6, we could solve for pH: pH = −log10 (concentration of hydronium ions) pH = −log10 (1 × 10−6) = −log10 (10−6) Let’s calculate the log first . . . Asking “10 to what power is 10−6?” is kind of like asking, “Who’s buried in Grant’s tomb?” or “What color was George Washington’s white horse?” Obviously, 10 to the −6th power is 10−6. See, this question looked ominous and menacing, but it was actually easier than any you’ve done so far! But before you start patting yourself on the back, remember to put the −6 back into the equation . . . pH = −log10 (1 × 10−6) = pH = −(−6) = 6. You could then look at a pH chart to reveal that this mystery solution is an acid! 36 visual math Exercise 6: Use a spiral to solve the logs below: log10 10,000 = ? log10 1,000 = ? pre-algebra: a review of number properties The commutative property holds for addition and multiplication, as shown below: number concepts and properties 37 Note that 2 − 5 does not equal 5 − 2. If you want to apply the commutative property, you must rewrite 2 − 5 as 2 + −5. Now the commutative property holds: 2 + −5 = −5 + 2. 38 visual math Another property that holds for addition and multiplication is the associative property. Note that (5 − 3) + 4 = 6, but 5 − (3 + 4) = 5 − 7, or −2. The associative property holds for addition. In order for you to apply it to subtraction, you need to change 5 − 3 + 4 into 5 + −3 + 4. number concepts and properties 39 40 visual math When trying to calculate math problems in your head, the distributive property comes in handy. Suppose you wanted to give out awards and you lined up medals on a table in a 20 by 14 arrangement. How many medals do you have? You can think of these medals as two groups . . . So, if you think of 20(14) as 20(10 + 4). The distributive property equates 20(10 + 4) with 20 • 10 + 20 • 4 . . . number concepts and properties 41 Thus, 20 • 10 + 20 • 4 = 200 + 80 = 280. Let’s say you had the expression 100 − 2(30 − 2) and you wanted to apply the distributive property. One way to deal with the two instances of subtraction is to change each “minus” into “plus a negative.” Thus, 100 − 2(30 − 2) = 100 + −2(30 + −2). Next, apply the distributive property . . . 100 plus −2 times 30 plus −2 times −2 equals 100 + (−2 • 30) + (−2 • −2 ), or 100 − 60 + 4 = 44. You may have noticed that if given 100 − 2(30 − 2), it would be easier to just start inside the parentheses to yield 100 − 2(28). In this case, it’s a viable option. But what if you had an unknown? Let’s say you had 12 − (x − 1). First, you could get rid of the subtractions by turning each “minus” into “plus a negative.” So, 12 − (x − 1) = 12 + −1(x + −1). [Note that there was an “invisible one” in front of the parentheses 12 − (x − 1) = 12 − 1 (x − 1) = 12 + −1(x + −1).] 42 visual math Next, apply the distributive property . . . 12 plus −1 times x plus −1 times −1 equals 12 + −1x + (−1 (−1), or 12 − x + 1. Once you get used to turning a “minus” into “plus a negative” and spotting “invisible ones” you will be able to account for these things in your head. Thus, upon seeing 12 − (x − 1), you will think: number concepts and properties 43 Exercise 7: As you think about each question, fill in the thought bubbles below. Note “I’m in the mood for pizza,” is not an acceptable answer. Suppose you are trying to calculate 18 × 11 in your head. You decide to use the distributive property and equate 18 × 11 with 18(10 + 1) . . . = Suppose you are given 15 − (x − 2). Which way would you prefer to look at the expression? Choose one way and solve . . . = 44 visual math = other fun things to do with numbers absolute value If you look at a point on the number line, measure its distance from zero, and consider that value as positive, you have just taken the absolute value. Let’s take the absolute value of 7. |7| = 7 Next, let’s calculate |−7|, which also equals 7. Exercise 8: What is |−11|? factorials When you take the factorial of a number, you just multiply that number by every positive whole number less than it. For example, 5 factorial, written 5! = 5 × 4 × 3 × 2 × 1. number concepts and properties 45 Exercise 9: 46 8! = × × × × × 6! = × × × × × 3! = × × visual math × × answers to chapter exercises Exercise 1: π is a real number (it is also an irrational number). It is a number with a never-ending decimal extension. 221, −31, and .675 are also a real numbers. They are not integers because they involve fractions and decimals. −13 and −555 are integers (in addition to being real numbers). Notice that −13 is also prime. It cannot go into the center circle labeled “prime” because those prime numbers are also whole numbers. −13 is not a whole number because it is negative. 1, 6, and 8,700 are real numbers, integers, and whole numbers. The center circle represents numbers that are real, integers, whole, and prime. 2, 17, and 79 can be classified as such. number concepts and properties 47 Exercise 2: Simplify the following. 32 • 34 • 3 = 32 • 34 • 31 = 32+4+1 = 37 = 2,187 710 = 710−5 = 75 = 16,807 75 (23b7)2 = 23•2b7•2 = 26b14 = 64b14 52 ÷ 5−2 = 52 − (−2) = 52+2 = 54 = 625 42 • 25 = (22)2 • 25 = 24 • 25 = 24+5 = 29 = 512 4-2 ÷ 4−5 = 4−2 − (−5) = 4−2 +5 = 43 = 64 Exercise 3: The expression of 6,871,235 in expanded notation is (6 1,000,000) + (8 100,000) + (7 10,000) + (1 1,000) + (2 100) + (3 10) + (5 1). Exercise 4: Complete the chart below. exponential form rewrite as a radical 3 1253 1 125 1 121 1212 48 visual math =5 2 121 6 646 1 64 −83 −8 1 solve 3 = 11 =2 = −2 Exercise 5: 23 − 7 × (5 − 8) ÷ 3 Parentheses 23 − 7 × (5 − 8) ÷ 3 23 − 7 × (−3) ÷ 3 23 − 7 × −3 ÷ 3 Exponents 23 − 7 × −3 ÷ 3 8 − 7 × −3 ÷ 3 Multiplication/Division You do the multiplication first because it appears first . . . 8 − 7 × −3 ÷ 3 8 − (−21) ÷ 3 Next, you do the division . . . 8 − (−21) ÷ 3 8 − (−7) Addition/Subtraction 8 − (−7) 8 + +7 15 number concepts and properties 49 Exercise 6: 104 = 10,000, so the answer is 4. 103 = 1,000, so the answer is 3 50 visual math Exercise 7: = (18 × 10) + (18 × 1) = 180 + 18 = 198 = 15 − x + 2 = 15 + −1x + (−1 • −2) = 17 − x = 15 − x + 2 = 17 x number concepts and properties 51 Exercise 8: On the number line below, mark the point −11. |−11| = 11 Exercise 9: 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 6! = 6 × 5 × 4 × 3 × 2 × 1 3! = 3 × 2 × 1 52 visual math chapter two Fractions and Decimals introduction to fractions Fractions are used to represent parts of a whole. You can think of the fraction bar as meaning “out of ” . . . fractions and decimals 53 You can also think of the fraction bar as meaning “divided by” . . . examples of fractions: 54 1 2 One out of two pieces is filled in . . . 1 3 One out of three pieces is shaded . . . 1 4 One out of four pieces is shaded . . . visual math 1 12 One out of twelve pieces is shaded . . . Although most people call the top part of the fraction the “top” and the bottom part of a fraction the “bottom,” the technical names are numerator and denominator. You are never allowed to have a zero in the denominator. Anything divided by zero is undefined. A proper fraction has a numerator that is smaller than its denominator. Examples are 12, 1235 , and 19090 . Improper fractions have numerators that are bigger than their denominators. Examples include 85, 9193 , and 52. fractions and decimals 55 A fraction that represents a particular part of the whole is sometimes referred to as a fractional part. For example, let’s say that a family has four cats and two dogs. What fractional part of their pets are cats? Since four out of the total six animals are cats, the fractional part of their pets 4÷2 2 that are cats is equal to 46. You can reduce this fraction to 23: 6÷2 = 3 What fractional part of the figure below is shaded? Let’s rearrange the shaded areas . . . Thus, 12 is shaded. 56 visual math Exercise 1: Use the data below to answer the following questions: ■ ■ ■ ■ What fractional part of the days listed were sunny? What fraction of the days were 40°? What fraction of the days were 55° or higher? What fractional part of the days listed were greater than 60°? adding and subtracting fractions In order to add or subtract fractions, you must have the same denominator. 3 4 7 numerator + numerator 1 = 1 1 + 1 1 = 1. denominator When you don’t have the same denominator you can convert the denominators so that you can add or subtract. You should find the least common denominator (LCD). Think of the least common denominator as a cookie cutter. Let’s say you are adding 13 and 14 of a cookie. fractions and decimals 57 You can use your cookie cutter to make both cookies have the same sized slices. It is important to pick out the right cookie cutter. Otherwise, you get a mess . . . If you just grab the fifths cutter and cut both our cookies, you get something that looks like this . . . Let’s try that again. You have 13 and 14. The least common multiple (LCM) of 3 and 4 is 12. Let’s try the twelfths cutter . . . 58 visual math You will cut both cookies into 12 slices . . . Now you add 142 + 132 to get 172. Note that once you pick out the new denominator that you need, you can multiply your fraction by a version of “1” and that gives you a new denominator. 1 3 × “1” = 13 × 44 = 142 1 4 × “1” = 14 × 33 = 132 fractions and decimals 59 converting improper fractions into mixed numbers Let’s add 34 and 58 . . . 