Preliminary Mathematics
|
|
- Maximilian Cook
- 7 years ago
- Views:
Transcription
1 Preliminary Mathematics The purpose of this document is to provide you with a refresher over some topics that will be essential for what we do in this class. We will begin with fractions, decimals, and percentages. Then, we will discuss solving equations and rewriting functions. Fractions Before discussing fractions, we must clarify two things. First, never, EVER, divide anything by 0. This is one of the most important rules in all of mathematics. Lastly, any whole number (such as, 2, -, -2, etc.) can be written as a fraction of the form x where x is the whole number. This section is perhaps the most important for what we will do in this class. Much of financial mathematics revolves around percentage rates which in turn rely on decimals which rely on fractions. Now, suppose you have a pizza it can be any kind you like and a friend you have to share it with. You know, maybe without knowing the proper notation or mathematics, that you will get half the pizza and your friend will get the other half (if it is done fairly). Now, what this also means is that you have divided one piece (the whole pizza) into 2 pieces. If we expressed this in fractions, we would have 2 = One Half. This is how we can think of fractions: the top part (the numerator) represents the number of pieces that are being divided up to supply however many people there are (the denominator). Perhaps you have heard of the term improper fractions where the numerator is greater than the denominator. In previous classes you may have been asked to convert such fractions into equivalent mixed fractions an expression involving an integer times a proper fraction. To convert an improper fraction into a mixed fraction, one must first determine how many times the denominator divides into the numerator without going over. The number from this step is the integer multiplier in the mixed fraction. The remainder of dividing the denominator into the numerator becomes the numerator in the proper fraction part of a mixed fraction while the denominator is the same as in the improper fraction. For example, if we wanted to write 6 as a mixed fraction we would have 5 since divides into 6 five times before we go over with a remainder of.
2 Example : Suppose you have 8 pizzas that need to be distributed amongst 5 people. (a) Express the amount that each person gets as a fraction. (b) Notice that your answer from the previous part is an improper fraction. Convert this improper fraction into a mixed fraction. For most instances, we will keep improper fractions as they are since they are much easier to work with. To truly see this, let us discuss the basic operations adding, subtracting, multiplying, and dividing. Contrary to how you learned these basic operations, we will start with multiplication and division of fractions.. Multiplying and Dividing Fractions.. Multiplication If we have two fractions, a b and c d (with b and d not 0), then their product a b c d = c d a b = ac bd. As you can see, regardless of the numerator and denominator, the product of two fractions is equal to the product of the numerators divided by the product of the denominators. Example : Find the product of and Division Dividing two fractions is just as easy, it only requires one extra step. If we wish to divide the fraction a by the fraction c, all we must do is invert make the numerator the denominator, b d and make the denominator the numerator c and multiply it by a. That is, d b a b c d = a b d c = ad bc. It is important to remember that, unlike multiplication, unless both numbers are the same and not zero, a b b a. One important use of multiplying and dividing fractions is cross-multiplication. Say we have a fraction like 5 and want to know if it is equal to another fraction, such as. 7 5 The easiest way to do this is to set them equal to each other, then multiply the numerator of one with the denominator of the other. For our example here, we would get 7 5 = 05 2
3 on one side and 5 = 05 on the other. Thus, the two fractions are equal. In class we will see why this works. In the solving equations section we will see why this is useful. Now, we are prepared to move on to adding and subtracting two fractions..2 Adding and Subtracting Fractions.2. Like Denominators If the denominators of the two fractions are the same, all we need do is add or subtract the numerators, while keeping the denominator the the same. For example: = 9 5. We will see in the next section that when the denominators are different, the procedure is more involved..2.2 Unlike Denominators If we have two fractions (proper or improper) with different denominators and want to add or subtract them, we run into a bit of difficulty. Continuing with the people and pizzas example, one fraction might represent how much one person can have if there are 8 pizzas split amongst friends, while another may represent the same allowance in the case of pizzas and 6 friends one fraction is thirds and one is sixths. In