6 8 + 58 = 181, which is an improper fraction. To convert 181 to a mixed number, you just divide 11 by 8 to get 1 with a remainder of 3. Since you’re dealing with eighths, you stick the remainder over 8, yielding 138. If you look at the two fractions below, you can see how easily 68 + 58 can be rearranged into a whole with three slices left over. 60 visual math Exercise 2: fractions and decimals 61 converting mixed numbers into improper fractions Let’s consider the mixed number 114. You could easily cut the whole pie into four quarters . . . Thus, you know that 114 = 54. To convert mixed numbers into improper fractions on the fly, you just multiply the whole number by the denominator, add this to the numerator, and stick this value over the same denominator . . . 62 visual math multiplying fractions You don’t need to worry about lowest common denominators (LCDs) or lowest common multiples (LCMs) when you multiply fractions. You just multiply the numerators and the denominators: 1 3 1×3 3 numerator × numerator 5 × 5 = = 5 × 5 = 2 5 denominator × denominator Sometimes, you can reduce before you multiply: 2 3 3 × 5 = 1 2 3 2 3 × 5 = 5 1 If you are asked to find the fraction of a number, just multiply that number by the fraction: of means 1 3 of 9 means 13 × 9 = 3. fractions and decimals 63 Suppose that from your available funds, you decide to transfer 13 of it into a checking account. Next, you withdraw 41 of what’s left in order to buy a computer. How much of your original funds are left? Half your funds are left. So if you started with $6,000, you’d have $3,000 left. If you started with $10,000, you’d have $5,000 left. You can look at the same question in another way. Notice that the fractions you will be working with are 13 and 14. Dividing the whole into 3 × 4, or 12 wedges will guarantee that you can take all the fractional parts away neatly. 64 visual math Exercise 3: Fill in the diagram on the next page to answer the following question: Fifteen new children just enrolled at a day camp. First, 15 of the children are placed into the preschool group. Next, 13 of the remaining campers are placed into the girls group. What fractional part of the new campers has yet to be placed? fractions and decimals 65 dividing fractions Pretend you had 12 a pie and you wanted to share it with somebody. It is clear that you would each get 14 of the pie (that is, if you were being fair ). 66 visual math What if you had two donuts and lots of mouths to feed? You might decide to cut the donuts into quarters so that you could divvy them out . . . You would then be able to feed eight mouths. You may remember that in order to divide one fraction by another, you need to flip the second fraction and then multiply the fractions: 3 5 ÷ 23 = First, take the reciprocal of 23 . . . Then, multiply: 35 × 32 = 190 Exercise 4: Look back at the pie example and the donut example and rewrite the division problems as multiplication problems: 1 ■ 2 ■ ÷2= 2 ÷ 14 = fractions and decimals 67 fraction word problems Your friend is an aspiring screenwriter and comes to you for advice. She is writing a scene in which a character named Dominick is obsessed with listening to his Yanni album over and over again. In desperation, his sister hides the record under a couch cushion when he leaves for work. Later, their mutual friend Alice sits on the couch, breaking the album into 8 equal pieces. To make matters worse, their neighbor Joey’s hamster makes off with a piece and eats 32 of it. When Dominick is apprised of the ill fate his record endured, he demands that Joey pay $12 for a new one. However, Joey replies that he should only have to pay for the piece his hamster ate. How much should Joey claim he owes? Joey’s hamster ate 23 of 18. 23 × 18 = 224, or 112. This means that he is financially obligated to pay 112 of the $12. 112 of $12 = 112 × $12 = $1. Exercise 5: Athena is wallpapering one wall of her room. The following diagram shows how much she has completed so far. She already used up 68 visual math $15 worth of wallpaper. Estimate how much more it will cost her to finish the wall. introduction to decimals Decimals are really just fractions in disguise. Or maybe it’s the fractions that are really decimals in disguise. You may never know who’s disguised as what in this crazy world of mathematical espionage. One thing is for sure; you can equate fractions and decimals. For example, the fraction 110 can be written as the decimal .1 The fraction 1100 is equal 1 ÷ 100 = .01 fractions and decimals 69 1 The fraction 1000 is equal 1 ÷ 1000 = .001 70 visual math Some decimals can only approximate fractions because they are irrational numbers. For example, π, 2, and 11 are equivalent to decimals that never terminate or repeat. Sometimes we approximate π as 272. π = 3.141592 . . . 2 = 1.414213 . . . 11 = 3.316624 . . . place value You already looked at the names of the places to the left of the decimal point in Chapter 1. Now, you will look at the names of the places to the right of the decimal point. operations with decimals The following diagram reviews basic operations involving decimals. These are the types of calculations you need to make when dealing with money, such as when balancing your checkbook. fractions and decimals 71 ■ ■ ■ 72 For addition and subtraction, you just need to line up the decimal points and add or subtract. When multiplying decimals, first you multiply in the usual fashion, and then count over the proper number of places. When dividing with decimals, you move the decimal point of the dividend and divisor the same number of places. (Recall: dividend ÷ divisor = quotient.) visual math converting fractions to decimals To convert a fraction to a decimal, just divide the top by the bottom. For example, 12 would equal 1 ÷ 2, or .5. Memorizing some of the decimal values of the more common fractions can come in handy: Fraction Decimal 1 2 .5 1 3 .3 2 3 .6 1 4 .25 3 4 .75 1 5 .2 converting decimals to fractions If you can say the name of the decimal, you can easily convert it to a fraction. For example, if you see .123, you can say “One hundred and twenty-three thou1 23 sandths,” which is the same as 1 00 0 . fractions and decimals 73 123 = 100 0 scientific notation In the old days scientists found themselves dealing with big, big numbers like 13,000,000,000 and really little numbers like .000000013. They were very sad because they did not like having to write so many zeros. It hurt their hands, wasted chalk, and took time away from playing practical jokes on each other. Then one day, a scientist came up with the idea to use “shorthand” for all those zeros. The “shorthand” incorporates the powers of ten . . . 74 visual math 100 =1 101 = 10 102 = 100 103 = 1,000 104 = 10,000 105 = 100,000 106 = 1,000,000 107 = 10,000,000 108 = 100,000,000 109 = 1,000,000,000 1010 = 10,000,000,000 1011 = 100,000,000,000 1012 = 1,000,000,000,000 1013 = 10,000,000,000,000 1014 = 100,000,000,000,000 one million one billion one trillion This meant that instead of writing 13,000,000,000, one could write 1.3 × 1010. Instead of writing .000000013, one could write 1.3 × 10−8. Then all the scientists were very happy and called this shorthand scientific notation. Fortunately, you do not have to memorize the chart above. The trick to expressing, say, 250,000,000,000,000 as 2.5 times a power of 10 is to start at the current decimal point and then count until you reach the place where you want to insert the new decimal point. You counted 14 places to the left, so 250,000,000,000,000 = 2.5 × 1014. Let’s divide 7.6 ÷ 104 by 200: fractions and decimals 75 7.6 × 104 = .038 × 104 200 The standard form dictates that you express .038 × 104 as 3.8 times a power of 10. But what power of ten? Note that the 104 is really a bunch of powers of ten neatly stored in exponential form. You can steal some powers of 10 in order to move our decimal over to the right. Notice that when you stole the 10 × 10, you subtracted from the exponent in 104, effectively turning .038 × 104 into 3.8 × 102. Similarly, 5,000 × 104 becomes 5 × 107. 76 visual math Sometimes you may have to move the decimal point to the left. Let’s convert 5,000 × 104 into standard scientific notation format. First, notice that 5,000 = 5 × 1,000. When you move the decimal point to the left (like when you go from 5,000 to 5.0) you donate some powers of 10, and you add to the exponent. fractions and decimals 77 To summarize: Exercise 6: Use the rules for dividing decimals and the rules for dividing exponents to solve the expressions below: 6.3 × 104 2.1 × 108 = ■ ■ Express the answer in scientific notation. Express the answer as a decimal. Hint: Divide the 6.3 by the 2.1 and the 104 by the 108 and then convert to standard form if necessary. 78 visual math Use the following diagram to convert 6,500 × 106 into standard scientific notation form. fractions and decimals 79 solutions to chapter exercises Exercise 1: 80 6 12 = 12 ■ 6 out of the 12 days were sunny. ■ 3 out of 12 days were 40°. ■ 5 out of 12 were 55° or higher. ■ Only one day was the temperature greater than 60°, so the fractional part is 112. visual math 3 12 = 14 5 12 Exercise 2: + 192 = 1132 . 13 ÷ 12 = 1 with a remainder of 1. Since you’re dealing with twelfths, you stick the remainder over 12, making the answer 1112. Note that in the diagram, you could also move 3 slices from the 142 and turn the 192 into a whole, or 1. You would then have 1 whole and 1 slice, or 1112: 4 12 fractions and decimals 81 Exercise 3: Exercise 4: 1 ■ 2 ■ 82 ÷ 2 = 12 ÷ 21 = 12 × 12 = 14 2 ÷ 14 = 2 × 41 = 8 visual math Exercise 5: First, estimate how much of the wall has been wallpapered so far. She has completed 31 and it cost her $15. To complete each additional third, it will cost $15. Since there are two-thirds left that need coverage, she will pay an additional 2 × $15 = $30. 6.3 × 104 6.3 104 Exercise 6: 2.1 × 108 = 2. 1 × 108 ■ 6.3 ÷ 2.1 = 3, and 104 ÷ 108 = 104 − 8 = 10−4, so you have 3 × 10−4. This answer is already in the standard form of scientific notation. ■ 1 3 × 10−4 = 3 × 1104 = 3 × 10,000 = 3 ten thousandths = .0003. To convert 3 × 10 −4 to a decimal, move the decimal 4 places to the left. (move 4 places to the left) fractions and decimals 83 6,500 × 106 = 6.5 × 109 You moved the decimal 3 places to the left and added 3 to the exponent. 84 visual math chapter three Ratios and Proportions ratios Ratios are a way of comparing numbers. If your basketball team has 3 boys for every girl, you can express this as a 3 to 1 ratio. You can express ratios in three ways: ■ In sentence form: There is a three to one ratio of boys to girls. ■ By using a colon: 3:1 By using a bar: 31 ■ ratios and proportions 85 Your team: Just like fractions, ratios can be reduced to smaller terms. Someone may count that your team has 12 boys and 4 girls. That person may comment that there is a 12 to 4 ratio of boys to girls. This is true because mathematically 142 reduces to 31. Notice that when dealing with a 3:1 ratio, the total is going to be a multiple of 3 + 1, or 4. Here you have 4 sets of 4 making a total of 16 teammates. Exercise 1: Use the diagram below to solve the following question. In a pond near Homer’s nuclear power plant, there are 3 one-eyed starfish to every 1 porcufish. If there are 30 one-eyed starfish, how many porcufish are there in the pond? 86 visual math You just planted yourself a garden. You decided to go with a 3:2 ratio of tulips to daisies. You have 36 tulips. How many flowers do you have in all? Your Garden: Thus, you have 12 groups of 5, or 60 flowers. Coincidentally, the day after you plant all of your flowers, your neighbor plants tulips and daisies in the same 3:2 ratio. You nonchalantly stroll past his house and quickly count that he has 14 daisies; how many flowers does he have in all? ratios and proportions 87 You know that when dealing with a 3:2 ratio, the total is going to be a multiple of 3 + 2, or 5. You just do not know how many sets of 5 the neighbor has . . . But, you do know that he has 14 daisies . . . 88 visual math Thus, he must have 5 × 7 = 35 flowers in all. ratios and proportions 89 Exercise 2: Use the diagram below to solve the following question. In an art classroom, there is a ratio of 7 pencils to 2 rolls of tape. If there are 108 pencils and rolls of tape combined, how many pencils are there? Now you’ll look at a trickier ratio question. This question involves a ratio that changes. There are currently 24 people at a costume party and there is a 2:1:1 ratio of Elvis impersonators to Napoleons to Caesars. I wish I was at this party. 90 visual math How many consecutive Napoleons must walk through the door in order for there to be a 2:2:1 ratio of Elvis impersonators to Napoleons to Caesars? The current 2:2:1 ratio means that the total is a multiple of 2 + 1 + 1 = 4. You know there are 24 people present, so there must be 6 groups of 4: We need to squeeze a new Napoleon into each set in order to generate the desired 2:2:1 ratio . . . Thus, 6 Napoleons must walk through the door. ratios and proportions 91 algebra & ratios Let’s look back at your team. As we reminisce, let’s contemplate the algebra behind solving ratio questions. Let’s look at the team again: Everybody’s favorite variable is x, so let’s take that 3:1 boy to girl ratio and say the number of boys is 3 times some number, or 3x. The number of girls is 1 times the same number, or 1x, and the total is then 4x. Here you can just look at the team picture to see that x equals 4 because there are 4 groups of 4. There are 3 × 4, or 12 boys. There are 1 × 4, or 4 girls. And there are 4 × 4, or 16 players in all. When you looked at the garden with the 3:2 ratio of tulips to daisies, you were given a situation in which you knew that the neighbor had 14 daisies and you wanted to figure out the total. 92 visual math You knew the total for a 3:2 ratio would be a multiple of 3 + 2, or 5. Using an x you could write: tulips + daisies = total 3x + 2x = 5x When you were given that there were 14 daisies, you were actually given the value of 2x. tulips + daisies = total 3x + 2x = 5x Knowing 2x = 14, means x = 7. If x is 7, then the total, or 5x = 5 × 7 = 35. Algebra comes in handy when working with large numbers . . . At Maynard High School there used to be a 2:3 ratio of males to females. After a group of males enrolled, the ratio changed to 4:3. If there are 350 students ratios and proportions 93 in the school now, how many people were in the school before the new males enrolled? To solve a question like this, you need to look at the before and after. Before you had a 2:3 ratio. The total was a multiple of 2 + 3, or 5. You do not know how many groups there were, but each group looked like this: Before: The males will be 2x and the females will be 3x, so that the 2:3 ratio is preserved. males + females = total 2x + 3x = 5x After, you have a 4:3 ratio, and the total is a multiple of 4 + 3, or 7. Since there are 350 students, you must have 350 ÷ 7 = 50 groups that look like this: If there are 50 groups that look like that, there must be 4 × 50 = 200 males. Also, there must be 3 × 50 = 150 females. Because you know that the number of females remained constant, you know that the 3x in the before scenario must equal 150. males + females = total 2x + 3x = 5x If 3x = 150, then x = 50. This means that the before total was 5x = 5 × 50 = 250. proportions Proportions are just two equal ratios. When written as a:b = c:d, we call a and d the extremes (they are on the end), and we call b and c the means (they are in the middle). 94 visual math Note that the product of the means equals the product of the extremes. We usually set proportion up by using ratios with bars. You can then cross multiply to solve for an unknown. When you set up a proportion, just be sure to align the units correctly. For example, suppose that a cat that is 1 foot tall casts a shadow that is 2 feet long. How tall is a lamppost that casts an 18-foot shadow? cat 1 ft = ? feet cat’s shadow 2 ft 18 ft lamppost lamppost’s shadow Cross multiply to get 18 × 1 = 2 × ?, or 18 = 2 × ?. Dividing both sides by 2, you get ? = 9 feet. ratios and proportions 95 Exercise 3: Many cell membranes have a pump which pumps 3 sodium ions out for every 2 potassium ions that it pumps into the interior of the cell. If 2,400 sodium ions were pumped out, how many potassium ions were pumped in? converting units Proportions can be used to convert units. Look at the chart below: 1 foot = 12 inches 1 cup = 8 ounces 3 feet = 1 yard 1 pint = 2 cups 1 mile = 5,280 feet 1 quart = 2 pints 1 min = 60 seconds 1 gallon = 4 quarts 1 hour = 60 minutes 1 meter = 100 cm 1 meter = 10 decimeters 1 meter = 1,000 millimeters If you know that 1 cup equals 8 ounces, you can easily figure out how many cups are in 64 ounces by setting up a proportion: 1 cup ? cup 8 o z = 64 oz 96 visual math Cross-multiplying, you get 64 • 1 = 8 • ?, or 64 = 8 • ?. Dividing both sides by 8, you get ? = 8. conversion factors Conversion factors are just a “version of 1” because the top and bottom are 1 ft equivalent. For example, 12 in is a conversion factor. You can multiply a quantity that you have by a conversion factor whenever you want to get rid of unwanted units. Why would you bother with conversion factors when you can just set up proportions? Consider this . . . A snail slides along at a rate of 2 cm per minute. How many meters does it move in an hour? Here you have to convert centimeters to meters and minutes to hours. You can do this easily with conversion factors: Notice that you strategically placed the centimeters in the bottom of the conversion factor so that it would cancel out the unwanted units on top. To get rid of the “unwanted” minutes on the bottom, you make a conversion factor that has minutes on top: ratios and proportions 97 120 m 1.2 m You multiply to get r . 100 hr = h cm 5m Exercise 4: Convert min into sec by using conversion factors: Convert units prior to setting up proportions! You should always convert your units before you set up a proportion. Suppose you knew that 8 inches of ribbon costs $.60, and you needed to order 20 feet of it. First, you would convert the 20 feet into inches. You know 1 ft = 12 in, so you multiply: 20 ft × 12 in = 240 inches 1 ft Now you know that you need a price for 240 inches. You set up a proportion: 8 inches 240 inches $.6 0 = $ ? 