order to determine how much a person would have if he had 8 and we must find a common denominator 6 for the two fractions and then express each as an equivalent fraction with that common denominator as the denominator. Admittedly, this seems like a lot, so lets tackle each step separately. Finding a common denominator is exactly what it sounds like: we need to find a number that is a multiple of both denominators. This can be done in two ways. The first is the easiest, but will usually require simplification if you want the fraction to be in lowest terms. In the first method, all we do is multiply the two denominators together, thereby creating a number which is a multiple of both. The other method is more exhaustive, but useful if you need the least common denominator. In the second method one must list the multiples of each denominator and find the first entry that is common to both lists. For small numbers, method two may be easier to do mentally. So, to continue our 8 and example, the 6 least common denominator would be 6 (why?). Alternatively, you could find a common denominator of 8 by multiplying the two values. Let us now progress to finding equivalent fractions. Given a fraction, like, an equivalent fractionis a fraction created by multiplying 2 the numerator and denominator by the same value. Thus, 2 = 6 4 = n 2n
4 for any n. Now we can combine these procedures to add/subtract fractions with unlike denominators. Two of the most useful tools in algebra are adding 0 and multiplying by. You may scoff and wonder why these are useful, but after this section you will see why. Suppose we have found a common denominator for two fractions (suppose 6, for definiteness). Now, if we take any number except for 0 and divide it by itself we get ; that is, r r = for any r 0. This fact becomes handy when we do the following: () Change each original fraction into an equivalent fraction that has our common denominator on the bottom; (2) We now have the same denominator for each fraction (was it magic?) and can add/subtract them as previously discussed. Finally, let s finish our example with these pizzas. 8 = 6 6 We found that 8 and have a common denominator of 6. Similarly, we have that 6. Thus, we can easily see that = = Decimals and Percentages 2. Understanding Decimal Places All of us have seen numbers like , but maybe not all of us have seen how decimal numbers relate to fractions and percentages. First, we need to understand that each place to the right of the decimal place has a specific representation just like the numbers to the left of the decimal place. Figure : A handy chart for decimal places As you can see in Figure, each place to the right has a name similar to each place to the left. By tenth, we mean, by hundredth we mean, etc. One key to decimal 0 00 numbers is that if we multiply a decimal number by a power of 0, the decimal place moves 4
5 to the right as many places as there are zeros. If we divide by a power of ten, the decimal moves to the left accordingly. For example, = , and = Fractions to Decimals and Back Again If we want to express as a decimal, we would have =. since it is three-tenths. 0 0 If we have a fraction like, we no longer have something like 0, 00, 000, etc. as a 2 denominator. But, if you recall equivalent fractions, we can write one half as 2 = 5 0 =.5. Thus, we can easily write fractions as decimals by finding an equivalent fraction that has a power of 0 in the denominator. But what if we can t do this? Say, for instance, we have the fraction. We know that equivalent fractions of one-third would be = 9 = 99 = 999 = 9999 =... Notice that no matter how far out we go, we won t get a power of 0. Thus, we say that =.... =. where the bar over the indicates that it repeats forever to the right Terminating Decimals In the event that the decimal number terminates becomes all 0 s after a certain point the process of going from a decimal to a fraction is even easier. First, determine the number of decimal places before the number terminates, call this number k. Then, the denominator of the fraction is k zeros { }} { 0 k = Then, the decimal until it terminates itself (after the decimal point) becomes the numerator. For example the decimal number 8 places { }} { { }} { =. } {{ } 8 zeros 5
6 Any numbers to the left of the decimal place take the role of the integer in a mixed fraction. Thus, 5.4 = Non-terminating Decimals What if the decimal number never terminates, like 0....? The key here is to see if there is a pattern repeating s, repeating 2, repeating 775, etc. that repeats continually. Notice here that the pattern must repeat forever, not just occur one or two or so times. If we have a decimal number that does have such a forever repeating pattern, say x = , we do the following: use the repeating part (87 in this case) as the numerator and the denominator is as many 9 s as there are digits in the pattern ( in this case). So, we have = *This section uses some steps that will be discussed in the Solving Equations section* What if there are digits between the decimal point and the pattern, as in the number x = ? Well, notice that 000x = If we now subtract x from 000x, we have 000x x = 999x = = 4.7. If we divide throughout by 999, we have x = 999x 999 = = As you can see now, we have a fractional form for our decimal number. 