8″ ($ ?) = ($.60)(240″) 8″ ($ ?) = 144″ $ ? = $18 98 visual math direct proportions When two quantities are directly proportional, it just means that if one quantity increases by a certain factor, the other quantity increases by the same factor. If one decreases by a certain factor, the other decreases by that same factor. For example, the amount of money you earn may be directly proportional to the amount of hours that you put in. This is true if you are in a situation where, if you work twice as long, you make twice as much. If you work 12 the amount you usually work in one week, you earn 12 the amount you usually make. In the equation Force = mass × acceleration (otherwise known as F = ma), you can see that for a given mass (like a manned car) the force of impact is directly proportional to the acceleration. Suppose a car hits a tree while traveling at a certain acceleration . . . How much greater would the force of impact be if the car was accelerating twice as fast? Well, you know that the mass is going to remain the same, so “lock” that number in the formula. In order for both sides to be equal, if you multiply one side of the equation by 2 (by doubling the acceleration), then you must also multiply the other side of the equation by 2 (hence, the force doubles as well) . . . ratios and proportions 99 Exercise 5: Which graph do you think could represent the relationship between dollars earned versus hours worked? Which one would represent the relation between force and acceleration? inverse proportions Two quantities are inversely proportional if an increase by a certain factor for one is accompanied by a decrease by that same factor for the other. We find a lot of these types of relationships in science, so let’s take a look at some physics. Once upon a time, physicists dreamed of a land where the flow of fluids was ideal—that is to say that fluids weren’t subjected to the perils of resistance, turbulence, and the like. These physicists loved this idea so much that they came up with a formula for ideal flow. The ideal flow, Q, which is a constant, is equal to the product of the cross-sectional area times the velocity: 100 visual math Q=A•V Now it’s your turn to imagine flowing fluids. Imagine that water is flowing through a pipe and suddenly the pipe widens to a cross-sectional area that is twice as big: What happens to the velocity of the water? If you increase A, then in order for Q to remain constant, what happens to V? Right! It decreases. ratios and proportions 101 In fact, A gets multiplied by 2, V gets multiplied by 12, and Q remains the same. Q = (A • 2) (V • 12) Q = AV • 2 • 12 Q = AV • 1 Q = AV Q is unchanged. Exercise 6: Which graph do you think could represent the relationship of A and V as described above? 102 visual math solutions to chapter exercises Exercise 1: Use the diagram to count the 30 one-eyed starfish. For each set of 3 starfish, there is 1 porcufish, so there must be 10 porcufish. ratios and proportions 103 Exercise 2: There is a 7:2 ratio, so the total is a multiple of 7 + 2, or 9. As a matter of fact, you were given a total of 108. This means there are 12 sets of 9. If you count up all of the pencils in these 12 sets, you get . . . Exercise 3: The pump pumps 3 sodium ions out for every 2 potassium ions that it pumps in. If you want to find how many potassium ions were pumped in when 2,400 sodium ions were pumped out, you just set up a proportion: 2,400 sodiums 3 sodiums ? potas siums 2 potas siums = Cross multiplying, you get 3 • ? = 2 • 2,400, or 3 • ? = 4,800. Dividing both sides by 3, you get ? = 1,600. Thus, 1,600 potassium ions were pumped in. 104 visual math Exercise 4: Using the facts 1 m = 100 cm and 1 min = 60 seconds, you can generate two conversion factors. Be sure to arrange your units so that you can cancel out ones you want to get rid of. You can get rid of meters by putting meters on the bottom of your conversion factor . . . You can get rid of minutes by putting minutes on top in our conversion factor . . . 813 cm 500 cm You multiply to get 60 sec = sec . ratios and proportions 105 Exercise 5: Dollars earned versus hours worked . . . Force versus acceleration . . . 106 visual math Exercise 6: You saw that as A increases, V decreases . . . ratios and proportions 107 chapter four Percents what is a percent? Percents are a way of creating a special ratio. When you see a number followed by a percent symbol—%, you just write a ratio comparing that number to 100. For example, 30% = 13000 . You can express a percent in two ways: as a fraction (just put the number over 100), or as a decimal (move the decimal point two places to the left). These two options are summarized on the following page: percents 109 For example: 40% = 14000 = .40 Let’s look at 25%. You stick 25 over 100 to get 12050 . Notice that 12050 reduces to 14. It is good to be familiar with some on the common fraction and decimal equivalents to percentages. Some are listed in the chart below: 110 Percent Fraction Decimal 25% 1 4 .25 50% 1 2 .50 75% 1 4 .75 visual math Exercise 1: Fill in the chart below. More than one response may be correct. When I see . . . I will write . . . 25% 1 4 32% 80% 100% 150% 500% taking the percent of a number Recall that “of ” means “multiply.” When you take the percent of a number, you multiply. Let’s say you are purchasing an item that usually costs $8, but is now on sale for 25% off. How much do you take off of the $8? You need to find 25% of $8? Remember that 25% = 12050 , or 14. So you are taking off 14 of $8. 1 4 of $8 means 14 × 8 = $2 off. percents 111 Notice that you are saving 25% of the original price, and you are paying 75% of the original price. There are always two ways of looking at the situation: In the diagram below, you see that a store is having a 20% off sale. Two shoppers are thinking about what this sign means to them: 112 visual math They are both looking at the situation differently, and they are both correct. Let’s say both women find a dress for $80. They want to do a quick mental calculation before deciding whether to buy it or not. Who’s going to make a decision quicker? Here is what they each think: First, the lady with the dog . . . And, now for the lady with the bags . . . The lady with the bags made a quicker calculation that did not require subtraction. Eliminating the need to subtract saves time when you are performing mental calculations. percents 113 Since we were discussing sales, you were thinking in terms of “I will save” versus “I will pay.” You can use this line of thinking in other situations as well. For example, if a container is 25% (or 14) full, then it is 75% (or 34) empty. Exercise 2: Evelyn found a dress she loves, selling for 40% off the original $85 price. If she lives in a state that doesn’t charge tax on clothes, how much is the dress? 114 visual math Exercise 3: Jim would like to use the image below for one of his graphics projects. He decides that he wants to reduce the length of the image by 25%, and keep the width the same. Shade in the area that represents how big the image will be after he reduces the length. percent of a percent When you take the percent of a percent, you just multiply. For example if you wanted to know what 40% of 20% of 600 was equal to, you would multiply: 40% of 20% of 600 .40 of .20 of 600 .40 × .20 × 600 = 48 percents 115 percent proportion Imagine you had a special Percent Thermometer that you could carry around with you that could instantly translate quiz scores to the equivalent grades out of 100? This device would instantly tell you the percentage that you got right: Well, there’s no magic thermometer that calculates percentages, but there’s an easy way to figure this out for yourself. You just set up a percent proportion: Your score out of 20 1290 = 10?0 Your score out of 100 Cross-multiply to get 19 × 100 = 20 × ?, or 1,900 = 20 × ?. Dividing both sides by 20 you get 95. Thus, your score is equivalent to 19050 , or 95%. unknown percent When you see the phrase “four percent,” how do you express this mathematically? That’s right, you write 1400. When you see the phrase “twenty-three percent,” how do you express this mathematically? You simply write, 12030 . 116 visual math When you see the phrase “what percent,” how do you express this mathematically? Hmmmm. Some of you are thinking, “I don’t know.” Well, you’re right: I don’t know “What percent” means: 10 0 Well, for the most part, we mathematicians don’t go around writing, “I don’t know” in the place of the numbers that we don’t know. We are partial to letting x represent the “unknowns” that we come across. “What percent” means: 1x00 Here’s an example: What percent of 250 is 30? Let’s break this down: “What percent” means: 1x00 “of 250” means: • 250 “is 30” means: = 30 So you have: x 100 • 250 = 30 x • 250 100 = 30 You cross-multiply to get: x • 250 = 30 • 100 x • 250 = 3,000 Dividing both sides by 250, you have: x = 12 percents 117 Exercise 4: What percent of 60 is 2? Break it down: “What percent” means: “of 60” means: “is 2” means: Put it all together and solve: percent change, percent increase, and percent decrease Here’s a secret. If you ever see a question involving percent change, percent increase, or percent decrease, you can use this multi-purpose formula. That’s right! No more stressing over terminology, folks. Too many people have been ripping their hair out over percent increase, percent decrease, and percent change! The madness must stop! Memorize this formula, and you can tackle all three types of questions as easily as a stampeding bull tackles the man with the red scarf. 118 visual math Use this formula for: ■ Percent Increase ■ Percent Decrease ■ Percent Change What you put in this formula: ■ The “change” is just the change in value. If something was $17 and now it is $10, the change is $7. ■ The “initial” is the initial value. If something was $17 and now it is $10, the initial value is $17. Sticking the change over the initial creates a ratio of the change in value to change the initial value. (The ratio is initial .) This is the premise of any percent change, percent increase, or percent decrease question. Once you know the ratio of the change to the initial, you can figure out how much of a change out of 100 this is equivalent to. How much of a change is it? I don’t know, you say. Hence, the “?” in the equation. change ? Your change out of initial initial = 100 Your change if out of 100 Let’s apply this formula. Jade invested $80 in tech stocks. By the next week, her stock was already worth $120. What is the percent increase in the stock value? percents 119 You use: change ? init ial = 10 0 Here the change is $120 − $80 = $40. The initial value is $80. You put these numbers into the formula: 4800 = 10?0 . Cross-multiplying, you get: 4,000 = 80 • ?. You divide both sides by 80 to get ? = 50. Thus, there was a 50% increase. You can see that this is true: 120 visual math Next, consider a percent decrease. The snowfall in your town was 120 inches throughout the year 2001, and only 90 inches throughout the year 2002. Calculate the percent decrease. You use: change ? init ial = 10 0 Here the change is 120 inches − 90 inches = 30 inches. Note that you don’t have to worry about the change being −30. You know that the snowfall decreased, so you know you are calculating a percent decrease. Don’t even bother with the negative sign. The initial value is 120 inches. You put these numbers into the formula: 30 ? = . Cross-multiplying, you get: 3,000 = 120 • ?. You divide both sides 120 100 by 120 to get ? = 25. Thus, there was a 25% decrease. percents 121 You can see that this is true: Exercise 5: Refer to the chart below in order to calculate the percent change in recalled trucks from 1998 to 1999. 122 visual math successive percent changes When you have a situation where there is a percent increase, followed by a percent decrease, followed by another percent decrease, be forewarned: There are no shortcuts when dealing with successive percent changes! What you think may be a shortcut may ultimately mean that you get a wrong answer! Let’s look at a 75% decrease followed by a 50% increase. Here, you solve this question the smart way: percents 123 Many see a 75% decrease followed by a 50% increase and think that they could just calculate a 25% decrease. This is WRONG: When taken successively, a 75% decrease followed by a 50% increase is the same as a 62.5% decrease (note that you are left with 37.5% of the original amount). The doomed “shortcut” method was to take a 25% decrease, and that yielded 75% of the original amount. Exercise 6: Use the grid below to demonstrate a 40% decrease, followed by a 50% increase. 124 visual math simple interest You calculate simple interest with a simple formula: Interest = Principal × Rate × Time What we put into the I = PRT formula: ■ The interest is the money you make on top of your initial investment. ■ The principal is your initial investment. In other words, it is the amount of money you started with. ■ The rate at which you earn money is expressed as a percent. You may earn money at a rate of 3%. This means you would put R = 3%, or better yet, R = .03 into the I = PRT formula. ■ The amount of time that your investment earns money is always expressed in years. Make sure you convert the time into years if it isn’t given in years. For example, if the rate of interest is 20% a year, what will the interest be on a $6,000 investment for 2 years? 1. Express the percent as a decimal (or fraction). Thus, R = 20% = .20 2. Make sure that time is in years. 3. Use the formula: I = PRT I = $6,000 × .20 × 2 = $2,400 percents 125 Exercise 7: Zoey earned $200 in 18 months at a rate of 10%. Find the principal. 1. Express the percent as a decimal (or fraction): 2. Make sure the time is in years: 3. Use the formula I = PRT: compound interest Sometimes you earn interest on your interest! For example, if you put money into an account that pays 5% interest compounded annually, 5% of your principal is added to your account after the first year. You would then have a new (and larger) principal that earns interest for the second year. Here is the compound interest formula: 126 visual math What we put into the A = P(1 + nr)nt formula: ■ A is the total amount ■ P is the original principal ■ r is the rate ■ n is the number of yearly compounds ■ t is time (in years) To find your “n” look out for these terms: ■ compounded annually means interest is paid each year ■ compounded semiannually means interest is paid two times per year ■ compounded quarterly means interest is paid four times per year ■ compounded monthly means interest is paid every month ■ compounded daily means interest is paid every day So, let’s look at an example. You open a savings account that pays 3% interest semiannually. If you put in $1,000 initially, how much do you have after 2 years? We use A = P(1 + nr)nt, and you substitute in the following values: P = 1,000 r = 3%, or .03 n = 2 (compounded semiannually means twice a year) t=2 A = P(1 + nr)nt = 1000(1 + .023 )2•2 = 1,000(1 + .015)4 = 1,000(1.015)4 = 1,000(1.06) – = 1,061.36 = 1,061.37 Always round money to the nearest cent. Thus, you’d have $1,061.37. percents 127 You ask: “What if I don’t want to memorize that scary formula?” Well, you have a few options: ■ ■ ■ ■ You can do the calculation “the long way.” For example you would know that after 12 a year, the $1,000 principal above would earn I = PRT, or I = 1,000 × .03 × 12 = $15. Now the account has $1015. In another 12 year you earn I = PRT = 1,015 × .03 × 12 = 15.23, and you would have $1,030.23. You would continue calculating in this manner until you completed two years worth of money making. You can find out if there is a reference sheet that may contain this formula (if you are taking a standardized test). You can use process of elimination on tests. Cross off any preposterous answers and try to pick one that would make sense. TIP: In doing a compound interest test question, you know that a lot of people would tend to accidentally solve it as if it were a simple interest question. And you can bet the test designers know this! So, you can cross off the answer that represents I = PRT (the simple interest formula) and pick an answer that is greater. Exercise 8: Evan opens a savings account that pays 5% interest quarterly. If he put in $2,000 initially, how much does he have after six months? algebraic percents Let’s say that Jaclyn buys a printer for D dollars and gets a 20% discount. How do you represent this mathematically? Well, if Jaclyn is getting a 20% discount, she must be paying 80% of the original price. What is the original price? D. So she is paying 80% of D. This is just .8 • D, or .8D. What if she was buying three items that cost D, E, and F dollars each, and she was getting the same 20% discount on her entire order? Well, without the discount, her cost would be (D + E + F ). non-discounted = (D + E + F ) 128 visual math Now you know that she will only have to pay 80% of the original price, so you multiply the non-discounted price by .8. discounted price = 80% of (D + E + F) discounted price = .8 • (D + E + F) = .8(D+E+F) Brian goes to a restaurant and wants to leave a tip that represents 20% of his bill. If his bill costs B dollars, how much will he spend on the cost of dinner and tip? Think it through: ■ He will pay the bill and leave a tip. ■ He will spend $B on the bill. ■ He will pay 20% of B for the tip, or .2B for the tip. ■ Bill + Tip = B + .2B ■ This adds to a 1.2B total. Mental Shortcut: ■ He is paying 100% of the bill (the whole bill) plus 20% of the bill, so he is paying 120% of the bill. ■ He pays 1.2B total. Consider this scenario: Jane’s store sells items for 50% more than she pays for them. If it costs her C dollars to purchase an item, how much will she charge for it? percents 129 Here’s a trickier question: How much water must be added to 2L of a 20% bleach−80% water mixture to yield a 10% bleach−90% water mixture? Note that you are only adding water to the initial 20% bleach−80% water mixture. This means that the amount of bleach is the same in both mixtures. The old mixture is 2 liters, and the new mixture is 2 liters plus some unknown amount, or (2 + ?) liters. You know that the amount of bleach that comprised 20% of the initial 2 liters is the same amount of bleach present when comprising 10% of the new (2 + ?) liter mixture: This means that 20% of the 2 liter mixture is the same as 10% of the (2 + ?) liter mixture. 20% of 2 liters = 10% of (2 + ?) liters .2 • 2 = .10 • (2 + ?) .4 = .10 • (2 + ?) .4 .10 =2+? 4=2+? 2=? Thus, 2L of water has to be added. 130 visual math solutions to chapter exercises Exercise 1: When I see . . . I will write . . . 25% 1 4 32% 32 8 10 0, 2 5, 80% 80 4 10 0 , 5, 100% 100 , 100 150% 150 3 , , 100 2 500% 500 , 100 .32 .8 1 1.5 5 percents 131 Exercise 2: If she is saving 40% off the price of the dress, then she is paying 60% of the price listed. 60% of $85 can be written mathematically as .60 × 85, which comes out to $51. Exercise 3: You want to reduce the length by 25%, or 14. The shaded area below represents the new dimensions of the image: 132 visual math Exercise 4: Given “What percent of 60 is 2?” you break it down as follows: x “What percent” means: 100 “of 60” means: • 60 “is 2” means: = 2 Put it all together and solve: 1x00 • 60 = 2 60x 100 =2 60x = 200 x = 313% Exercise 5: In order to calculate the percent change in recalled trucks from 1998 to 1999, use the formula: change ? init ial = 10 0 Here the change is 6,000 − 4,000 = 2,000. The initial value is 4,000. 