2. Percentages and Decimals We are now prepared to discuss percent (from the Latin per centum meaning of one hundred ) values. Just like fractions and decimals, percentages provide an answer to how much of something we have. Maybe we have half of it (50%) maybe we have five of it (500%) maybe we have one-tenth of it (0%). To transition from percent to decimals all we have to do is move the decimal point to the left 2 places and drop the percent symbol: 97.85% = To convert from a decimal number to a percentage, all we do is move the decimal point 2 places to the right and add the percent symbol:.4056 =.4056%. 6
7 If we have to go from a fraction to a percentage rate, we simply need to first convert to a decimal and then to a percentage. Solving Equations When handling non-terminating decimals we performed some algebraic operations that you may have forgotten. An equation is algebraic if it involves one or more unknown values which we call variables x, y, z,... for example. When it comes to actually solving for values, we will only have one variable, while our financial formulas will be rewritten for different variables using the operations we learn here. Now, let us review the basic rules/procedures for solving an equation such as (x + 2) = 8. Now, to this equation, we can do the following: () Add or subtract the same value to BOTH sides; (2) Multiply or divide BOTH sides by the same value; () Distribute, factor, combine like terms to ONE side. Notice that some operations can be done to one side without having to be done to the other, while others must be done to both sides. To better understand this, think of the equation as a balanced scale. We want to not only keep it in balance at all times, but we also want to eventually find out what our mystery weight is (the variable x). Rule () above can be thought of as rearranging items on one of the sides. Since nothing is being added or taken away, that scale is balanced. Rules () and (2) change the weight on one side, so to keep it balanced, we have to do the same thing to the other side. Now, before you get too worried, we won t be doing much algebra at all in this class, but being able to rewrite formulas is a must in financial mathematics and that is where the algebra will be used. Well, to end the section, let s solve this equation for x. First, using R(), we can add to both sides so that we have = 0 { }} { (x + 2) + = 9 }{{} 8 + = 9 Well, using R(2) here we can divide both sides by this will cancel the out of our product on the left and obtain 9 = (x + 2) {}}{ =. We now have a pretty easy problem: we need a number that when added to 2 gives us. Well, after we use R() and subtract 2 from both sides we see that x =. 7.
8 There is one important thing to mention here: canceling terms in a fraction is allowed if they are multiplied. Recall what we learned with equivalen fractions if we multiply the top and bottom by the same thing, we have the same fraction. To better see this, let us do two examples. First, given the fraction (x + 2) 6 = (x + 2) 2 we can cancel the in the top and bottom since it is common in both. But, if we have the fraction x we cannot cancel a out since is not common to both 2 and x (it is certainly a part of x). So, if ever you are about to cancel terms in a fraction, make sure they can in fact be canceled. One easy way to check this is to see if the value can be factored out, as in x + 6 = 4 Rewriting Formulas can be multiplied by each { }} { (x + 2). In this section, we will simply do one example to see how what we just learned can be used. Conside the following formula (will be introduced in Chapter ): M = P + P RT. It is not important what the letters stand for. If I were to ask you to find a formula for P, though, all you would need to do is solve the equation for P. Well, on the right hand side, I can factor out P since it is common to both terms, and get M = P ( + RT ). Recall that a number divided by itself is equal to. Now, if I use R(2), I can divide both sides by ( + RT ) to get M ( + RT ) = P ( + RT ) ( + RT ) = P. We now have a formula for P, as desired. We will use this procedure a lot in the latter part of the class, but knowing these skills we be useful throughout the quarter. 8
9 5 Recommended Exercises For practice, work on the Cumulative Review on pages from Salzman, 8th edition. This won t be collected as homework, but it will help you to see how well you understand the material, and will also assist you in knowing what you may need help on. If your struggles occur with the algebra, do not be too worried as we will only utilize that material occasionally. Being comfortable with percents, decimals, and fractions however is an absolute must in this class. 9
Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.
Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material
More informationFractions to decimals
Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of
More informationParamedic Program Pre-Admission Mathematics Test Study Guide
Paramedic Program Pre-Admission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page
More informationMath Circle Beginners Group October 18, 2015
Math Circle Beginners Group October 18, 2015 Warm-up problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd
More information+ = has become. has become. Maths in School. Fraction Calculations in School. by Kate Robinson
+ has become 0 Maths in School has become 0 Fraction Calculations in School by Kate Robinson Fractions Calculations in School Contents Introduction p. Simplifying fractions (cancelling down) p. Adding
More information47 Numerator Denominator
JH WEEKLIES ISSUE #22 2012-2013 Mathematics Fractions Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3 etc.). The preferred way to represent these partial numbers (rational
More information3.3 Addition and Subtraction of Rational Numbers
3.3 Addition and Subtraction of Rational Numbers In this section we consider addition and subtraction of both fractions and decimals. We start with addition and subtraction of fractions with the same denominator.
More informationNUMBER SYSTEMS. William Stallings
NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html
More informationActivity 1: Using base ten blocks to model operations on decimals
Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division
More informationFractions. If the top and bottom numbers of a fraction are the same then you have a whole one.
What do fractions mean? Fractions Academic Skills Advice Look at the bottom of the fraction first this tells you how many pieces the shape (or number) has been cut into. Then look at the top of the fraction
More informationThe gas can has a capacity of 4.17 gallons and weighs 3.4 pounds.
hundred million$ ten------ million$ million$ 00,000,000 0,000,000,000,000 00,000 0,000,000 00 0 0 0 0 0 0 0 0 0 Session 26 Decimal Fractions Explain the meaning of the values stated in the following sentence.
More informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationPart 1 Expressions, Equations, and Inequalities: Simplifying and Solving
Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words
More informationNumeracy Preparation Guide. for the. VETASSESS Test for Certificate IV in Nursing (Enrolled / Division 2 Nursing) course
Numeracy Preparation Guide for the VETASSESS Test for Certificate IV in Nursing (Enrolled / Division Nursing) course Introduction The Nursing course selection (or entrance) test used by various Registered
More information5.4 Solving Percent Problems Using the Percent Equation
5. Solving Percent Problems Using the Percent Equation In this section we will develop and use a more algebraic equation approach to solving percent equations. Recall the percent proportion from the last
More information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
More informationMULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.
1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with
More informationSunny Hills Math Club Decimal Numbers Lesson 4
Are you tired of finding common denominators to add fractions? Are you tired of converting mixed fractions into improper fractions, just to multiply and convert them back? Are you tired of reducing fractions
More informationPre-Algebra Lecture 6
Pre-Algebra Lecture 6 Today we will discuss Decimals and Percentages. Outline: 1. Decimals 2. Ordering Decimals 3. Rounding Decimals 4. Adding and subtracting Decimals 5. Multiplying and Dividing Decimals
More informationMaths Workshop for Parents 2. Fractions and Algebra
Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)
More informationDecimals and other fractions
Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very
More informationTHE BINARY NUMBER SYSTEM
THE BINARY NUMBER SYSTEM Dr. Robert P. Webber, Longwood University Our civilization uses the base 10 or decimal place value system. Each digit in a number represents a power of 10. For example, 365.42
More informationWelcome to Basic Math Skills!
Basic Math Skills Welcome to Basic Math Skills! Most students find the math sections to be the most difficult. Basic Math Skills was designed to give you a refresher on the basics of math. There are lots
More informationMultiplication and Division with Rational Numbers
Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is
More informationBinary Number System. 16. Binary Numbers. Base 10 digits: 0 1 2 3 4 5 6 7 8 9. Base 2 digits: 0 1
Binary Number System 1 Base 10 digits: 0 1 2 3 4 5 6 7 8 9 Base 2 digits: 0 1 Recall that in base 10, the digits of a number are just coefficients of powers of the base (10): 417 = 4 * 10 2 + 1 * 10 1
More informationThis explains why the mixed number equivalent to 7/3 is 2 + 1/3, also written 2
Chapter 28: Proper and Improper Fractions A fraction is called improper if the numerator is greater than the denominator For example, 7/ is improper because the numerator 7 is greater than the denominator
More informationBase Conversion written by Cathy Saxton
Base Conversion written by Cathy Saxton 1. Base 10 In base 10, the digits, from right to left, specify the 1 s, 10 s, 100 s, 1000 s, etc. These are powers of 10 (10 x ): 10 0 = 1, 10 1 = 10, 10 2 = 100,
More informationPREPARATION FOR MATH TESTING at CityLab Academy
PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST
More informationIntroduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman
Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman hundredths tenths ones tens Decimal Art An Introduction to Decimals Directions: Part 1: Coloring Have children
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationStanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions
Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in
More informationnorth seattle community college
INTRODUCTION TO FRACTIONS If we divide a whole number into equal parts we get a fraction: For example, this circle is divided into quarters. Three quarters, or, of the circle is shaded. DEFINITIONS: The
More informationAll the examples in this worksheet and all the answers to questions are available as answer sheets or videos.
BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com Numbers 3 In this section we will look at - improper fractions and mixed fractions - multiplying and dividing fractions - what decimals mean and exponents
More informationNumerator Denominator
Fractions A fraction is any part of a group, number or whole. Fractions are always written as Numerator Denominator A unitary fraction is one where the numerator is always 1 e.g 1 1 1 1 1...etc... 2 3
More information2.3 Solving Equations Containing Fractions and Decimals
2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions
More informationPAYCHEX, INC. BASIC BUSINESS MATH TRAINING MODULE
PAYCHEX, INC. BASIC BUSINESS MATH TRAINING MODULE 1 Property of Paychex, Inc. Basic Business Math Table of Contents Overview...3 Objectives...3 Calculator...4 Basic Calculations...6 Order of Operation...9
More informationSolution Guide Chapter 14 Mixing Fractions, Decimals, and Percents Together
Solution Guide Chapter 4 Mixing Fractions, Decimals, and Percents Together Doing the Math from p. 80 2. 0.72 9 =? 0.08 To change it to decimal, we can tip it over and divide: 9 0.72 To make 0.72 into a
More informationMultiplying Fractions
. Multiplying Fractions. OBJECTIVES 1. Multiply two fractions. Multiply two mixed numbers. Simplify before multiplying fractions 4. Estimate products by rounding Multiplication is the easiest of the four
More informationDecimal Notations for Fractions Number and Operations Fractions /4.NF
Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions.
More informationSession 7 Fractions and Decimals
Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,
More informationWSMA Decimal Numbers Lesson 4
Thousands Hundreds Tens Ones Decimal Tenths Hundredths Thousandths WSMA Decimal Numbers Lesson 4 Are you tired of finding common denominators to add fractions? Are you tired of converting mixed fractions
More informationSample Fraction Addition and Subtraction Concepts Activities 1 3
Sample Fraction Addition and Subtraction Concepts Activities 1 3 College- and Career-Ready Standard Addressed: Build fractions from unit fractions by applying and extending previous understandings of operations
More informationContents. Subtraction (Taking Away)... 6. Multiplication... 7 by a single digit. by a two digit number by 10, 100 or 1000
This booklet outlines the methods we teach pupils for place value, times tables, addition, subtraction, multiplication, division, fractions, decimals, percentages, negative numbers and basic algebra Any
More informationSection 4.1 Rules of Exponents
Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells
More informationBasic numerical skills: FRACTIONS, DECIMALS, PROPORTIONS, RATIOS AND PERCENTAGES
Basic numerical skills: FRACTIONS, DECIMALS, PROPORTIONS, RATIOS AND PERCENTAGES. Introduction (simple) This helpsheet is concerned with the ways that we express quantities that are not whole numbers,
More informationCharlesworth School Year Group Maths Targets
Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve
More informationChapter 1: Order of Operations, Fractions & Percents
HOSP 1107 (Business Math) Learning Centre Chapter 1: Order of Operations, Fractions & Percents ORDER OF OPERATIONS When finding the value of an expression, the operations must be carried out in a certain
More informationDr Brian Beaudrie pg. 1
Multiplication of Decimals Name: Multiplication of a decimal by a whole number can be represented by the repeated addition model. For example, 3 0.14 means add 0.14 three times, regroup, and simplify,
More informationUsing Proportions to Solve Percent Problems I
RP7-1 Using Proportions to Solve Percent Problems I Pages 46 48 Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving
More informationIrrational Numbers. A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers.