2,000 1 ? You put these numbers into the formula: 4,000 = 100 . You reduce to get 2 = ? . Cross-multiplying, you get: 100 = 2 • ?. Divide both sides by 2 to get 100 ? = 50. Thus, there was a 50% change. Since you went from 4,000 to 6,000, you know that the change was +50%. percents 133 You can see that this is true: Exercise 6: First decrease by 40%. Next, increase by 50%. The final result is 90% of the original value. 134 visual math Exercise 7: You know Zoey earned $200 (that’s the Interest) in 18 months at a rate of 10%. Find the principal. 1. Express the percent as a decimal (or fraction): .1 2. Make sure the time is in years: 18 mo = 1.5 years 3. Use the formula I = PRT I=P×R×T 200 = P × .1 × 1.5 200 = .15P P = $ 1,333.33 Exercise 8: Evan opens a savings account that pays 5% interest quarterly. He put in $2,000 initially, so to find how much he has after 6 months, you use: A = P(1 + nr)nt Make sure to convert the 6 months into 12 yr. A = P(1 + nr)nt 1 = 2,000(1 + .045 )4•2 = 2,000(1 + .0125)2 = 2,000(1.0125)2 = 2,000(1.0251563) = 2,050.3125 = 2,050.31 percents 135 chapter five Algebra variables Variables are just numbers in disguise. It has always been a traditional pastime for young numbers to pretend that they were secret agents. These numbers tried to conceal their identity, first with sunglasses . . . algebra 137 . . . but that didn’t work out so well. Then they tried using paper bags, but they kept bumping into things. So, finally they decided to take on a new identity altogether! And hence, using a LETTER to represent a NUMBER became the standard in the world of numerical espionage. Numbers tried really hard to keep their true identities hidden, but cunning sleuths were always able to track down the clues they left behind. The sleuths could then rearrange these clues and deduce the true identity of the secret agents. 138 visual math Once you realize that these variables are just numbers in disguise, you’ll understand that they must obey all the rules of mathematics, just like the numbers that aren’t disguised. This can help you figure out what number the variable at hand stands for. simplifying equations and expressions Let’s look at an expression that doesn’t have any variables: 3(2) + 4(2) Would this be the same as 7(2)? You quickly solve both expressions and say Yes! But what underlying mathematical property tells us that this will be true? It’s the distributive property: 2(7) = = 2(3) + 2(4) algebra 139 If you saw 2(7), you probably wouldn’t change it to 2(3+4) and apply the distributive property, would you? Of course not! But understanding this mathematical concept will help you when you see operations with variables. For example, if you see 3x + 4x, you can think: And you would feel secure in saying that 3x + 4x = 7x; x is really just a number in disguise, and it must obey all the properties that govern numbers. Just as 3(2) + 4(2) = 7(2), 3x + 4x = 7x. When you combine the 3x and the 4x into 7x, you are “combining like terms.” What does this mean? 3x and 4x are considered like terms because they both involve x. Here, the 3 and the 4 are called coefficients of the x term. If you see 3x + 4x, you just combine them, or add them to get 7x. A term with no coefficient actually has a coefficient of one: Always combine like terms: Combine the x-terms with x-terms, and x -terms with x2-terms. Combine y-terms with y-terms, combine nondisguised numbers with the other non-disguised numbers, and so forth. 2 140 visual math For example, if you see something like 2x + 3x2 + 4x2 + 5x, you can immediately simplify the expression by spotting like terms: You would then combine like terms to get 7x + 7x2. Exercise 1: Simplify the following expression. 2x + 3x2 + 5x − x2 + x + 7x3 Sometimes you can simplify an equation by multiplying the entire equation by a certain number. This is mathematically acceptable because you are just generating an equivalent multiple of the equation. Suppose you had 2x = 2. algebra 141 Now let’s take a multiple of 2x = 2. Let’s multiply the whole equation by 2: What if you had multiplied the 2x = 2 equation by 3? Let’s look at a situation where multiplying the entire equation by a certain number would come in handy. For example, let’s look at: 1 1 2 4x + 6 = 3 To get rid of the fractions, you can multiply the entire equation by 12: 12 (14x + 1 6 = 23) You distribute the 12 to get: 12 • 14x + 12 • 16 = 12 • 23 3x + 2 = 8 142 visual math This is a much simpler equation than the one you started with. x 3x 2x Exercise 2: (3) + (1 0 ) − (5) is equivalent to 7x a. 1 5 31x b. 3 0 8x c. 1 8 7x d. 3 0 solve for x When you want to solve for x, you want to get x all by itself. We call this “isolating the variable.” In order to preserve the equality of the given equation, you need to be sure that you are doing the same thing to both sides of the equation. This means that you should perform corresponding operations on both sides of the equals sign. If you subtract 2 from the left side, you need to subtract 2 from the right side. If you divide the left side by 3, you must divide the right side by 3. For example, let’s look at 2x + 7 = 15. You will solve for x. You have 2x + 7 = 15. In order to get x by itself, first get rid of the 7. This means you will subtract 7 from both sides. 2x + 7 = 15 −7 −7 2x = 8 Next, divide both sides by 2 in order to get x by itself: 2x 8 = 2 2 x=4 algebra 143 Exercise 3: Solve for x. Given 7x + 2 = 5x + 14, what is the value of x? inequalities Inequalities contain the greater than, less than, greater than or equal to, or less than or equal to symbols. When you solve the inequalities for x, you can figure out a range of numbers that your unknown is “allowed” to be. This symbol . . . Means . . . > “Greater than” ≥ “Greater than or equal to” < “Less than” ≤ “Less than or equal to” Now, there is one rule that you need to remember when dealing with inequalities: When you multiply or divide by a negative number, you need to reverse the sign. For example: −5x + 3 > 28 can also be expressed as which of the following? a. x < −351 b. x > −351 c. x > −5 d. x < −5 144 visual math This type of question is a lot like the “Solve for x” questions that you did above. The goal here is to isolate your x. First you will subtract 3 from both sides. −5x + 3 > 28 −3 −3 −5x > 25 When you multiply or divide by a negative number you need to reverse the sign. So when you divide by -5, you get: 25 −5x > −5 −5 x < −5, which is choice d. On a number line, this answer looks like: Here’s a trick to avoid having to worry about flipping the sign: Just move your terms in a manner such that you will end up with a positive coefficient on your variable. Let’s look at −5x + 3 > 28 again. Our x coefficient is negative, so you will add 5x to both sides. −5x + 3 > 28 +5x +5x 3 > 28 + 5x Next, you subtract 28 from both sides: 3 > 28 + 5x −28 −28 3 − 28 > 5x −25 > 5x algebra 145 You divide both sides by 5 to yield: −5 > x Let’s look at another example. Suppose you had x2 + 12 > 16 and you wanted to represent this on a number line. First, try to isolate x. You can get rid of the 12 by subtracting 12 from both sides: x2 + 12 > 16 −12 −12 x2 > 4 If x2 is greater than 4, then what do you know about x? What number squared equals 4? Both + 2 and −2, when squared yield 4. Any x greater than 2 will yield an x2 greater than 4. Any x less than −2 will yield an x2 greater than 4: Exercise 4: Graph the solution to x2 + 12 ≥ 42 on the number line below. 146 visual math english to equation In English we can say, “Felicia is 3 years older than Samantha.” You can write this mathematically as: ■ ■ ■ Felicia is 3 years older than Samantha F= +3 S You can put this all together: F = S + 3. Let’s say Felicia is 3 years younger than Christina. How can you express this mathematically? ■ ■ ■ Felicia is 3 years less than Christina F= −3 C You can put this all together: F = C − 3. Notice that this means that if you take Christina’s age and subtract 3 years, you will end up with Felicia’s age. Let’s say that there is 1 more sister. Melissa is twice as old as Samantha. How can you express this? ■ ■ Melissa is twice as old as Samantha M= S×2 You put this all together to get M = S × 2, or M = 2S. Exercise 5: How can you represent the following phrase mathematically? Erik has 3 CDs less than Danny. a. D = E − 3 b. D = 3 − E c. E = 3 + D d.E = D − 3 algebra 147 Let’s look at a trickier example: Joe only owns 12 more than half the amount of CDs stacked on his dresser. The rest were borrowed from a friend. If there are a total of 52 CDs in the stack, which equation represents the amount of CDs that he borrowed, B? a. B = 12 + (12 • 52) b. B = 52 − 12 c. B = 12 • 52 − 12 d.B = 52 − (12 + 12 • 52) First, realize that there are 52 CDs total, and that some are Joe’s and some are the ones he borrowed. So the basic idea would be: 52 total CDs = # Joe’s + # Joe borrowed. You know you should call the borrowed CDs B, and if you similarly call the number of Joe’s CDs J, you know 52 = J + B. Because you know that you need to find B, rearrange this equation by subtracting J from both sides: 52 = −J J+B −J 52 − J = B Hence, you know that B = 52 J. But none of the answers have a J! This means you need to be more specific about J. What do we know about J, or the number of CDs that Joe owns? Well, the question states that: “Joe only owns 12 more than half the amount of CDs stacked on his dresser.” You need to express this statement mathematically. If Joe owns12 more than half the total amount, and you