Irrational Numbers A. Rational Numbers 1. Before we discuss irrational numbers, it would probably be a good idea to define rational numbers. Definition: Rational Number A rational number is a number that
More informationExponents. Exponents tell us how many times to multiply a base number by itself.
Exponents Exponents tell us how many times to multiply a base number by itself. Exponential form: 5 4 exponent base number Expanded form: 5 5 5 5 25 5 5 125 5 625 To use a calculator: put in the base number,
More informationIntegers, I, is a set of numbers that include positive and negative numbers and zero.
Grade 9 Math Unit 3: Rational Numbers Section 3.1: What is a Rational Number? Integers, I, is a set of numbers that include positive and negative numbers and zero. Imagine a number line These numbers are
More informationJobTestPrep's Numeracy Review Decimals & Percentages
JobTestPrep's Numeracy Review Decimals & Percentages 1 Table of contents What is decimal? 3 Converting fractions to decimals 4 Converting decimals to fractions 6 Percentages 6 Adding and subtracting decimals
More informationSolving Systems of Two Equations Algebraically
8 MODULE 3. EQUATIONS 3b Solving Systems of Two Equations Algebraically Solving Systems by Substitution In this section we introduce an algebraic technique for solving systems of two equations in two unknowns
More information0.8 Rational Expressions and Equations
96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to
More informationIntroduction to Fractions
Section 0.6 Contents: Vocabulary of Fractions A Fraction as division Undefined Values First Rules of Fractions Equivalent Fractions Building Up Fractions VOCABULARY OF FRACTIONS Simplifying Fractions Multiplying
More information1. The Fly In The Ointment
Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 5: Dividing Decimal Fractions, Part 2. The Fly In The Ointment The meaning of, say, ƒ 2 doesn't depend on whether we represent
More informationSequences. A sequence is a list of numbers, or a pattern, which obeys a rule.
Sequences A sequence is a list of numbers, or a pattern, which obeys a rule. Each number in a sequence is called a term. ie the fourth term of the sequence 2, 4, 6, 8, 10, 12... is 8, because it is the
More informationFractions Packet. Contents
Fractions Packet Contents Intro to Fractions.. page Reducing Fractions.. page Ordering Fractions page Multiplication and Division of Fractions page Addition and Subtraction of Fractions.. page Answer Keys..
More informationUnit 11 Fractions and decimals
Unit 11 Fractions and decimals Five daily lessons Year 4 Spring term (Key objectives in bold) Unit Objectives Year 4 Use fraction notation. Recognise simple fractions that are Page several parts of a whole;
More information3 cups ¾ ½ ¼ 2 cups ¾ ½ ¼. 1 cup ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼. 1 cup. 1 cup ¾ ½ ¼ ¾ ½ ¼
cups cups cup Fractions are a form of division. When I ask what is / I am asking How big will each part be if I break into equal parts? The answer is. This a fraction. A fraction is part of a whole. The
More informationUseful Number Systems
Useful Number Systems Decimal Base = 10 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Binary Base = 2 Digit Set = {0, 1} Octal Base = 8 = 2 3 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7} Hexadecimal Base = 16 = 2
More informationBinary Adders: Half Adders and Full Adders
Binary Adders: Half Adders and Full Adders In this set of slides, we present the two basic types of adders: 1. Half adders, and 2. Full adders. Each type of adder functions to add two binary bits. In order
More informationAccuplacer Arithmetic Study Guide
Accuplacer Arithmetic Study Guide Section One: Terms Numerator: The number on top of a fraction which tells how many parts you have. Denominator: The number on the bottom of a fraction which tells how
More informationNegative Integer Exponents
7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5
More informationIntegrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
More informationFree Pre-Algebra Lesson 55! page 1
Free Pre-Algebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can
More informationMATH-0910 Review Concepts (Haugen)
Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,
More informationUNDERSTANDING ALGEBRA JAMES BRENNAN. Copyright 2002, All Rights Reserved
UNDERSTANDING ALGEBRA JAMES BRENNAN Copyright 00, All Rights Reserved CONTENTS CHAPTER 1: THE NUMBERS OF ARITHMETIC 1 THE REAL NUMBER SYSTEM 1 ADDITION AND SUBTRACTION OF REAL NUMBERS 8 MULTIPLICATION
More informationSolving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More information2.6 Exponents and Order of Operations
2.6 Exponents and Order of Operations We begin this section with exponents applied to negative numbers. The idea of applying an exponent to a negative number is identical to that of a positive number (repeated
More informationBalancing Chemical Equations
Balancing Chemical Equations A mathematical equation is simply a sentence that states that two expressions are equal. One or both of the expressions will contain a variable whose value must be determined
More informationComputer Science 281 Binary and Hexadecimal Review
Computer Science 281 Binary and Hexadecimal Review 1 The Binary Number System Computers store everything, both instructions and data, by using many, many transistors, each of which can be in one of two
More informationMathematics. Steps to Success. and. Top Tips. Year 5
Pownall Green Primary School Mathematics and Year 5 1 Contents Page 1. Multiplication and Division 3 2. Positive and Negative Numbers 4 3. Decimal Notation 4. Reading Decimals 5 5. Fractions Linked to
More information6 3 4 9 = 6 10 + 3 10 + 4 10 + 9 10
Lesson The Binary Number System. Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base- system. When you
More informationCOMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012
Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about
More informationCAHSEE on Target UC Davis, School and University Partnerships
UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,
More informationYOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!