know that the total is 52, then he owns 12 more than 12 of 52. Remember that more than means plus, and of means multiply. Mathematically, you know J = 12 + 12 • 52. Now write 12 + 12 • 52 in place of J in the equation B = 52 J. So, the answer is d. 148 visual math substitution Sometimes algebra is just a matter of sticking, or “substituting,” numbers in for the right variables. For example, you are told that a = b + c. If b = 5 and c = 7, what is the value of a? You simply substitute 5 for b and 7 for c to get: a=5+7 = 12 2 b +b when b = 1 and a = 2? Example: What is the value of a Here you put 1 in for each b you see in the equation. You put 2 in for the a in the denominator: b2 + b 12 + 1 1+1 2 a = 2 = 2 = 2 = 1 Exercise 6: What is the value of the expression 5x2 + 2xy3 when x = 3 and y = −2? a. −3 b. 3 c. −93 d.93 function tables Function tables portray a relationship between two variables, such as an x and a y. It is your job to figure out exactly what that relationship is. Let’s look at a function table: algebra 149 x y 0 — 1 4 2 5 3 — 4 7 Notice that some of the data was left out. Don’t worry about that! You can still figure out what you need to do to our x in order to make it our y. You see that x = 1 corresponds to y = 4; x = 2 corresponds to y = 5; and x = 4 corresponds to y = 7. Did you spot the pattern? Our y value is just our x value plus 3. Exercise 7: The table below shows the relationship between two variables: x and y. Write an equation that demonstrates this relationship. x y 0 0 1 — 2 4 3 — 4 16 algebraic formulas Two algebraic formulas that you should be familiar with are: ■ D=R×T ■ w×d=W×D 150 visual math The first formula is sometimes called the Constant Rate Equation, and D = R × T means: Distance = Rate × Time Distances may be given in miles, meters, feet, etc. Rates may be given in meters per second (m/s), miles per hour (mi/hr or mph), feet per second (ft/sec), and so forth. Just be sure to check that the units you are using will work together. For example, it “works” when you multiply mi/hr by a time in hours, because the hours cross out to yield the distance in miles: mi hr × hr = mi A dog is walking at a rate of 5 meters per minute. 80 meters away, a turtle is walking right at him at a rate of 1 meter a minute. How long before they meet? It is easy to solve this kind of question if you draw a diagram: You know that the initial distance apart, 80 m, is equal to D1 + D2. 80 = D1 + D2 You know D1 = R1 × T and D2 = R2 × T, so you can rewrite the above equation as: 80 = (R1 × T) + (R2 × T) You know the dog walks at 5 m/min (this is R1) and the turtle walks at 1 m/min (this is R2), so put these two rates into the equation: 80 = 5T + 1T 80 = 6T algebra 151 Dividing both sides by 6 yields T = 1313 minutes. One-third of a minute = 13 • 60 minutes = 20 seconds. The answer can also be written as 3 minutes and 20 seconds. Suppose Tamara is standing on the corner with her two pups, and one gets 4m loose. The pup takes off at a rate of sec . By the time Tamara notices that her pup broke free, two seconds have passed. She starts running after the dog at a rate of 6 m/sec. How far will she have to run until she catches the dog? This is a slightly confusing situation, but let’s draw a diagram to visualize what is happening. So far we know: 4m The fugitive pup runs at se c. The pup gets a 2 second head start. Since D = R × T, you know that the pup has a D = 4 × 2 = 8 m head start. 6m Tamara runs at a rate of se c. So you draw: You want to know the amount of time it will take for Tamara to travel the distance labeled D2. You can see that D2 = D1 + 8. Use the formula D2 = R2 × T, and replace D2 with D1 + 8. 152 visual math D2 = R2 × T D1 + 8 = R2 × T 4T + 8 = 6T 8 = 2T 4=T It will take her 4 seconds. Exercise 8: Train 1 is traveling eastbound at 60 mph. On an adjacent track to the east, Train 2 is traveling at 50 mph in a westbound direction. If the two trains departed from their respective stations at the same time and were initially 220 miles apart, how long will it take for them to pass each other? The second equation that you should know deals with balancing a fulcrum. For a balanced fulcrum, w × d = W × D which means: Weight1 × Distance1 = Weight2 × Distance2 Suppose your 180-pound uncle is sitting in his 20-pound easy chair on a balanced fulcrum. He is 4.5 feet from the pivot point. On the other side is an anvil. It has been placed 3 feet away from the pivot point. How much does it weigh? Let’s look at a diagram: algebra 153 We will call the anvil’s weight w, and you will put all the other information into the formula: w×d=W×D w × 3 = 180 + 20 × 4.5 w × 3 = 200 × 4.5 3w = 900 w = 300 Thus, the anvil weighs 300 pounds. This equation works for a balanced fulcrum. If you moved the anvil to a point 4 feet away from the pivot point, then w × d would be 300 × 4 = 1,200. This is greater than W × D, which you calculated as 900. In this case since wd > WD, the plank will tilt to the left: factoring Sometimes expressions are easier to deal with when you “pull out” common factors from the terms. For example, if you had 6xy2 + 3xy, you could notice that each term is divisible by 3. Each term is also divisible by x. And each term is divisible by y as well. 154 visual math You can “pull out” a 3xy from each term to yield 3xy(2y + 1). You can check your work by distributing the 3xy: = 3xy • 2y + 3xy • 1 = 6xy2 + 3xy Exercise 9: Factor the expression 20x2y + 15x + 10xy. foil “Foil” is an acronym for “First, Outer, Inner, Last, “ which is the way in which you combine terms within two sets of parentheses. Let’s look at the following question: Which answer choices mathematically express the product of 2 more than x and 3 less than twice x? a. 3x2 + 7x + 6 b. 3x2 − 7x − 6 c. 3x2 + x − 6 d.3x2 + x + 6 algebra 155 You are asked to find the product so you know that you will be multiplying. What exactly are you multiplying? Well, one of the quantities given is “2 more than x,” which is just (x + 2). The second quantity given is “3 less than twice x,” which can be expressed mathematically as (2x − 3). When you multiply (x + 2) by (2x − 3), you get: (x + 2) (2x − 3) This would be a perfectly good answer except for one problem: It is not one of your choices! So after muttering comments about the question designer under your breath, you’ll realize that you need to change the way you write your answer. Specifically, you must expand your current expression. You expand out (x + 2) (2x − 3) by using FOIL. FOIL is just an acronym for FIRST, OUTER, INNER, and LAST. It describes the order in which you multiply your two sets of parentheses: (x + 2) (2x − 3) = 2x2 − 3x + 4x − 6. This simplifies to 3x2 + x − 6, which is choice c. Exercise 10: Which answer mathematically expresses the product of five more than x and 1 more than twice x? a. x2 + 11x + 5 b. 2x2 + x + 5 c. 2x2 + 10x + 5 d.2x2 + 11x + 5 156 visual math reverse foil In the above examples you used FOIL to expand expressions. Sometimes you will see an expression in expanded form, and you will need to perform a “reverse FOIL” in order to solve for your unknown. In other words, you are finding the factors of the given expression. What are the solutions to x2 + 6x − 16 = 0 The expression x2 + 6x − 16 = 0 can be factored into two sets of parentheses: (x ± ?)(x ± ?) = 0 Because the coefficient of the x2 term is 1, you know that the sum of the 2 missing numbers is 6 (the coefficient of the x term) and the product of the 2 missing numbers is -16 (the lone number). The two numbers that satisfy these conditions are -2 and 8. Fill in the parentheses: (x 2)(x + 8) = 0 You have two quantities that, when multiplied, yield zero as the answer. Simply put, you have: something something = 0 If the answer is zero, then you know that one of those quantities (one of those “somethings”) has to be zero. So you set both of those “somethings” equal to 0. (x 2)(x + 8) = 0 x − 2 = 0| x + 8 = 0 | x = 2| x = −8 Thus, the solutions are x = 2 and x = −8. algebra 157 Exercise 11: If x is a positive number, and x2 − 8x − 9 = 0, what is the value of x? a. −1 b. 3 c. 9 d.both b & c e. both a & c simultaneous equations When you are dealing with equations that have more than one variable, instead of isolating your variable, sometimes it is easier to use simultaneous equations. All you need to do is arrange your two equations on top of one another. Be sure that the like terms are written in the same order. You can add the equations together, or you can subtract them. If you are strategic, you can figure out what to do in order to calculate the answer to the question. Always pay close attention to what the question “wants.” Let’s look at an example: If 2x + y = 13, and 5x − y = 1, what is the value of x? a. −2 b. 1 c. 2 d.3 Let’s set up a simultaneous equation. Keep in mind that the question asks us for the value of x. Arrange your two equations on top of each other, and in this case, you will add them together: 2x + y = 13 + 5x − y = 1 7x = 14 x= 2 158 visual math Note that adding them together was a good choice, because you got rid of the y variables. 