DETAILED SOLUTIONS AND CONCEPTS - DECIMALS AND WHOLE NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! YOU MUST
More informationHFCC Math Lab Arithmetic - 4. Addition, Subtraction, Multiplication and Division of Mixed Numbers
HFCC Math Lab Arithmetic - Addition, Subtraction, Multiplication and Division of Mixed Numbers Part I: Addition and Subtraction of Mixed Numbers There are two ways of adding and subtracting mixed numbers.
More informationSquaring, Cubing, and Cube Rooting
Squaring, Cubing, and Cube Rooting Arthur T. Benjamin Harvey Mudd College Claremont, CA 91711 benjamin@math.hmc.edu I still recall my thrill and disappointment when I read Mathematical Carnival [4], by
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationScope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B
Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is first introduced
More informationRadicals - Multiply and Divide Radicals
8. Radicals - Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals
More informationUnit 6 Number and Operations in Base Ten: Decimals
Unit 6 Number and Operations in Base Ten: Decimals Introduction Students will extend the place value system to decimals. They will apply their understanding of models for decimals and decimal notation,
More informationReview of Scientific Notation and Significant Figures
II-1 Scientific Notation Review of Scientific Notation and Significant Figures Frequently numbers that occur in physics and other sciences are either very large or very small. For example, the speed of
More informationMathematics Practice for Nursing and Midwifery Ratio Percentage. 3:2 means that for every 3 items of the first type we have 2 items of the second.
Study Advice Service Student Support Services Author: Lynn Ireland, revised by Dave Longstaff Mathematics Practice for Nursing and Midwifery Ratio Percentage Ratio Ratio describes the relationship between
More informationCalculation Policy Fractions
Calculation Policy Fractions This policy is to be used in conjunction with the calculation policy to enable children to become fluent in fractions and ready to calculate them by Year 5. It has been devised
More informationACCUPLACER. Testing & Study Guide. Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011
ACCUPLACER Testing & Study Guide Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011 Thank you to Johnston Community College staff for giving permission to revise
More informationCHAPTER 4 DIMENSIONAL ANALYSIS
CHAPTER 4 DIMENSIONAL ANALYSIS 1. DIMENSIONAL ANALYSIS Dimensional analysis, which is also known as the factor label method or unit conversion method, is an extremely important tool in the field of chemistry.
More informationOrder of Operations More Essential Practice
Order of Operations More Essential Practice We will be simplifying expressions using the order of operations in this section. Automatic Skill: Order of operations needs to become an automatic skill. Failure
More informationFRACTIONS OPERATIONS
FRACTIONS OPERATIONS Summary 1. Elements of a fraction... 1. Equivalent fractions... 1. Simplification of a fraction... 4. Rules for adding and subtracting fractions... 5. Multiplication rule for two fractions...
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More informationRational Number Project
Rational Number Project Fraction Operations and Initial Decimal Ideas Lesson : Overview Students estimate sums and differences using mental images of the 0 x 0 grid. Students develop strategies for adding
More information