2x + y = 13 + 5x − y = 1 7x = 14 x= 2 Now you know x = 2, choice c. Exercise 12: If 3x + y = 40 and x − 2y = 4, what is the value of y? a. −535 b. 4 c. 535 d.8 Hint: You want to find the value of y, so try to get rid of x. algebra 159 solutions to chapter exercises Exercise 1: Combine like terms: 2x + 3x2 + 5x - x2 + x + 7x3 = 2x + 3x2 + 5x − x2 + x + 7x3 = 8x + 3x2 − x2 + 7x3 = 8x + 3x2 x2 + 7x3 = 8x + 2x2 + 7x3 Exercise 2: d. Presented with the question: x 3x 2x (3) + (1 0 ) − (5) is equivalent to 7x a. 1 5 31x b. 3 0 8x c. 1 8 7x d. 3 0 First, notice that all the answer choices are expressed as fractions. That’s your clue that you need to find a common denominator for these terms. The denominators are 3, 10, and 5, so 30 would be a great common denominator. To turn x3 into something over 30, you are multiplying top and bottom by 10. This first term becomes 1300x . Now look at the second term, 31x0 . To turn this term into something over 30, you’ll multiply top and bottom by 3, yielding 9x 2x . Next, to turn into something over 30, multiply top and bottom by 6, 30 5 12x and you get 30 . The original expression (x3) + (31x0 ) − (25x) is now (1300x ) + (93x0 ) − (1320x ) = 73x0 . 160 visual math Exercise 3: Given 7x + 2 = 5x + 14, you are told to find the value of x. The first thing you want to do is isolate your variable. This means you want to combine your x terms on one side of the equation and your numbers on the other side of the equation. Below you will subtract 5x from both sides in order to combine x terms: 7x + 2 = − 5x 5x + 14 − 5x 2x + 2 = 14 Now you will subtract 2 from both sides in order to isolate the x term. 2x + 2 = 14 −2 2x −2 = 12 Finally, divide both sides by 2 to get x = 6. Exercise 4: First, simplify the expression x 2 + 12 ≤ 42 by squaring the 4: x 2 + 12 ≤ 16 x 2 + 12 ≤ 16 Subtract 12 from both sides. − 12 x2 − 12 ≤ 4 − 2 ≤ x ≤ 2 Numbers between −2 and 2 (inclusive) will yield numbers less than or equal to 4 when they are squared. algebra 161 Exercise 5: Let’s look at the phrase: Erik has 3 CDs less than Danny. 3 CDs less than Danny would be D − 3. If Erik had 3 CDs less than Danny, you know: Danny’s − 3 = Erik’s So, Erik has E = D − 3, choice d. Exercise 6: To calculate the value of the expression 5x2 + 2xy3 when x = 3 and y = −2, you just substitute the given values for the variables. Substituting x =3 and y = −2 into the equation 5x2 + 2xy3 you get: 5(3)2 + 2(3)(−2)3 = (5)(9) + (2)(3)(−8) = 45 + (6)(−8) = 45 + (−48) = −3, or choice a. Exercise 7: x y 0 0 1 — 2 4 3 — 4 16 When x is 0, y is 0. When x = 2, y = 4. And when x = 4, y = 16. What is the pattern? y is simply x2. Write y = x2. Exercise 8: You know Train 1 is traveling eastbound at 60 mph and Train 2 is traveling at 50 mph in an westbound direction. The trains were initially 162 visual math 220 miles apart. We need to figure out how long it will be before they pass each other. First, draw a diagram: The initial distance between the 2 stations will equal the sum of the distances traveled by each train for the period of time in question. Total Distance = D1 D2 220 miles = R1T R2T 220 = (60)(t) + (50)(t) 220 = 110t 2=t It will take 2 hours. Exercise 9: Each term in the expression 20x2y + 15x + 10xy is divisible by 5x. You pull out a 5x to yield: 5x(4xy + 3x + 2y) Exercise 10: d. The fact that you see x2 in every answer choice is a clue that FOIL needs to be used. So what goes in each set of parentheses? The first pair of parentheses will be filled by “five more than x,” which is represented mathematically as (x + 5). The second pair of parentheses will be filled by “1 more than twice x,” which is simply (2x + 1). The question asks you to calculate a product, thus you multiply using FOIL: (x + 5)(2x + 1) = algebra 163 = 2x2 + x + 10x + 5. This simplifies to 2x2 + 11x + 5. Exercise 11: Given x is a positive number, and x2 − 8x − 9 = 0, you can actually cross off choices a and e because you know that x is positive. a. −1 b. 3 c. 9 d.both b & c e. both a & c Next, the expression x2 − 8x − 9 = 0 can be factored into two sets of parentheses: (x ± ?)(x ± ?) = 0 Again, because the coefficient of the x2 term is 1, you know that the sum of the 2 missing numbers is −8 (the coefficient of the x term) and the product of the 2 missing numbers is −9 (the lone number). The two numbers that satisfy these conditions are −9 and 1. Fill in the parentheses and set each set of parentheses equal to 0: (x 9)(x + 1) = 0 x − 9 = 0| x + 1 = 0 | x = 9| x = −1 But be careful! The question told us that x is positive! This means that only 9 is correct, choice c. 164 visual math Exercise 12: b. Here you will set up a simultaneous equation. Note that the question asks you to find y, so it would be nice to make a simultaneous equation that allows you to cross out x. The first given equation has a 3x in it. The second equation has an x in it. If you multiply the entire second equation by 3, you will be able to subtract it from the first and there will be no more x term! First, multiply x − 2y = 4 by 3 to get 3x − 6y = 12. Next, let’s line up our equations: 3x + y = 40 −(3x − 6y = 12) 7y = 28 y= 4 Note that you got 7y because y minus a negative 6y = y plus 6y. 3x + y = 40 −(3x − 6y = 12) 7y = 28 y= 4 algebra 165 chapter six Geometry and Measurement the metric system The metric system is a very well thought out way to measure distances, volumes, and masses. Once you understand what the prefixes mean and what the basic terms used for measurement are, you will have no problem dealing with these units. For example, meters (m) are used to measure length. 1 m = 1,000 millimeters = 1,000 mm 1 m = 100 centimeters = 100 cm 1 m = 10 decimeters = 10 dm geometry and measurement 167 Here are other prefixes you will see: The prefix . . . Means . . . Example . . . milli 1 1,0 00 1 1 milliliter is 1,0 00 of a liter centi 1 100 deci 1 1 0 of 1 decigram is 110 of a gram deca 10 times 1 decameter is 10 meters hecto 100 times 1 hectoliter is 100 liters kilo 1,000 times 1 kilometer is 1,000 meters of of 1 of a meter 1 centimeter is 100 So what is a liter, a meter, and a gram? A liter is used to measure volume. A gram is used to measure mass. And a meter is used to measure length. Term Used to measure Liter Volume Gram Mass Meter Length advanced conversions 1 acre = 43,560 square feet 1 liter = 1,000 cubic centimeters You may already be used to the customary system. Here are some units that you should know: 168 visual math Customary Units 1 foot = 12 inches 1 cup = 8 fluid ounces 3 feet = 1 yard 1 pint = 2 cups 1 mile = 5,280 feet 1 quart = 2 pints 1 pound = 16 ounces 1 gallon = 4 quarts 1 ton = 2,000 pounds operations with mixed measures A mixed measure is part one unit and part another unit. For example, “12 feet 5 inches” is part feet and part inches. When adding or subtracting mixed measures, you just need to align the units and perform the operation at hand. You can then rename units as necessary. Example: Ryan’s band played for 1 hour and 35 minutes. Ray’s band played for 1 hour and 40 minutes. When combined, the two performances lasted how long? 1 hr + 35 min + 1 hr + 40 min 2 hr + 75 min Rewrite 75 minutes as 60 min + 15 min. 2 hr + (60 min + 15 min) Rename 60 minutes as 1 hour. 2 hr + 1 hr + 15 min 3 hr + 15 min geometry and measurement 169 using proportions to convert units Proportions can be used to convert units. Look at the equivalents below: 1 cup = 8 ounces 1 meter = 100 cm 1 hour = 60 minutes 1 min = 60 seconds If you know that 1 cup equals 8 ounces, you can easily figure out how many cups are in 64 ounces by setting up a proportion: 1 cup ? cup 8 o z = 64 oz Cross multiplying, you get 64 • 1 = 8 • ?, or 64 = 8 • ?. Dividing both sides by 8 you get ? = 8. Exercise 1: Chris lays two planks of wood end to end. If one plank is 6 yd., 2 ft., 8 in. long, and the other is 7 yd., 1 ft., 5 in. long, how long are they when combined? rays and angles Terminology: ■ A ray is part of a line that has one endpoint. (It extends indefinitely in one direction.) ■ An acute angle is less than 90°. ■ A right angle equals 90°. ■ A straight line is 180°. ■ An obtuse angle is greater than 90°. ■ Two angles are supplementary if they add to 180°. ■ Two angles are complementary if they add to 90°. ■ If you bisect an angle, you cut it exactly in half. ■ Vertical angles are formed when two lines intersect; the opposite angles are equal. 170 visual math Exercise 2: Refer to the figure below when completing the following chart. An example of . . . Would be . . . Vertical angles An acute angle A right angle Complementary angles Supplementary angles parallel lines Facts: ■ A line that crosses the pair of parallel lines will generate corresponding angles that are equal. (For example, angles p and k are in corresponding positions, so they are equal.) ■ Alternate interior angles are equal. (m & q are alternate interior angles, so they are equal. This is also true for p and n.) ■ Alternate exterior angles are equal. (This means k = s, and r = l.) geometry and measurement 171 Example: If line segments A B and C D are parallel, what is the value of y? Here, you see that the 50° angle beneath C D corresponds to the angle beneath AB . You could then add in a new 50° angle: You see that the “new” 50° angle and y are supplementary—they form a straight line. Straight lines are 180°, so you know that 50 + y = 180. Subtracting 50 from both sides yields y=130°. Exercise 3: If, in the figure below O , P Q R , T S and U V are parallel, what is the value of angle b? 172 visual math interior angles of common shapes The interior angles of a triangle add up to 180°. The interior angles of any four-sided figure, or quadrilateral, add up to 360°. ■ ■ ■ ■ ■ square rectangle parallelogram rhombus trapezoid The central angles of a circle add up to 360°. For a polygon with n sides, the interior angl